structured/0709.0751.structured.json

{"paper_meta":{"paper_id":"arxiv:0709.0751","title":"0709.0751","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0709.0751v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VIII: On canonical\nbases for the nonstandard quantum groups\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\n(Technical Report TR-2007-15\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nOctober 27, 2018\nAbstract\nThis article gives conjecturally correct algorithms to construct canon-\nical bases of the irreducible polynomial representations and the matrix\ncoordinate rings of the nonstandard quantum groups in GCT4 and\nGCT7, and canonical bases of the dually paired nonstandard deforma-\ntions of the symmetric group algebra therein. These are generalizations\nof the canonical bases of the irreducible polynomial representations\nand the matrix coordinate ring of the standard quantum group, as\nconstructed by Kashiwara and Lusztig, and the Kazhdan-Lusztig ba-\nsis of the Hecke algebra. A positive (#P-) formula for the well-known\nplethysm constants follows from their conjectural properties and the\nduality and reciprocity conjectures in [GCT7].\n1\n\n1\nIntroduction\nLet H be a complex connected classical reductive group, X = Vμ(H) its\nirreducible polynomial representation with highest weight μ, G = GL(X),\nand ρ : H →G the representation map. Given a highest weight π of H\nand λ of G, the plethysm constant aπ\nλ,μ is defined to be the multiplicity of\nVπ(H) in Vλ(G), considered an H-module via ρ. A fundamental problem\nin representation theory is to find a positive (#P-) formula (rule) for the\nplethysm constant [GCT7, St] akin to the Littlewood-Richardson rule. Mo-\ntivated by this problem, the article [GCT7] constructs a quantization ρq of\nthe homomorphism ρ in the form\nρq : Hq →GH\nq ,\n(1)\nwhere Hq is the standard (Drinfeld-Jimbo) quantum group [Dri, Ji, RTF]\nassociated with H and GH\nq is the new (possibly singular) quantum group,\ncalled the nonstandard quantum group associated with ρ. In the standard\ncase, i.e., when H = G, this specializes to the standard quantum group, and\nin the Kronecker case, i.e., when H = GL(V ) × GL(W), X = V ⊗W with\nthe natural H action, this specializes to the nonstandard quantum group in\n[GCT4]. Also constructed in [GCT7] is a nonstandard quantization BH\nr (q)\nof the group algebra C[Sr], Sr the symmetric group, whose relationship with\nGH\nq is conjecturally similar to that of the Hecke algebra with the standard\nquantum group.\nThis article gives conjecturally correct algorithms for constructing canon-\nical bases of the irreducible polynomial representations and the matrix co-\nordinate ring of GH\nq (Section 2) and a canonical basis of BH\nr (q) (Section 3).\nWe call these nonstandard canonical bases. They are generalizations of the\ncanonical bases of the irreducible polynomial representations and the matrix\ncoordinate ring of the standard quantum group, as constructed by Kashi-\nwara and Lusztig [Kas1, Kas2, Lu1, Lu2], and the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. A positive (#P-) formula for the plethysm con-\nstant follows from their conjectural properties (Sections 2.4 and 3.4), which\nare akin to those of the standard canonical basis, and the conjectural duality\nand reciprocity between GH\nq and BH\nr (q); cf. [GCT7].\nExperimental evidence (Section 5) suggests that these algorithms should\nbe correct. But we can not prove this formally, nor the required properties of\nthe nonstandard canonical bases. Mainly because we are unable to deal with\nthe complexity of the minors of the nonstandard quantum group. Specifi-\ncally, in contrast to the elementary formula for the Laplace expansion of a\n2\n\nminor of the standard quantum group–which is akin to the classical Laplace\nexpansion at q = 1–the Laplace expansion of a minor of a nonstandard\nquantum group is highly nonelementary; cf. [GCT7]. Its coefficients de-\npend on the multiplicative structural constants of a canonical basis akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nas per Kashiwara and Lusztig. In the Kronecker case, these constants are\nconjecturally polynomials in Q[q, q−1] with nonnegative coefficients, and in\ngeneral, polynomials with a conjectural relaxed form of this property. To\nprove this and to get explicit formulae for the minors in the nonstandard\nsetting, one needs explicit interpretations for these structural constants in\nthe spirit of the interpretation based on perverse sheaves for the Kazhdan-\nLusztig polynomials [KL2] and the multiplicative structural constants of the\ncanonical basis of the Drinfeld-Jimbo enveloping algebra [Lu2]. Thus even\nto get explicit formulae for the minors of the nonstandard quantum group\na (nonstandard) extension of the theory of perverse sheaves [BBD], and the\nunderlying Riemann hypothesis over finite fields [Dl2] seems necessary.\nMinors of the standard quantum group are in a sense the simplest (basic)\ncanonical basis elements in its matrix coordinate ring. That the simplest\ncanonical basis elements for the nonstandard quantum group–namely, its\nminors–are already so nonelementary in contrast to the standard case indi-\ncates the possible difficulties that may be encountered in proving correctness\nof the algorithms given here for constructing nonstandard canonical bases.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for the help in explicit\ncomputations in MATLAB.\nNotation: We use the symbols π and μ to denote labels of irreducible\nrepresentations of the standard quantum group and the symbols α, α0, α1, . . .\nto denote labels of irreducible representations of the nonstandard quantum\ngroup. Thus objects with subscripts π and μ are standard and the objects\nwith subscripts α, α0, . . . are nonstandard.\n2\nNonstandard canonical basis for GH\nq\nIn this section we describe a conjecturally correct algorithm for constructing\nthe canonical basis of the matrix coordinate ring of the nonstandard quan-\ntum group GH\nq . We follow the same terminology as in [Kli] for the basic\nquantum group notions.\nFor the sake of simplicity, let us assume that H = GL(V ). Let Hq =\n3\n\nGLq(V ) denote its standard (Drinfeld-Jimbo) quantization [Dri, Ji, RTF],\nand Mq(V ) the standard quantization of the matrix space M(V ).\nLet\nO(Mq(V )) be the coordinate ring of Mq(V ). We call it the matrix coordinate\nring of GLq(V ). The coordinate ring O(GLq(V )) of GLq(V ) is obtained by\nlocalizing O(Mq(V )) at the quantum determinant of GLq(V ). Let H denote\nthe Lie algebra of H, and Uq(H) the Drinfeld-Jimbo universal enveloping\nalgebra of Hq = GLq(V ).\nTo quantize the homomorphism ρ : H →G = GL(X) as in (1), the arti-\ncle [GCT7] constructs a nonstandard matrix coordinate ring O(MH\nq (X)) of\na (virtual) nonstandard matrix space MH\nq (X), and then defines the nonstan-\ndard quantized universal enveloping algebra U H\nq (G) by dualization. The non-\nstandard quantum group GH\nq is the virtual object whose universal enveloping\nalgebra is U H\nq (G). The construction also yields natural bialgebra homomor-\nphisms from Uq(H) to U H\nq (G) and from O(MH\nq (X)) to O(Mq(V ), thereby\ngiving the desired quantizations of the homomorphisms U(H) →U(G)\nand O(M(X)) →O(M(V )). This is what is meant by the quantization\n(1) of the representation map ρ.\nThe determinant of GH\nq\nmay vanish,\nand hence, we cannot, in general, define its coordinate ring O(GH\nq ) by lo-\ncalizing O(MH\nq (X)). Fortunately, this does not matter since O(MH\nq (X))\nstill has properties similar to that of the standard matrix coordinate ring\nO(Mq(V )). Specifically, it is cosemisimple. This means all (finite dimen-\nsional) polynomial representations of GH\nq , by which we mean corepresen-\ntations of O(MH\nq (V )), are completely reducible. A nonstandard quantum\nanalogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\n(2)\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)). Fur-\nthermore, the nonstandard enveloping algebra U H\nq (G) is a bialgebra with a\ncompact real form (∗-structure).\nThe goal is to construct a canonical basis for the matrix coordinate ring\nO(MH\nq (X)) akin to the canonical basis of the standard matrix coordinate\nring O(Mq(V )) as per Kashiwara and Lusztig [Kas1, Kas2, Lu1, Lu2].\n2.1\nThe standard setting\nWe begin by reviewing the basic scheme of Kashiwara and Lusztig for con-\nstructing a canonical basis of the matrix coordinate ring O(Mq(V )). The\n4\n\ncanonical basis of the coordinate ring O(GLq(V )) is obtained by localizing\nat the determinant.\nFollowing Kashiwara, we first define a balanced triple. Let A and ̄A be\nthe ring of rational functions in q regular at q = 0 and q = ∞, respectively.\nLet V be a Q(q)-vector space, L0 a sub-A-module (A-lattice) of V , L∞a\nsub- ̄A-module ( ̄A-lattice) of V , and VQ a sub-Q[q, q−1]-module of V such\nthat\nV ∼= Q(q) ⊗Q[q,q−1] VQ ∼= Q(q) ⊗A L0 ∼= Q(q) ⊗ ̄\nA L∞.\nWe say that (VQ, L0, L∞) is a balanced triple if any of the following three\nequivalent conditions hold:\n(a) E = VQ ∩L0 ∩L∞→L0/qL0 is an isomorphism.\n(b) E →L∞/q−1L∞is an isomorphism.\n(c) Q[q, q−1] ⊗Q E →VQ, A ⊗Q E →L0, ̄A ⊗Q E →L∞are isomorphisms.\nLet R = O(Mq(V )).\nKashiwara constructs an A-submodule (lattice)\nL = L(R) ⊂R, an involution −of R, a Q[q, q−1]-submodule RQ ⊂R, and\na basis B of L/qL such that (RQ, L, ̄L) is a balanced triple and, letting G\ndenote the inverse of the isomorphism RQ ∩L ∩ ̄L →L/qL, {G(b) | b ∈B}\nis the canonical basis of R. The pair (L, B), called the (upper) crystal base\nof R, has the following form. By the q-analogue of the Peter-Weyl theorem\nfor the standard quantum group,\nR = ⊕πV ∗\nq,π ⊗Vq,π\n(3)\nas a bi-GLq(V )-module, where Vq,π = Vq,π(V ) is the irreducible polynomial\nrepresentation of GLq(V ) with highest weight π. Let (Lπ, Bπ) denote the\nupper crystal base of Vq,π. Then\n(L, B) = ⊕π(L∗\nπ, B∗\nπ) ⊗(Lπ, Bπ)\n(4)\nwith appropriate normalization.\nWe now describe a construction of the upper crystal base (Lπ, Bπ) that\ncan be generalized to the nonstandard setting.\nLet r be the size of the partition π. Choose any embedding ρ : Vq,π →\nV ⊗r such that the highest weight vector of the image V ρ\nq,π = ρ(Vq,π) be-\nlongs to the A-lattice L(V )⊗r of V ⊗r, where L(V ) denotes the lattice of\nV generated by its standard basis {vi}. We also assume that the highest\nweight vector does not belong to qL(V )⊗r. Choose a Hermitian form on\nV ⊗r so that its monomial basis {vi1 ⊗· · · vir} is orthonormal. Let V ρ,⊥\nq,π de-\nnote the orthogonal complement of V ρ\nq,π. Since GLq(V ) has a compact real\n5\n\nform Uq(V )–i.e., the unitary compact subgroup in the sense of Woronowicz\n[W]–it follows that V ρ,⊥\nq,π is a GLq(V )-module. Thus V ⊗r = V ρ\nq,π ⊕V ρ,⊥\nq,π as a\nGLq(V )-module. Let\nLρ\nπ = L(V )⊗r ∩V ρ\nq,π\nand\nLρ,⊥\nπ\n= L(V )⊗r ∩V ρ,⊥\nq,π .\nIt follows from Kashiwara’s work [Kas1] that L(V )⊗r = Lρ\nπ ⊕Lρ,⊥\nπ\n.\nLet B(V ) = {bi = ψ(vi)} denote the basis of L(V )/qL(V ), where ψ :\nL(V ) →L(V )/qL(V ) is the natural projection. Let B(V )⊗r = {bi1 ⊗· · · ⊗\nbir} denote the monomial basis of B(V )⊗r. Given b ∈B(V )⊗r, let\nb =\nX\ni1,...,ir\nf(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients f(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\nIt follows from the works of Kashiwara [Kas1] and Date et al [DJM]\nthat Lρ\nπ/qLρ\nπ has a unique basis Bρ\nπ (up to scaling by constant multiples)\nsuch that the monomial supports of its elements are disjoint–in fact, one\ncan choose ρ so that the monomial support of each basis element consists of\njust one distinct monomial. This basis can be made completely unique by\nappropriate normalization. Then (Lρ\nπ, Bρ\nπ) coincides with the upper crystal\nbase of Vq,π as constructed by Kashiwara. Furthermore, this crystal base\ndoes not depend on the embedding ρ (up to isomorphism). Hence, we let\n(Lπ, Bπ) = (Lρ\nπ, Bρ\nπ) for any ρ as above. Kashiwara [Kas1] also shows that\nLπ and Bπ ∪{0} are invariant under certain crystal operators ̃ei and ̃fi\ncorresponding to the simple roots of Hq.\nThis scheme for constructing the upper crystal base (Lπ, Bπ) crucially\ndepends on the existence of a compact real form Uq(V ) ⊆GLq(V ) = Hq.\nEven the existence of a compact real form of the standard Drinfeld-Jimbo\nenveloping algebra Uq(H) suffices here.\n2.2\nNonstandard triple\nWe now generalize the preceding scheme to the nonstandard setting using\nthe compact real form of U H\nq (G), whose existence is proved in GCT7. The\ngoal is to construct an analogous triple for the matrix coordinate ring S =\nO(MH\nq (X)) of GH\nq . It will turn out that this triple need not be balanced as\n6\n\nin the standard case. We shall describe in Section 2.3 how a canonical basis\ncan be constructed from such a triple despite the lack of balance.\nWe begin by recalling that the q-analogue of the Peter-Weyl theorem (2)\nin the nonstandard setting need not hold over Q(q) unlike in the standard\nsetting. It holds only over an appropriate algebraic extension K of Q(q)\n[GCT7]–thinking of q as a transcendental. It will be convenient to assume\nin what follows that K is actually an algebraic extension ̃Q(q), where ̃Q is\nthe algebraic closure of Q. We let AK and ̄AK be the subrings of algebraic\nfunctions in K that are regular at q = 0 and q = ∞, respectively. Let KQ\nbe the integral closure of Q[q, q−1] in K. Clearly, KQ ∩AK ∩ ̄AK = ˆQ, where\nˆQ denotes the integral closure of Q in ̃Q. In what follows, we let ˆQ, AK, ̄AK\nand KQ play the role of Q, A, ̄A and Q[q, q−1] in Section 2.1. Thus by a\nlattice at q = 0 we mean an AK-lattice, by a lattice at q = ∞a ̄AK-lattice,\nby a Q-form, a KQ-form (module).\nSimilarly, instead of Q-modules and\nQ(q)-modules, we will be considering ˆQ-modules and K-modules. That is,\nin what follows we shall assuming that the underlying base field is K, instead\nof Q(q).\nNow let us describe the construction of the upper crystal base of an\nirreducible polynomial representation Wq,α of GH\nq . Let Xq = Vq,μ denote the\nquantization of X = Vμ; i.e, the Hq-module with highest weight μ. Since\nthe underlying field is K, Xq is a K-module. Let {zi} denote its standard\nq-orthonormal Gelfand-Tsetlin basis. Let {xi} denote the rescaled version\nof Gelfand-Tsetlin basis (as described in Section 7.3.3 of [Kli]) so that there\nare no square roots in the explicit formulae for the action of the generators\nof the Drinfeld-Jimbo algebra UqH on this basis. Alternatively, we can let\n{xi} be the standard (upper) canonical basis [Kas1, Lu2] of Xq as an Hq-\nmodule. We assume that limq→0(zi −xi) = 0; this can always be arranged.\nIn what follows, we sometimes denote Xq by X. What is meant should be\nclear from the context. It is shown in [GCT7] that Wq,α can be embedded\nin X⊗r for an appropriate r. We choose an embedding ρ : Wq,α →X⊗r as\nfollows.\nIf r = 1, Wq,α = X, so this is trivial. Otherwise, choose any β of degree\nr−1 so that Wq,α occurs as a GH\nq -submodule of Wq,β⊗X. By semisimplicity,\nthe latter is completely reducible as a GH\nq -module:\nWq,β ⊗X = ⊕βjWq,βj,\n(5)\nwhere we assume that this is also a decomposition as a K-module. Choose\nany j such that Wq,βj ∼= Wq,α. This fixes an embedding of Wq,α in Wq,β ⊗X.\n7\n\n(It is a plausible conjecture that the decomposition ( 5) is multiplicity free.\nThis would be a conjectural analogue of Pieri’s rule in the nonstandard\nsetting. It would imply that the embedding of Wq,α in Wq,β ⊗X is unique.\nBut this is not required here.). By induction, we have fixed an embedding\nof Wq,β in X⊗r−1. This fixes an embedding ρ of Wq,α in X⊗r (among many\npossible choices). Let W ρ\nq,α = ρ(Wq,α) be its image.\nChoose a Hermitian form on X⊗r so that its Gelfand-Tsetlin basis {zi1 ⊗\n· · · ⊗zir} is orthonormal. Let W ρ,⊥\nq,α denote the orthogonal complement of\nW ρ\nq,α. Since U H\nq (G) has a compact real form [GCT7] such that X⊗r\nq\nis its\nunitary representation with respect to this Hermitian form, it follows that\nW ρ,⊥\nq,α is a GH\nq -module. Thus X⊗r = W ρ\nq,α ⊕W ρ,⊥\nq,α as a GH\nq -module. Let\nLρ\nα = L(X)⊗r ∩W ρ\nq,α\nand\nLρ,⊥\nα\n= L(X)⊗r ∩W ρ,⊥\nq,α .\nThen, in analogy with Kashiwara’s work mentioned above:\nProposition 2.1 L(X)⊗r = Lρ\nα ⊕Lρ,⊥\nα .\nProof: The r.h.s. is clearly contained in the l.h.s. To show the converse it\nsuffices to show that Lρ\nα and Lρ,⊥\nα\nare projections of the lattice L(X)⊗r onto\nW ρ\nq,α and W ρ,⊥\nq,α respectively. Let us show this for Lρ\nα, the other case being\nsimilar. Clearly, the projection of L(X)⊗r onto W ρ\nq,α contains Lρ\nα. We only\nhave to show that the projection ˆy of any y ∈L(X)⊗r onto W ρ\nq,α also belongs\nto the lattice L(X)⊗r, and hence to Lρ\nα. Since y ∈L(X)⊗r, its length |y|\nw.r.t. the preceding Hermitian form tends to a well defined nonnegative real\nnumber as q →0. Since, the projection y →ˆy is orthonormal, the length\n|ˆy| of ˆy is at most |y|, and hence also tends to a well defined nonnegative\nreal number as q →0. This means ˆy is regular at q = 0 and hence belongs\nto L(X)⊗r. Q.E.D.\nLet B(X) = {bi = φ(xi)} denote the basis of the ̃Q-module L(X)/qL(X),\nwhere φ : L(X) →L(X)/qL(X) is the natural projection (the bi’s in this\nsection are different from the bi’s in Section 2.1). Let B(X)⊗r = {bi1 ⊗· · ·⊗\nbir} denote the monomial basis of L(X)⊗r. Given b ∈B(X)⊗r, let\nb =\nX\ni1,...,ir\ng(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients g(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\n8\n\nIn analogy with the work of Kashiwara and Date et al mentioned above,\nit may be conjectured that:\nConjecture 2.2 (Existence of (local) crystal basis)\nThe ̃Q-module Lρ\nα/qLρ\nα has a unique basis Bρ\nα (up to scaling by constant\nmultiples, and which can be made completely unique by appropriate normal-\nization) such that:\n1. The monomial supports of its elements are disjoint,\n2. Lρ\nα and Bρ\nα ∪{0} are invariant under Kashiwara’s crystal operators ̃ei\nand ̃fi for Hq (which are well defined since Uq(H) is a subalgebra of\nU H\nq (G)), and (Lρ\nα, Bρ\nα) is a local crystal basis, in the sense of Kashi-\nwara, of Wq,α as an Hq-module\nFurthermore, this crystal base does not depend on the embedding ρ (up to\nisomorphism).\nThis conjecture has been supported by experimental evidence; cf.\nSec-\ntion 5.5.\nAssuming this, we let (Lα, Bα) = (Lρ\nα, Bρ\nα) for any ρ as above.\nIt is\ncalled the upper crystal base of Wq,α.\nIn standard setting, the embedding ρ : Vq,π →V ⊗r in Section 2.1 can\nbe chosen so that the support of each basis element in Bρ\nπ consists of just\none monomial. That is, so that each b ∈Bρ\nπ is a monomial in B(V )⊗r. In\nthe nonstandard setting, it is not always possible to choose an embedding\nρ : Wq,α →X⊗r so that the support of each basis element in Bρ\nα in Conjec-\nture 2.1 consists of just one monomial; cf. Section 5.5 for a counterexample.\nIn view of the nonstandard q-analogue of the Peter-Weyl theorem (2),\nS = O(MH\nq (X)) has a natural upper crystal base\n(L(S), B(S)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(6)\nat q = 0 (with appropriate normalization). Let SQ be the KQ-forms (KQ-\nsubring) of S generated by the entries ui\nj of the generic nonstandard quantum\nmatrix u; recall (cf. [GCT7]) that S is the quotient of C⟨u⟩modulo appro-\npriate quadratic relations over the entries ui\nj’s of u. We define an involution\n−over S by a natural generalization of its definition in the standard setting.\nSpecifically, S is a O(MH\nq (X))-bi-comodule. Via the homomorphism from\n9\n\nO(MH\nq (X)) to O(Mq(V )), S is also O(Mq(V ))-bi-comodule; i.e., an Hq-bi-\nmodule. In the spirit of [Kas2], for any u and v in S with Hq-bi-weights\n(λr, λl) and (μr, μl), let uv = q(λr,μr)−(λl,μl) ̄v ̄u, where ( , ) denotes the usual\ninner product in the Hq-weight space. We let ̄ui\nj = ui\nj, and ̄q = q−1. This\ndefines −on S completely.\nApplying −to (L(S), B(S)), we get an upper crystal base ( ̄L(S), ̄B(S))\nat q = ∞. In analogy with the standard setting, we can now ask:\nQuestion 2.3 Is the triple (SQ, L(S), ̄L(S)) balanced? In other words, is\nthe map ψ : E = SQ ∪L(S) ∪ ̄L(S) →L(S)/qL(S) of ˆQ-modules an isomor-\nphism.\nIf it were, we could have defined the canonical basis of S by a globalization\nprocedure very much as in the standard setting, i.e., as ψ−1(B(S)). But, as\nit turns out, this need not be so; cf. Section 5.3. Specifically, for a given\nb ∈B(S), the fibre ψ−1(b) need not be singleton. This is the major difference\nfrom the standard setting that makes construction of the canonical basis of\nS in the nonstandard setting much more complex. We turn to this in the\nnext section.\n2.3\nNonstandard globalization via minimization of degree\ncomplexity\nWe now give a conjectural procedure for choosing an unambiguously defined\ncanonical element yb ∈ψ−1(b), b ∈B(S). The set {yb |b ∈B(S)} will then\nbe the canonical basis of S.\nFix r. Let Sr denote the degree r component of S. Let A = AH\nr\n=\nAH\nr (q) be the nonstandard q-Schur algebra [GCT7], which is the dual S∗\nr =\nHom(Sr, K) of Sr. A polynomial irreducible GH\nq -module Wq,α of degree r\nis an irreducible A-module, and conversely every irreducible A-module is\nof this form. Furthermore, the q-analogue of the Peter-Weyl theorem also\nholds for A:\nA = ⊕αW ∗\nq,α ⊗Wq,α.\n(7)\nFor reasons given later (cf. Remark 1 below), it will be more convenient\nto construct the canonical basis of AH\nr first. The canonical basis of Sr will\nthen be defined to be its dual.\n10\n\nLet\n⟨⟩: A ⊗Sr →K\nbe the natural pairing. The lattice L(A) is defined to be the dual lattice of\nL(Sr):\nL(A) = {a ∈A | ⟨a, L(Sr)⟩⊆AK}.\nThe automorphism −of A is defined by:\n⟨ ̄a, s⟩= ⟨a, ̄s⟩−.\nWe define ̄L(A) by applying −to L(A). We define the Q-form (i.e. KQ-\nform) AQ of A by:\nAQ = {a ∈A | ⟨a, Sr,Q⟩⊆KQ},\nwhere Sr,Q = Sr ∩SQ.\nWe define the basis B(A) of L(A)/qL(A) to be\nthe dual of B(S). Thus (L(A), B(A)) is the local crystal basis of A as per\nConjecture 2.2 and we have an analogue of (6):\n(L(A), B(A)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(8)\nThis defines the triple (AQ, L(A), ̄L(A)) for A. It need not be balanced\nin the standard sense, just like the triple (SQ, L(S), ̄L(S)) above. Now we\ndescribe the conjectural construction of a canonical basis of A.\nEach component B∗\nα ⊗Bα of B(A) has the left and right action of Kashi-\nwara’s crystal operators for Hq (cf. Conjecture 2.2). Thus, by [Kas1], we\nget a crystal graph on B∗\nα ⊗Bα, whose each connected component intutively\ncorresponds to an irreducible Hq-bi-submodule V ∗\nq,μ1 ⊗Vq,μ2 of the compo-\nnent W ∗\nq,α ⊗Wq,α in the Peter-Weyl decomposition of A (7). With each\nelement b ∈B(A) that occurs in such a connected component, we associate\nthe triple T(b) = (α, μ1, μ2). We call it the type of b. The types T(b)’s can\nbe partially ordered as follows.\nFirst, put a partial order ≤on the labels α of polynomial irreducible A-\nmodules Wq,α as follows. Consider Wq,α as an Hq-module. Let μ(α) denote\nthe highest weight in Wq,α as an Hq-module. There may be several highest\nweight vectors in Wq,α, since Wq,α need not be irreducible as an Hq-module.\nThat is fine. Let ≤denote the usual partial order on the highest weights of\nHq-modules: μ1 ≤μ2 iffμ2 −μ1 ∈P\nτ Nτ where τ ranges over the simple\npositive roots of H. We say α ≤α′ iffμ(α) ≤μ(α′).\n11\n\nNow observe that each component W ∗\nq,α⊗Wq,α in the Peter-Weyl decom-\nposition of A is an Hq-bimodule; i.e., has a left and right action of Hq. With\neach irreducible Hq-bimodule in this component isomorphic to Vq,μ1 ⊗Vq,μ2,\nwhere μ1 and μ2 are highest left and right weights of Hq, we associate the\ntype T = (α, μ1, μ2). Put a partial order, which we shall again denote by\n≤, on the types T as per the partial order ≤on the individual components.\nThe type T(b) associated with each b ∈B(A) above is similar to this type.\nSo this also puts a partial order on the types T(b)’s.\nNext fix a b ∈B(A). Let T = T(b) be its type. We shall associate a\ncanonical basis element yb with each such b by induction on its type using\nthe preceding partial order. The set {yb} will then be the required canonical\nbasis of A.\nLet A≤T denote the span of all Hq-bimodules in A of types less than or\nequal to T as per ≤. Let\nL(A≤T) = L(A) ∩A≤T .\nWe define ̄L(A)≤T , and A≤T\nQ\nsimilarly. Consider the natural projection\nψT : A≤T\nQ\n∩L(A)≤T ∩ ̄L(A)≤T →L(A)≤T /qL(A)≤T .\nLet ψ−1\nT (b) be the fibre of b. If this fibre were to contain a unique element,\nthen we can simply let yb be this unique element. But this need not be\nso, because the triple (A≤T\nQ , L(A)≤T , ̄L(A)≤T ) need not be balanced. So\nwe have to resolve the ambiguity in some canonical way. Towards this end,\nwe shall associate with each element in AQ ∩L(A) a complexity measure,\ncalled its degree complexity. We shall then define yb to be the element in\nψ−1\nT (b) of minimum degree complexity–it would be conjecturally unique; cf.\nConjecture 2.5 below.\nThis scheme is in the spirit of [KL1] where each\nelement of the Kazhdan-Lusztig basis of the Hecke algebra is defined to be\nan element of minimum degree in a certain sense.\nSo let us define the degree complexity of an element y ∈AQ ∩L(A).\nSince X⊗r\nq\nis a represention of A = AH\nr (q), we have the injection\nη : A ֒→Z = End(X⊗r\nq ) = (X⊗r\nq )∗⊗X⊗r\nq .\nLet L(Z) = L(X⊗r\nq )∗⊗L(X⊗r\nq ) be the lattice associated with Z. The Q-form\n(or rather KQ-form) ZQ is defined similarly. Then\nProposition 2.4 The embedding η injects the Q-form AQ into ZQ. Fur-\nthermore, assuming Conjecture 2.2, η also injects the lattice L(A) into L(Z).\n12\n\nThe proof is easy. (To be filled in).\nFix the upper canonical basis {xi} of Xq as an Hq-module. Let {x∗\ni } be\nthe dual canonical basis of X∗\nq . This fixes the upper canonical basis CB(Z)\nof Z, whose each element is of the form\nzi1,...,ir;j1,...,jr = x∗\ni1 ⊗· · · ⊗x∗\nir ⊗xj1 ⊗· · · ⊗xjr.\nIt is also a basis of the Q-form ZQ and the lattice L(Z). Now given any\ny ∈A, let w = η(y). Express w in the canonical basis of Z:\nw = η(y) =\nX\nz\na(y, z)z,\n(9)\nwhere z ranges of the basis elements in CB(Z). Since w ∈L(Z) ∩ZQ, each\na(y, z) ∈AK ∩KQ. This means it is integral over Q[q] and hence has a well\ndefined degree d(y, z) at q = 0 (the same as the order of its pole at q = ∞);\nif a(y, z) = 0, we define d(y, z) = −∞. We define the degree complexity d(y)\nof y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We put a partial order\non the degree complexity as follows.\nLet U = Uq(H) be the Drinfeld-Jimbo enveloping algebra of Hq, U −the\nsubalgebra generated by its generators Fi’s. For any string ν = ν1, ν2, . . . of\npositive integers, let U −\nν be the subspace of U −spanned by the words in Fi’s\nin which each Fi occurs occurs νi times. Given a canonical basis element x\nof Xq, we define its length to be |x| = P\ni νi, where x ∈U −\nν x0, and x0 is\nthe highest weight vector of Xq. Order the canonical basis elements of Xq\nas per the reverse order on their lengths; so that x0 is the highest element\nin this order.\nPut a similar order on X∗\nq .\nThis puts an induced partial\norder on the elements of the canonical basis CB(Z) of Z. We let < denote\nthe strict less than relation as per this partial order. Given y and y′, and\nletting w = η(y), w′ = η(y′), we say that d(y) ≤d(y′) if for every z: either\nd(y, z) ≤d(y′, z), or for some ̄z < z d(y, ̄z) < d(y′, ̄z).\nConjecture 2.5 (Minimum degree) The fibre ψ−1\nT (b) contains a unique\nelement yb of minimum degree complexity. Minimum means d(yb) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (b).\nWe call yb the canonical basis element associated with b, and the set {yb}\nthe canonical basis CB(A) of A. The canonical basis CB(Sr) = {xb} of Sr\nis defined to be its dual. The canonical basis CB(S) of S is ∪rCB(Sr).\nRemark 1: The reader may wonder why we defined the canonical basis of A\nfirst, and that of Sr later, as its dual. Can we define the canonical basis of Sr\n13\n\ndirectly? The algorithm for Sr would be similar as above. The main problem\nis to define the degree complexity of an element x ∈L(Sr) ∩SQ. We have\na natural projection from Z = End(X⊗d\nq ) to Sr, but not a natural injection\nthat injects Sr,Q = Sr ∩SQ into ZQ. So the analogue of Proposition 2.4 does\nnot hold.\nRemark 2: In the standard settting, Kashiwara and Lusztig give an efficient\nscheme for constructing each canonical basis element of the standard matrix\ncoordinate ring O(Mq(V )). We do not have here an analogous efficient algo-\nrithm for constructing the canonical basis element yb in Conjecture 2.5. In\nthe standard setting an efficient cosntruction was possible because the stan-\ndard Drinfeld-Jimbo universal enveloping algebra has an explicit presenta-\ntion in terms of generators and defining relations. This explicit presentation\nis crucially used in the construction of the standard canonical basis and also\nin proving correctness of the construction.\nThe nonstandard universal algebra does not have an analogous explicit\npresentation as yet; cf. [GCT7]. For this we need explicit formulae for the\ncoefficients of the Laplace relation in [GCT7] among the simplest nonstan-\ndard canonical basis elements in S, namely nonstandard minors, since as\ndiscussed in [GCT7], it is the mother relation in the representation theory\nof the nonstandard quantum group (just as in the standard setting). That\nis, we need explicit interpretation for these coefficients in the spirit of the ex-\nplicit interpretation for the coefficients of the Kazhdan-Lusztig polynomials\nin terms perverse sheaves. This is the basic core problem that needs to be\nsolved to prove that the preceding algorithm for constructing the nonstan-\ndard canonical basis is correct and to give an explicit, efficient construction\nof yb. Furthermore, if explicit presentation of the nonstandard universal al-\ngebra is so nonelementary, as against the elementary explicit presentation of\nthe standard (Drinfled-Jimbo) enveloping algebra, then the task of proving\ncorrectness may be formidable.\n2.4\nProperties of the nonstandard canonical basis\nIt may be conjectured that the nonstandard canonical bases CB(S) and\nCB(A) have properties akin to the standard canonical basis of O(Mq(V )):\n2.4.1\nCellular decomposition\nConjecture 2.6 (Cell decomposition) The refined Peter-Weyl theorem,\nakin to the one proved by Lusztig [Lu2] in the standard setting, holds for\n14\n\nCB(S) and CB(A).\nThis means the left, right and two-sided cells in O(MH\nq (X)) with re-\nspect to CB(S) yield irreducible left, right, and two-sided (polynomial)\nrepresentations of GH\nq . And furthermore, the left sub-cells of each left cell\nwith respect to the restricted Hq-action yield irreducible Hq-representations.\nThe left cell of O(MH\nq (X)) is defined as follows.\nGiven b ∈CB(S), let\n∆(b) = P\nb′,b′′ cb′,b′′\nb\nb′ ⊗b′′, where ∆denotes comultiplication. Then we say\nthat b′′ ←L b if cb′,b′′\nb\nis nonzero for some b′. Let <L denote the transitive\nclosure of ←L. Using <L we define left cells in a natural way. The right\nand two-sided cells are defined similarly. The left, right and two-sided sub-\ncells with respect to the Hq-action are defined similarly. The definitions for\nCB(A) are similar.\nBy restricting the canonical basis to any left cell corresponding to an\nirreducible polynomial representation Wq,α of GH\nq , we get the canonical basis\nof Wq,α; here the choice of the left cell would conjecturally not matter up to\nscaling.\n2.4.2\nPositivity in the Kronecker case\nConjecture 2.7 (Positivity) In the Kronecker case–i.e. when H = GL(V )×\nGL(W), X = V ⊗W with the natural H-action–each coefficient g(q) of any\ncanonical basis element in CB(AH\nr ) in the basis CB(Z) (cf. eq.(9)) is a\npositive polynomial in q.\nSimilarly, each multiplicative or comultiplicative structural constant of\nCB(AH\nr ) is of the form\n+−(q −1\nq)af(q), where a is a nonnegative integer\nand f(q) is a −-invariant positive and unimodal polynomial in q and q−1.\nThe same also for CB(S).\nHere by a positive polynomial we mean a polynomial with nonnegative\nrational coefficients. By unimodality of the (−-invariant) polynomial f(q),\nwe mean its coefficients f−k, . . . , fk satisfy the condition\nf−k ≤f−k+1 ≤· · · ≤f−1 ≤f0 ≥f1 ≥· · · fk.\nBy multiplicative structural constants, we mean the coefficients mb′′\nb,b′ in the\nexpansion\nbb′ =\nX\nb′′\nmb′′\nb,b′b′′,\n15\n\nfor b, b′ ∈CB(S), and with b′′ ranging over the elements in CB(S). Comul-\ntiplicative structural constants are defined similarly.\nFor experimental evidence for the dual nonstandard algebra BH\nr (q), see\nSection 5.1.\nPresumably, the nonnegative coefficients of g(q) and f(q) may have a\ntopological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of f(q) may be a conse-\nquence of some result akin to the Hard Lefschetz theorem in the spirit of\nthe results on unimodal sequences in [St].\nThe Kronecker case is fundamental because O(MH\nq (X)) therein is a non-\nflat deformation of O(M(X)), though CH\nq [X] is a flat deformation of C[X].\nThus to prove the positivity Conjecture 2.7, some nonstandard extension\nof the theory of perverse sheaves and the work surrounding the Riemann\nhypothesis over finite fields [BBD, Dl2] that can deal with nonflat, noncom-\nmutative varieties like O(MH\nq (X)) may be neeeded.\n2.4.3\nNonstandard positivity and saturation\nIn the Kronecker case, the braided symmetric algebra CH\nq [X] is a flat de-\nformation of C[X]; i.e. dim(CH,r\nq\n[X]) = dim(Cr[X]). But when CH\nq [X] is a\nnonflat deformation the situation is much more complex. To see this, let c be\na coefficient of any canonical basis element in CB(AH\nr ) in the basis CB(Z).\nAs observed after eq.(9), it is integral over Q[q]. Hence every coefficient of\nits minimal polynomial fc is a polynomial in q with rational coefficients. In\nthe spirit of Conjecture 2.7, one may ask if the coefficients of this polynomial\nare always nonnegative. Unfortunately, this need not be so; cf. Section 5.3\nfor counterexamples in the dual setting of BH\nr (q). The following is a relaxed\nversion of Conjecture 2.7 for the general case.\nConjecture 2.8 (Saturation) Each coefficient a(q) of fc is a saturated\npolynomial (in the terminology of [GCT6]); this means a(1) is a positive\nrational if a(q) is not an identically zero polynomial.\nSimilarly, each coefficient b(q) of the minimal polynomial of a multipli-\nciative or comultiplicative structural constant of CB(AH\nr ) can be expressed\nin the form\nb(q) = (−1)e(q −1\nq )e′c(q)\n(10)\nfor some nonnegative integers e, e′, where\n16\n\n1. e is chosen so that the middle term of c(q) is positive; here by the\nmiddle term we mean the coefficient of qi for the smallest i such that\nthis coefficient is nonzero, and\n2. c(q) is a saturated polynomial in q and q−1–this again means c(1) is a\npositive rational if c(q) is not an identically zero polynomial.\nIn the context of the plethysm problem, one is finally interested in the\nbehaviour of b(q) at q = 1 (cf. Section 6), so this relaxed saturation form\nof positivity should be sufficient; see also [GCT6] for the importance of\nsaturation in the context of the flip in GCT. A stronger positivity conjecture\nthat would specialize to Conjecture 2.7 in the Kronecker case is:\nConjecture 2.9 (Nonstandard Positivity) Each polynomial c(q) in Con-\njecture 2.8 is almost positive and unimodal.\nThat is, it is of the form\nc0(q) + c1(q), where, if c(q) is not identically zero,\n1. c0(1) >> |c1(1)|, where >> means much greater as r →∞, and more\ngenerally,\n2. c0(q) is a dominant positive unimodal polynomial, and c1(q) is a very\nsmall error-correction polynomial. Specifically,\n||c1(q)||/||c0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwhere μ and π are as in the plethysm problem (cf. begining of Sec-\ntion 1), ⟨\n⟩denotes the bitlength-of-specification function, and poly(\n)\nmeans polynomial of a fixed (constant) degree in the specified bitlengths,\nand ||\n|| denotes the L2-norm of the coefficient vector of the polyno-\nmial.\nSee Section 5.3 for experimental evidence for Conjectures 2.8-2.9 in the\ndual setting of BH\nr (q).\nPresumably, the nonnegative coefficients of such c0(q) may again have\na topological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of c0(q) may again be a con-\nsequence of some result akin to the Hard Lefschetz theorem. The correction\npolynomial c1(q) may also have a topological interpretation that depends on\na cohomological measure of nonflatness of CH\nq [X]. In the Kronecker case,\nwhen CH\nq [X] is a flat deformation of C[X], this correction would then van-\nish, and Conjecture 2.9 would reduce to Conjecture 2.7. Furthermore, the\n17\n\nconjectural nonnegative value of c(1) may also have an interpretation akin\nto the representation-theoretic interpretation for the values of the Kazhdan-\nLusztig polynomials at q = 1.\nIf nonstandard extension of the work surronding the Riemann hypothe-\nsis over finite fields as needed to prove the positivity Conjecture refcposkro-\nneckerleft in the Kronecker case can be found, that may open the way for\ninvestigating the more complicated nonstandard form of positivity in Con-\njecture 2.9.\n3\nNonstandard canonical basis of BH\nr\nLet BH\nr = BH\nr (q) be the nonstandard quantization of the symmetric group\nring C[Sr] in [GCT7]. In this section, we describe an analogous conjecturally\ncorrect algorithm for constructing a nonstandard canonical basis E(r) of BH\nr .\nIn the standard setting–i.e., when H = G–this basis would conjecturally\nspecialize to the Kazhdan-Lusztig basis of the Hecke-algebra, though the\nspecialized algorithm here is different from the algorithm in [KL1].\nSince BH\nr (q) is semisimple [GCT4, GCT7], by the Wedderburn structure\ntheorem\nBH\nr (q) =\nM\nα\nT ∗\nq,α ⊗Tq,α,\n(11)\nwhere Tq,α ranges over the irreducible representations of BH\nr , assuming that\nthe underlying base field is a suitable algebraic extension of Q(q1/2). We\nshall denote this base field by K–it is the same as the base field K in\nSection 2.2, except that the role of q there is played by q1/2 here.\nLet\nAK, ̄AK, KQ, ˆQ be as in Section 2.2, with the role of q played by q1/2.\nWe assume that H is the general linear group GL(V ) or a product of\ngeneral linear groups. In this case (cf. [GCT7]), BH\nr is a −-invariant sub-\nalgebra of a suitable Hecke-algebra or a product of Hecke algebras, where\n−denotes the usual bar-automorphism on the Hecke algebra [KL1] (ana-\nlogue of −in Section 2). Let Pi’s and Qi’s denote the rescaled positive and\nnegative generators of BH\nr as defined in [GCT7] (denoted by p+,H\nX,i and q+,H\nX,i\ntherein) so that they belong to the usual Z[q1/2, q−1/2]-form on the ambient\nHecke algebra (or the ambient product of Hecke algebras). Let BH\nr,Q denote\nthe KQ-form of BH\nr generated ring-theoretically by Pi’s, or equivalently Qi’s.\nThe goal is to construct an AK-lattice R(r) ⊆BH\nr so that the canonical\nbasis E(r) of BH\nr\ncan then be constructed by a nonstandard globalization\n18\n\nprocedure on the triple (BH\nr,Q, R(r), ̄R(r)) analogous to the one Section 2.3.\nJust as in Section 2.2, it will turn out that this triple need not be balanced.\nWe will resolve the ambiguity caused by lack of balance using the notion of\nminimum degree complexity very much as in Section 2.3.\n3.1\nNonstandard Gelfand-Tsetlin basis of BH\nr (q)\nIn the construction of Kazhdan-Lusztig polynomials as described in [KL1,\nSo], the lattice in the Hecke algebra is constructed using its standard mono-\nmial basis. But in general BH\nr (q) does not have a naturally defined monomial\nbasis; see [GCT4] for an example. So we need a different way to construct\nthe lattice R(r). The construction here will be analogous to the construction\nof the lattice in the standard matrix coordinate ring O(Mq(V )) based on its\nGelfand-Tsetlin basis. This construction in O(Mq(V )) is different from the\none defined in Section 2.1. We shall recall it using the same notation as\nin Section 2.1. It is based on the observation that the Gelfand-Tsetlin ba-\nsis of an irreducible Hq-representation Vq,μ (after rescaling as described in\nsection 7.3.3 of [Kli]) is its local crystal basis: i.e., it is an A-basis of the\nlattice Lμ ⊆Vq,μ, and that its projection in Lμ/qLμ is equal to the basis\nBμ (whose elements have disjoint monomial supports as described before).\nThis observation was in fact the starting point for the theory of local crystal\nbasis [Kas1]. Let us denote the (rescaled) Gelfand-Tsetlin basis of Vq,μ by\nGTq,μ. The Gelfand-Tsetlin basis of O(Mq(V )) is defined as per the standard\nPeter-Weyl theorem ( 3):\nGT(O(Mq(V ))) =\n[\nμ\nGT ∗\nq,μ ⊗GTq,μ.\n(12)\nLet LGT be the lattice generated by this Gelfand-Tsetlin basis, and BGT the\nprojection of the Gelfand-Tsetlin basis on LGT /qLGT . Then (LGT , BGT )\ncoincides with the standard crystal base (L, B) of O(Mq(V )) in Section 2.1.\nThe algebra BH\nr (q) has a natural analgoue of the Gelfand-Tseltin basis,\nwhich can then be used to construct the lattice R(r). We begin by describing\nthis basis for an irreducible representation Tq,α of BH\nr (q).\nWe proceed by induction on r, the case r = 1 being easy. The following\nis a conjectural analogue of the standard Pieri’s rule in this setting:\nC1’: Tq,α has a multiplicity-free decomposition as a BH\nr−1-module.\n(This conjecture is not really necessary as long as there is a natural\nway to resolve the ambiguity caused by multiplicity).\nBy induction, we\n19\n\nhave defined a basis for each irreducible BH\nr−1(q)-submodule of Tq,α. Putting\nthese bases together, we get the sought nonstandard Gelfand-Tsetlin basis\nCα of Tq,α.\nAssuming multiplicity-free decomposition, such a basis is unique, up to\nscaling factors, which will be fixed in the course of the algorithm below.\nEach element x ∈Cα can be indexed by a nonstandard Gelfand-Tsetlin\ntableau, which is an analogue of the standard Gelfand-Tsetlin tableau [Kli]\nin this setting. It is defined to be the tuple (αr, αr−1, . . .), with αr = α, of the\nclassifying labels–which we shall call types–of the irreducible BH\ni -submodules\nTq,αi containing x, where Tq,αi ⊂Tq,αi+1.\nWe define the nonstandard Gelfand-Tsetlin basis C(r) of BH\nq (r) as per\nthe decomposition (11):\nC(r) =\n[\nα\nC∗\nα ⊗Cα.\nEach element of C(r) is indexed by a nonstandard Gelfand-Tsetlin bi-tableau\nas per this decomposition. We shall denote the element of C(r) indexed by\na nonstandard Gelfand-Tsetlin bitableau T by cT .\n3.2\nLocal crystal base\nThe sought lattice R(r) ⊆BH\nr (q) will be generated by the elements of C(r)\nafter scaling them appropriately in the course of the algorithm below. Let\nus assume at the moment that this scaling has already been given to us,\nand thus R(r) is fixed. Let bT denote the image of cT under the projection\nψ : R(r) →R(r)/q1/2R(r). Let B(r) = {bT } be the basis of R(r)/q1/2R(r).\nThen (R(r), B(r)) is the analgoue of the local crystal base in the standard\nsetting.\n3.3\nNonstandard globalization via minimization of degree\ncomplexity\nThe elements of C(r) need not be −-invariant.\nNext we globalize C(r)\nto get a −-invariant canonical basis E(r) of BH\nr\nin the spirit of Kazhdan\nand Lusztig [KL1], with the role of the standard basis in [KL1, So] played\nby the nonstandard Gelfand-Tsetlin basis of BH\nr (q) here. As already men-\ntioned, the main difference from the standard setting of Hecke algebra is\nthat (BH\nr (q), R(r), ̄R(r)) need not be balanced. This is the main problem\n20\n\nthat needs to be addresssed. The nonstandard globalization procedure here\nis analogous to the one in Section 2.3. It goes as follows.\n(1) In Section 2.3, we described a partial order ≤on the types (classifying\nlabels) of the irreducible modules Wq,α of AH\nr (q). By the nonstadard duality\nconjecture [GCT7], this induces a partial order ≤on the types (classifying\nlabels) of the (paired) irreducible modules Tq,α of BH\nr (q). (In the standard\nsetting, this procedure would yield a partial order on the partitions of size\nr, with the partition containing a single row of size r at the top of the order\nand the partition containing a single column of size r at the bottom of the\norder.)\n(2) This induces a lexicographic partial order ≤on the nonstandard Gelfand-\nTsetlin tableaux, since they are just tuples of types, and also on nonstandard\nGelfand-Tsetlin bitableau which index the basis elements C(r).\n(3) Let B≤T be the span of the basis elements cT ′ ∈C(r) such that T ′ ≤T.\nLet R≤T = R(r) ∩B≤T , ̄R≤T = ̄R(r) ∩B≤T, and B≤T\nQ\n= BH\nr,Q ∩B≤T . Then\nthe triple (B≤T\nQ , R≤T , ̄R≤T ) need not be balanced. To define a canonical\nbasis element eT associated with T, we associate a degree complexity with\neach element y ∈BH\nr,Q in the spirit of Section 2.3.\nThis is done as follows. Since we are assuming that H is GL(V ) or a\nproduct of general linear groups, BH\nr is a subalgebra of a product of Hecke\nalgebras, say Z = Hk1(q) ⊗· · · ⊗Hkl(q), where Hj(q) denotes the Hecke\nalgebra with rank j. Furthermore,\nBH\nr,Q ⊆ZQ = Hk1,Q × · · · × Hkl,Q,\nwhere Hj,Q denote the KQ-form of Hj(q) obtained by tensoring its usual\nQ[q1/2, q−1/2]-form with KQ. Consider the Kazhdan-Lusztig basis KL(Z)\nof Z formed by taking the product of the Kazhdan-Lusztig bases of its Hecke\nalgebra factors. Express y ∈BH\nr,Q in terms of KL(Z):\ny =\nX\nz\na(y, z)z,\n(13)\nwhere z ranges over the elements in KL(Z). Then each coefficient a(y, z) ∈\nKQ. Let d(y, z) denote the degree of a(y, z); i.e., the order of its pole at\nq = ∞.\nIf a(y, z) = 0, we define d(y, z) = −∞.\nWe define the degree\ncomplexity d(y) of y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We\nput a partial order on degree complexities as follows. Put a partial order\n≤on the Kazhdan-Lusztig basis of the Hecke algebra Hj(q) as per the\nreverse order on the (reduced) lengths of the permutation indices of the\n21\n\nbasis elements–so 1 is the highest element as per this order. This also puts\na partial order on KL(Z). Let < denote the strict less than relation as per\nthis partial order. Given y and y′, we say that d(y) ≤d(y′) if for every z:\neither d(y, z) ≤d(y′, z), or for some ̄z < z, d(y, ̄z) < d(y′, ̄z).\nConsider the natural projection\nψT : R≤T ∩ ̄R≤T ∩B≤T\nQ\n→R≤T /q1/2R≤T .\nLet ψ−1\nT (bT ) be the fibre of bT ∈B(r). The following is the analogue of\nConjecture 2.5 in this context (with different interpretation for b, y, ψ etc.\nfrom there):\nConjecture 3.1 (Minimum degree) The fibre ψ−1\nT (bT ) contains a unique\nelement eT of minimum degree complexity. Minimum means d(eT ) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (bT ).\nWe call eT the canonical basis element associated with T, and E(r) =\n{eT } the canonical basis BH\nr (q).\nSo far we have not discussed how to scale the nonstandard Gelfand-\nTsetlin basis of BH\nr (q) to get the lattice R(r). To complete the algorithm,\nit remains to fix this scaling.\nLet {c′\nT } denote the nonstandard Gelfand-Tsetlin basis of BH\nr (q) before\nscaling. The scaled cT will be of the form qaT c′\nT for some rational aT . We\nhave to determine all aT ’s. Assume that aT ′, T ′ < T, have been fixed. For\nany rational a, let ca,T = qac′\nT . Let Ra,≤T be the lattice generated by ca,T\nand cT ′, T < T, and ̄Ra,≤T obtained by applying −to it. Consider the\nprojection\nψa,T : Ra,≤T ∩ ̄Ra,≤T ∩B≤T\nQ\n→Ra,≤T /q1/2Ra,≤T .\nLet ba,T be the image of ca,T under the projection Ra,≤T /q1/2Ra,≤T . Let\nψ−1\na,T (ba,T ) be its fibre. The following is the strengthened form of Conjec-\nture 2.5.\nConjecture 3.2 (Minimum degree) There exists a unique aT and eT ∈\nψ−1\naT ,T(baT ,T) such that for any a and any y ∈ψ−1\na,T (ba,T ) d(eT ) ≤d(y). That\nis, eT is the unique element of minimum degree complexity over all choices\nof a.\n22\n\nThis fixes aT . Furthermore, ψT = ψaT ,T , bT = baT ,T , and eT in Conjec-\nture 3.1 is the same as here.\nIf instead of the order ≤among the classifying labels α’s of Tq,α, we\nuse its reverse order, and in the definition of degree complexity use the\nopposite of the Kazhdan-Lusztig basis (obtained by replacing Qi by Pi), we\nget another canonical basis Eopp(r) of BH\nr (q), which we shall call its opposite\ncanonical basis.\nTo prove Conjectures 3.1-3.2, we need to know relations among the gen-\nerators Pi’s of BH\nr explicitly, just as we know the relations among the gen-\nerators of the Hecke algebra explicitly. This is not known at present. See\n[GCT7] for the problems that arise in this context.\nEach element c of the Kazhdan-Lusztig basis of the Hecke algebra can\nbe expressed in the form\nc = c0 +\nX\nj>0\na(j)cj,\nwhere each cj is a monomial in the generators of the Hecke algebra, a(j) ∈\nQ[q1/2, q−1/2] and the length of each cj, j > 0, is smaller than that of\nc0. This need not be so in the nonstandard setting: there can be several\nmonomials of maximum length with nontrivial coefficients in any monomial\nrepresentation of a nonstandard canonical basis element; cf. Section 5.3 and\nFigure 11 therein for an example.\n3.4\nConjectural properties\nIt may be conjectured that the canonical bases E(r) and Eopp(r) have prop-\nerties akin to those of the Kazhdan-Lusztig basis of the Hecke algebra.\n3.4.1\nCellular decomposition\nConjecture 3.3 (Cell decomposition) Analogue of the cell decomposi-\ntion property of the Kazhdan-Lusztig basis also holds for E(r) and Eopp(r).\nSpecifically this means the following. Let us define the left, right and\ntwo-sided cells of BH\nr\nwith respect to the canonical basis E(r) very much\nas in Section 2.4. Then it may be conjectured that they yield irreducible\nleft, right and two-sided representations of BH\nr . The conjecture for Eopp(r)\nis similar.\n23\n\nBy restricting E(r) to any left cell corresponding to an irreducible BH\nr -\nmodule Wq,α, we get the canonical basis of Wq,α; here the choice of the left\ncell would conjecturally not matter (up to scaling).\n3.5\nPositivity in the Kronecker case\nThe following is an analogue of Conjecture 2.7 here.\nConjecture 3.4 (Positivity) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13).\nIn the Kronecker case, c is of the form\n+−(q1/2 −q−1/2)af(q), where a\nis a nonnegative integer and f(q) is a −-invariant positive and unimodal\npolynomial in q1/2 and q−1/2.\nThe same for Eopp(r).\nFor experimental evidence see Sections 5.1.2 and 5.2.\n3.5.1\nNonstandard positivity and saturation\nThe general case is much more complex as in Section 2.4.3. The following\nis an analogoue of Conjecture 2.8.\nConjecture 3.5 (Saturation) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13). Let fc be its minimal polynomial with\ncoefficients in Q(q1/2).\nThen each coefficient s(q) of fc can be expressed in the form\ns(q) = (−1)e(q1/2 −q−1/2)e′g(q)\n(14)\nfor some nonnegative integers e, e′, where\n1. e is chosen so that the middle term of g(q) is positive; and\n2. g(q) is a saturated polynomial in q1/2 and q−1/2.\n24\n\nAnalogue of the stronger Conjecture 2.9 in this case is:\nConjecture 3.6 (Nonstandard Positivity, informal) Each polynomial\ng(q) above is almost positive and unimodal; i.e. of the form g0(q) + g1(q),\nwhere, if g(q) is not identically zero,\n1. g0(1) >> |g1(1)|, and more generally,\n2. g0(q) is a dominant positive unimodal polynomial, and g1(q) is a very\nsmall error-correction polynomial. Specifically,\n||g1(q)||/||g0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwith the terminology as in Conjecture 2.9.\nFor experimental evidence, see Section 5.4.1.\nHere, g(1) and the conjectural nonnegative coefficients of g0(q) may have\na representation-theoretic/topological/cohomological interpretation akin to\nthat sought for the analogous quantities in Section 2.4.3.\n3.5.2\nQuasi-cellular decomposition\nConjecture 3.7 The opposite canonical basis Eopp also has the following\nquasi-cellular decomposition property.\nFor this we define a quasi-subcellular decomposition of each left or right\ncell with respect to Eopp(r). Specifically, given e′, e′′ belonging to the same\nleft cell, express\nee′ =\nX\ne′′\nǫ(e, e′, e′′)(q1/2 −q−1/2)δ(e,e′,e′′)de′′\ne,e′e′′,\n(15)\nwhere e′′ ∈E(r), the sign ǫ(e, e′, e′′) is either 1 or −1, δ(e, e′, e′′) is a non-\nnegative integer, and de′′\ne,e′ is a −-invariant saturated polynomial in q1/2 and\nq−1/2 as per the saturation Conjecture 3.5. We say that e′′ ∝L e′ if, for some\ne, de′′\ne,e′ in (15) is nonzero and δ(e, e′, e′′) is zero; i.e., if, for some e, e′′ occurs\nwith nonzero coefficient in the expansion of ee′ specialized at q = 1. Let ≺L\ndenote the transitive closure of ∝L. Using ≺L we define left quasi-subcells\nof a left-cell of Eopp(r). It may be conjectured that each left quasi-subcell\nof Eopp(r) yields an irreducible representation of the symmetric group Sr at\n25\n\nq = 1. That is, when the BH\nr -representation Y corresponding this left cell\nis specialized at q = 1, so as to become a representation Yq=1 of C[Sr], the\npartial order on its left quasi-sub-cells induces a composition series of Yq=1\nwhose factors are irreducible representations of C[Sr].\nFix one such quasi-subcell C of Eopp(r). Let Sλ(C) be the irreducible\nrepresentation (Specht module) of C[Sr] that is isomorphic to the factor\nin correspondence with C in this composition series of Yq=1, where λ(C)\nis a partition depending on C. The canonical basis elements in C, after\nspecialization at q = 1 and projection, yield a basis of Sλ(C). It may be con-\njectured that this basis coincides with the Kazhdan-Lusztig basis of Sλ(C)\n(up to rescaling).\nBy the Kazhdan-Lusztig basis of Sλ(C), we mean spe-\ncialization at q = 1 of the Kazhdan-Lusztig basis of the quantized Specht\nmodule Sq,λ(C) of the Hecke algebra Hr(q).\nBut Conjecture 3.7 need not hold for E(r); cf. Section 5.3 for a coun-\nterexample. This is analgous to the fact that the refined Peter-Weyl theorem\nin [Lu2] for the coordinate ring of the standard quantum group Hq holds\nonly for the ordering ≤(as defined in Section 2.3) among the labels (high-\nest weights) of irreducible Hq-modules–there is no canonical basis of the\nstandard coordinate ring which admits refined Peter-Weyl theorem for the\nopposite of the order ≤.\n4\nInternal definition of degree complexity\nWe give here an internal definition of degree complexity which may be used\nin place of the definition in Section 3.3 during the construction of the canon-\nical basis. By internal, we mean it is based only on the structure of BH\nr (q)\nand does not depend on its embedding in the external ambient algebra Z\nthere. This notion of degree complexity does not coincide with the one in\nSection 3.3, but the canonical basis constructed using this definition may be\nconjectured to be the same as the one constructed therein.\nLet B[i] ⊆BH\nr (q) be the span of the monomials in Qj’s of length j, and\nB[< i] of length < i. We say that a given set of monomials in Qj’s of length\ni is independent if the images of these monomials in B[i]/B[< i] are linearly\nindependent. An expression\na =\nX\nm\namm,\n(16)\nwhere m ranges over monomials in Qj’s and am ∈KQ, is called valid\n26\n\nif, for each i, the monomials m of length i with am ̸= 0 in this expres-\nsion are independent.\nAssume that a is −-invariant, so that each am is\n−-invariant.\nThe degree complexity ˆd(a) of a is defined to be the tuple\n⟨ˆdl(a), . . . , ˆdi(a), . . . , ˆd0(a)⟩where ˆdi(a) denotes the maximum degree (at\nq = 0) of am for any m of length i, and l is the maximum length of m\nwith am ̸= 0 in the expression (16); by definition, ˆdi(a) = −∞if there is\nno m in (16) of length i with am ̸= 0. We order these degree complexities\nlexicographically. The degree complexity ˆd(b) of an element b ∈BH\nr,Q(q) is\ndefined to be the minimum degree complexity of its any valid expression.\nIt may be conjectured that if this definition of degree complexity, with the\nlexicographic ordering as above, is used in place of the definition of degree\ncomplexity in Section 3.3, the algorithm therein still works correctly and\nconstructs the same canonical basis E(r).\nFor the opposite canonical basis Eopp(r), one can similarly use the inter-\nnal definition as above with Pi in place of Qi.\nThe definition of degree complexity in this section is not as satisfactory\nas in Section 3.3 because BH\nq (r) does not a natural monomial basis [GCT4].\nHence to find the degree complexity of an element, one has to consider all\nits monomial expressions, finite but huge in number. It will be interesting\nto know if there is a more efficient internal definition.\n5\nExperimental evidence for BH\nr (q)\nIn this section we shall verify the conjectures in this paper for two nontriv-\nial special cases of the nonstandard algebra B = BH\nr (q). The nonstandard\ncanonical bases of BH\nr (q) in these cases were computed with the help of a\ncomputer using the algorithm in Section 3 and the notion of degree com-\nplexity as in Section 3.3.\nFirst, some notation. Given a string σ = i1 · · · ik of positive integers,\nwe let Pσ denote the monomial Pi1 · · · Pik; Qσ is defined similarly. Given a\n−-invariant polynomial g(q) ∈Q[q, q−1], we define the vector ag associated\nwith g(q) as follows. Express g(q) in the form\n+−(q−1/q)eh(q), where e is the\nmaximum possible. Let h−l, . . . , h0, . . . , hl be the coefficients of h(q). Then\nag is defined to be [h0, . . . , hl]. In particular if h(q) is (positive) unimodal,\nthen aq is a (positive) nonincreasing sequence. The vector associated with\na −-invariant polynomial in q1/2 and q−1/2 is defined similarly.\n27\n\n5.1\nKronecker problem: n = 2, r = 3\nConsider B = BH\n3 (q) in the special case of the Kronecker problem for n = 2\nand r = 3. Thus H = Gl2 ×Gl2, and G = Gl4 with H embedded diagonally.\nLet Pi, i = 1, 2, be as in Section 3 and [GCT4]. The nonstandard canonical\nbasis of B was computed in [GCT4] by an ad hoc method for r = 3, but it\ncoincides with the one computed by the algorithm here. It is as follows. Let\nc1 = q6 + 2q5 + 3q4 + 4q3 + 3q2 + 2q + 1\nq3\n,\nc2 = q4 + q3 + 4q2 + q + 1\nq2\n,\nb1 = −(q2 + 1)2/q2, and b2 = (q + 1)2/q.\nThen the opposite canonical basis Eopp(3) of B consists of the following\nten elements:\nΣ = c1P1 −c2P121 + P12121,\nγi\n1 = b1P1 + P121,\ni = 1, 2,\nγi\n12 = b1P12 + P1212,\ni = 1, 2,\nγi\n2 = b1P2 + P212,\ni = 1, 2,\nγi\n21 = b1P21 + P2121,\ni = 1, 2,\nμ = 1.\n(17)\nThe canonical basis E(3) is obtained by susbstituting Qi for Pi. In what\nfollows, we shall only consider Eopp(3).\n5.1.1\nCellular and quasi-cellular decomposition\nThe basis Eopp(3) has a cellular decomposition, in accordance with Conjec-\nture 3.3, with the following right cells:\nUσ\n=\n{Σ}\nV1\n=\n{γ1\n1, γ1\n12}\nV2\n=\n{γ2\n1, γ2\n12}\nW1\n=\n{γ1\n2, γ1\n21}\nW2\n=\n{γ2\n2, γ2\n21}\nUμ =\n=\n{μ}.\nThe left cell decomposition is similar. The representation of B supported\nby Uσ is the trivial one dimensional representation. The representation sup-\nported by V1 or W1 is isomorphic; let us call it χ1. Similarly, the representa-\ntion supported by V2 or W2 is isomorphic; let us call it χ2. Then χ1 and χ2\n28\n\nare two nonisomorphic two-dimensional representations of B which special-\nize at q = 1 to the two-dimensional Specht module of the symmetric group\nS3 corresponding to the partition (2, 1). Thus quasi-cellular decomposition\n(Conjecture 3.7) holds trivially here.\n5.1.2\nPositivity\nCoefficients of the elements of W1 and W2 in the Kazhdan-Lusztig basis of\nH3(q) ⊗H3(q) ⊇BH\n3 (q) are shown in Figure 1 (with the Kazhdan-Lusztig\nbasis symmetrized and appropriately ordered as described in [GCT4]); the\nfirst column shows the coefficients of γ1\n1, the second of γ1\n12, and so on. It\ncan be observed that all coefficients are positive, and unimodal polynomials\nin Q[q, q−1]. The cofficients of other canonical basis elements can be found\nin [GCT4]; they too are positive, unimodal polynomials. This verifies the\npositivity Conjecture 3.4 for the structural coefficients of the canonical basis.\nA few typical nonzero multiplicative structural constants of the canonical\nbasis are shown in Figure 2, where the coefficient of bb′ with respect to the\nbasis element b′′ is denoted by c(b, b′; b′′). It can be seen that each constant\nis a polynomial of the form\n(−1)a(q1/2 −q−1/2)bf(q1/2, q−1/2),\nwhere f is a positive unimodal polynomial. It was verified with computer\nthat all multiplicative structural constants are of this form. This verifies the\npositivity Conjecture 3.4 for the multiplicative structural constants as well.\n5.2\nKronecker case, H = SL2, r = 4\nFor the Kronecker case, H = Gl2 × GL2, G = GL4, and r = 4, we could\ncompute just one canonical basis element Σ (akin to Σ in Section 5.1) corre-\nsponding to the trivial one dimensional representation of BH\n4 (q). Symbolic\ncomputations needed to compute other canonical basis elements turned out\nto be beyond the scope of MATLAB/Maple on an ordinary workstation.\nThe coefficients of Σ in the Kazhdan-Luztig basis of H4(q)⊗H4(q) ⊃BH\n4 (q)\nwere computed in MATLAB/Maple. There are 576 coefficients in total. Fig-\nures 3-5 show the vectors associated with distinct nonzero coefficients among\nthese. They can be seen to be positive and nonincreasing in accordance with\nConjecture 3.4.\n29\n\n(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n8\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n8 (1+q)2\nq\n8\n8 (1+q)2\nq\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n(q2+6 q+1)(1+q)2\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2\n \n \nFigure 1: Coefficients of the elements of W1 and W2 in the symmetrized\nKazhdan-Lusztig basis, as computed in [GCT4]\n30\n\n5.3\nH = sl2, G = sl4\nNow we study the nonstandard algebra B = BH\n3 (q), when H = Gl2, X is its\nfour dimensional irreducible representation, and G = GL(X) = Gl4. It is\na 21-dimensional algebra whose explicit presentation is given in Section 7.1\nof [GCT7]. We follow the notation as therein. Let Pi and Qi be as defined\nin the begining of that section. The monomials Pσ, where σ ranges over\nstrings in 1 and 2 of length k ≤10 with no consecutive 1’s or 2’s, form a\nbasis of B. This algebra has one trivial one-dimensinal representation, and\nfive nonisomorphic two-dimensional representations, so that\n21 = 1 + 22 + 22 + 22 + 22 + 22.\nLet\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(18)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x]\nbe the algebraic extension of Q(q) obtained by adjoining x. It is shown in\n[GCT7] that B admits a complete Wederburn-structure decomposition over\nK, but not Q(q). In what follows, we assume that B is defined over this\nbase field K.\nThe nonstandard canonical bases E(3) and Eopp(3) of B were computed\nin MATLAB/Maple using the algorithm in Section 3. They are as follows.\nLet Ui, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the\nmatrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni\n \n,\nwhere u1\n1 is as specified in Figure 6, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 7-9. Elements\nare specified in these figures by giving their nonzero coefficients in the {Qσ}\nbasis; the coefficient for Qσ is shown in front of σ. Let u2\ni , 1 ≤i ≤5, be the\nelement obtained from u1\ni by interchanging Q1 and Q2. Let u12\ni\n= u1\ni Q2,\nand u21\ni\n= Q2u1\ni , for 1 ≤i ≤5. Let u0 = 1 (this definition of u0 is different\nfrom that in [GCT7]). Then u0 and the entries of ui form the canonical\nbasis E(3) of BH\n3 (q). The left cells of E(e) are {u0} and the columns of ui.\nThe right cells are {u0} and the rows of ui. The representation supported\n31\n\nby {u0} is the trivial one-dimensional representation; let us denote it by\nΣ. The representations supported by the columns or rows of ui are two-\ndimensional representations of B, distinct for each i; let us denote them by\nχi.\nThe left cell {u0} is at the top of the ≤L partial order and the left\ncells corresponding to the columns Ui are at depth 1 from the top in this\npartial order (and mutually incomparable). The situation for the right cells\nis similar.\nLet\nvi =\n v1\ni\nv12\ni\nv21\ni\nv2\ni\n \n,\nwhere vα\ni is obtained from uα\ni by substituting Pσ for Qσ in the expression of\nuα\ni in the {Qσ} basis. Let v0 be the element whose coefficients in the {Pσ}\nbasis are as shown in Figures 10-11. Then v0 and the elements of vi form the\nopposite canonical basis Eopp(3) of BH\n3 (q). The left cells are {v0} and the\ncolumns of vi. The right cells are {v0} and the rows of vi. The left cell {v0}\nis at the bottom of the ≤L partial order and the left cells corresponding to\nthe columns of vi at height 1 from the bottom (and mutually incomparable).\nThe situation for the right cells is similar.\nLet Σ′ be the trivial one-dimensional representation of the subalgebra\nBH\n2 (q) ⊂B = BH\n3 (q) generated by P2. Let μ′ be the other one-dimensional\nrepresentation of BH\n2 (q) that specializes to the signed one-dimensional rep-\nresentation of the symmetric group S2 at q = 1. Then the nonstandard\nGelfand-Tsetlin tableau T(b)’s associated with the basis elements b’s of E(3)\nare as follows.\nIf b = u0, then T(b) = [Σ, Σ′]. If b = u1\ni , then T(b) = [χi, Σ′]. If b = u21\ni ,\nthen T(b) = [χi, μ′].\nIf b = u2\ni , then T(b) = [χi, Σ′].\nIf b = u12\ni , then\nT(b) = [χi, μ′].\nThe nonstandard Gelfand-Tsetlin tableau associated with the basis ele-\nments in Eopp(r) are similar.\n5.3.1\nViolation of standard balance\nLet R(3) be the AK-lattice generated by E(3), ̄R(3) = (R(3))−. Let Ropp(3)\nand ̄Ropp(3) be defined similarly. Then it turns out that the triple (BH\n3,Q, R(3), ̄R(3))\nassociated with the canonical basis E(3) is balanced, but the triple (BH\n3,Q, Ropp(3), ̄Ropp(3))\nassociated with the opposite canonical basis Eopp(3) is not balanced. Specif-\nically, the fibre ψ−1\nT (0) ̸= {0} when T = T(b) is the nonstandard tableau\nassociated with b = v2\ni , for any i (it is zero for all other b’s).\n32\n\nFor example, with the help of computer it was found that the Q-module\nψ−1\nT (0), for b = v2\n5, T = T(b), is generated by the two elements w and x\nspecified in Figures 12-15, which give their nonzero cofficients in the {Pσ}\nbasis. Clearly w belongs to the KQ form BH\n3,Q since the coefficients belong\nto Q[q, q−1]. It is −-invariant, since the coefficients are −-invariant. It can\nbe verified that w ∈qRopp(3). Specifically, it can be shown that\nw = a0v0 + c1v1\n5 + c12v12\n5 + c2v2\n5 + c21v21\n5 ,\nwhere the coefficient vector [a0, c1, c12, c2, c21] is the following\n \n \n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)\n \n \nThis means ψT (w) = 0. It can be verified that w belongs to B≤T . Similarly,\nx is a −-invariant element of BH\n3,Q ∩B≤T that belongs to qRopp(3).\n5.4\nNonstandard globalization via minimization of degree\ncomplexity\nThus each element in ψ−1\nT ( ̄b), b = v2\n5, ̄b = ψ(b), T = T(b), is a linear combi-\nnation of b, w and x. It easy to see from the explicit formulae in Figures 12-15\nand Figure 9 that b = v2\n5 is the element of minimum degree complexity in\nψ−1\nT ( ̄b), where the degree complexity is defined internally as in Section 4.\n(Remember that v2\n5 is obtained from u1\n5 in Figure 9 by substituting Pi for Qi\nand then interchanging P1 and P2.) The same is also true for all b’s. This\nverifies the minimum degree conjecture (Conjecture 3.1); Conjecture 3.2 can\nalso be verified similarly.\nWe could not use the external definition of degree complexity as in Sec-\ntion 3.3 here, since the smallest product of Hecke algebras containing BH\n3 (q)\nis Z = H3(q)⊗9 with dimension 10077696 = 69. It is impossible to carry out\nsymbolic computations in an algebra of this size in MATLAB/Maple.\n33\n\nc(γ1\n1; γ1\n1; γ1\n1) = c(γ12\n1 ; γ1\n1; γ1\n1) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n2; γ1\n1; γ1\n21) = −(q2 + q + 1) ∗(q −1)2/q2;\nc(γ1\n21; γ1\n1; γ1\n21) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n1; γ1\n1; Σ) = c(γ2\n1; γ1\n1; Σ) = c(γ1\n2; γ1\n1; Σ) = c(γ2\n2; γ1\n1; Σ) = 1/q ∗(1 + q)2;\nc(γ1\n12; γ1\n12; Σ) = c(γ1\n21; γ1\n12; Σ) = (1 + q)2 ∗(6 ∗q + 5 ∗q2 + 5)/q2;\nc(γ2\n21; γ2\n21; γ2\n21) = (1 + q2)2 ∗(q2 + q + 1) ∗(q −1)2/q4;\nc(γ2\n21; γ2\n21; Σ) = c(γ2\n12; γ2\n21; Σ) = (1 + q)2 ∗(2 ∗q2 + q + 1) ∗(q2 + q + 2)/q3.\nFigure 2: Multiplicative structural constants of the canonical basis of BH\n3 (q)\nin the Kronecker case, n = 2, r = 3\n34\n\n10\n4\n3\n44\n21\n7\n22\n17\n7\n1\n50\n30\n15\n2\n20\n12\n4\n14\n7\n3\n44\n31\n14\n5\n19\n12\n4\n1\n6\n5\n3\n1\n94\n64\n29\n4\n88\n65\n28\n7\n39\n24\n8\n1\n80\n45\n17\n2\n40\n32\n16\n4\n28\n21\n10\n3\n75\n45\n19\n5\n38\n31\n16\n5\n1\n11\n8\n4\n1\n122\n69\n23\n2\n62\n49\n23\n5\nFigure 3: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (ignoring a positive, unimodal factor)\n35\n\n126\n92\n47\n12\n2\n158\n93\n33\n4\n153\n93\n35\n7\n78\n63\n32\n9\n1\n160\n125\n62\n19\n2\n72\n48\n20\n4\n150\n120\n64\n24\n5\n69\n47\n21\n6\n1\n22\n19\n12\n5\n1\n212\n163\n74\n17\n244\n191\n92\n25\n2\n111\n72\n28\n5\n102\n71\n33\n9\n1\n104\n88\n52\n20\n4\n100\n85\n52\n22\n6\n1\n30\n23\n13\n5\n1\n128\n106\n59\n20\n3\n316\n251\n126\n37\n4\n306\n246\n128\n42\n7\n141\n95\n41\n10\n1\nFigure 4: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q), continued\n36\n\n144\n120\n68\n24\n4\n41\n31\n17\n6\n1\n344\n213\n79\n12\n375\n237\n91\n17\n162\n117\n59\n19\n3\n222\n183\n100\n33\n5\n204\n173\n104\n42\n10\n1\n192\n140\n72\n24\n4\n60\n53\n36\n18\n6\n1\n220\n191\n124\n58\n18\n3\n264\n188\n92\n28\n4\n82\n72\n48\n23\n7\n1\n280\n244\n160\n76\n24\n4\n83\n66\n41\n19\n6\n1\n750\n612\n328\n108\n17\n345\n237\n107\n28\n3\n324\n279\n176\n78\n22\n3\n113\n89\n54\n24\n7\n1\n106\n96\n71\n42\n19\n6\n1\n528\n452\n280\n120\n32\n4\nFigure 5: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (continued)\n37\n\nσ\nCoefficient\n1\n1/2 (q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(q2 + 1)4\n×(x + 3 q28 + 4 q24 −2 q22 + 10 q20 −2 q18 + 4 q16 + 3 q12)/q40\n121\n−1/2 (q2 + 1)2(2 q18 −295 q28 −516 q36 + x + 210 q26 + 3 q56 −3 q54 + 47 q46 + 9 q52 −q48\n+2 q50 −84 q24 −295 q40 + 604 q34 + 462 q30 −xq2 + 47 q22 −9 q20x + 19 q10x −q26x + q28x −3 q14\n+q24x + 4 q22x + 30 q14x + 462 q38 −516 q32 + 19 q18x + 210 q42 −q20 + 9 q16 −24 q16x −24 q12x\n−9 q8x + 4 q6x + q4x −84 q44 + 3 q12)/q36\n12121\n1/2 (q18 −2 q28 + 22 q36 + x + 45 q26 + 2 q46 + 3 q48 + 22 q24 + 24 q40 + 45 q34 + 92 q30 + 18 q22 + q20x\n+6 q10x + 2 q14 + q14x + 18 q38 −2 q32 + q42 + 24 q20 + 9 q16 + q16x + q6x + q4x + 9 q44 + 3 q12)/q30\n1212121\n−1/2 (22 q20 + 6 q16 + 6 q24 + 2 q26 + 2 q14 + 2 q30 + 2 q10 + 3 q28 −2 q22 −2 q18 + 3 q12 + x)/q20\n121212121\n1\nFigure 6: Coefficients of u1\n1\n38\n\nσ\nCoefficient\n1\n(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4/q32\n121\n−(1 −4 q10 + 14 q8 −30 q14 + 44 q28 + 73 q16 + 3 q2 + 14 q32 −30 q26 + 73 q24 + 3 q38\n+102 q20 −53 q18 + q40 + 44 q12 −53 q22 −4 q30 + 5 q4 + 5 q36)(q2 + 1)2(q4 −q2 + 1)2/q26\n12121\n(3 + 72 q18 + 14 q28 + 3 q36 + 20 q26 + 10 q24 + 2 q34 + 2 q30 + 36 q22 + 14 q8 + 7 q4\n+2 q2 + 7 q32 + 2 q6 −10 q20 −10 q16 + 10 q12 + 20 q10 + 36 q14)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + q4 + 3 q2 + 4 q6 + q20 + 4 q14 + 3 q18 + q16)/q10\n121212121\n1\nFigure 7: Coefficients of u1\n3\n39\n\nσ\nCoefficient\n1\n(q2 + 1)2(q4 −q2 + 1)2(q4 + 1)2(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2/q30\n121\n−(1 + 75 q18 −49 q28 + 42 q36 + 206 q26 −q46 + q52 + 7 q48 −49 q24 + 40 q40 + 75 q34 + 158 q30 + 158 q22\n+22 q8 + 7 q4 + 17 q14 + 17 q38 + q32 −q6 + q20 + 42 q16 + 22 q44 + 40 q12)q26\n12121\n(3 + 80 q18 + 10 q28 + 3 q36 + 26 q26 −3 q24 + q34 + 5 q30 + 52 q22 + 10 q8 + 5 q4 + 52 q14 + q2 + 5 q32\n+5 q6 −19 q20 −19 q16 −3 q12 + 26 q10)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + 3 q2 + 5 q6 + q20 + 5 q14 + 3 q18)/q10\n121212121\n1\nFigure 8: Coefficients of u1\n4\n40\n\nσ\nCoefficient\n1\n(q2 + 1)2(q4 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q4 −q2 + 1)4/q26\n121\n−(q36 + 3 q34 + 10 q32 + 19 q30 + 33 q28 + 53 q26 + 64 q24 + 91 q22 + 84 q20 + 116 q18 + 84 q16 + 91 q14\n+64 q12 + 53 q10 + 33 q8 + 19 q6 + 10 q4 + 3 q2 + 1)(q4 −q2 + 1)2/q22\n12121\n(80 q16 + 3 q26 + 26 q24 + 4 q22 + q32 + 7 q28 + 50 q20 + 3 q6 + 3 q2 + 50 q12 + 1 + 3 q30 + 7 q4 −14 q18\n−14 q14 + 4 q10 + 26 q8)/q16\n1212121\n−(3 + 5 q12 −2 q10 + 14 q8 + 5 q4 + q2 −2 q6 + q14 + 3 q16)/q8\n121212121\n1\nFigure 9: Coefficients of u1\n5\n41\n\nσ\nCoefficient\n∅\n(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n×(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2\n×(q4 + 1)2(q −1)4(q + 1)4(q2 + 1)4(q4 −q2 + 1)5/q46\n2\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n1\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n12\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n21\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30\n+2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q31\n121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52\n−77 q22 + 2 q48 −2 q46 −2 q44)/q31\n1212\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18\n+q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\n2121\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16\n−110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\nFigure 10: First nine coefficients of v0 in {Pσ} basis\n42\n\nσ\nCoefficient\n21212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n12121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n212121\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n2121212\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n1212121\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n12121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n21212121\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n212121212\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n121212121\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n1212121212\n1\n2121212121\n1\nFigure 11: Last twelve coefficients of v0 in {Pσ} basis\n43\n\nσ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(q4 + 1)2(q2 −q + 1)2(1 + q2 + q)2(q −1)4(q + 1)4(q2 + 1)2\n(q4 −q2 + 1)5)/q40\n2\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40\n−14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n1\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10\n+q40 −14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n12\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34\n−q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n21\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34 −q32\n−3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n1212\n((q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q20\nFigure 12: A −-invariant element w in qRopp(3): the first eight coefficients\n44\n\nσ\nCoefficient\n2121\n(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)/q20\n21212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n12121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n121212\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n212121\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n2121212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n1212121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n12121212\n1\n21212121\n1\nFigure 13: A −-invariant element w in qRopp(3): the last nine coefficients\n45\n\nσ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q2 + 1)3(q4 −q2 + 1)5)/q45\n2\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56\n−4 q58 + 12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n1\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56 −4 q58\n+12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n12\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8\n−q46 + q12 −2 q10 −3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50\n−3 q28 −8 q34 −10 q22 −q32 −3 q38 + q40))/q35\n21\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8 −q46 + q12 −2 q10\n−3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50 −3 q28 −8 q34 −10 q22\n−q32 −3 q38 + q40))/q35\n212\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n121\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n1212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26\n+25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\nFigure 14: A −-invariant element x in qRopp(3): the first eight coefficients\n46\n\nσ\nCoefficient\n2121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\n21212\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n12121\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))/q15\n212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))//q15\n2121212\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n1212121\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n12121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n21212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n212121212\n−((q4 + 1)(q4 −q2 + 1))/q4\n121212121\n−((q4 + 1)(q4 −q2 + 1))/q4\nFigure 15: A −-invariant element x in qRopp(3): the last eleven coefficients\n47\n\n5.4.1\nNonstandard positivity\nNow we shall describe evidence for Conjecture 3.6 in the case under consid-\neration.\nSo far we are assuming that Pi’s are as defined in the begining of Section\n7.1 in [GCT7]. But as explained towards the end of that section, the actual\nPi’s differ from these (chosen for convenience and simplicity) by a positive,\nunimodal factor ˆfp(q) ∈Q[q, q−1] given there. As it turns this does not\nmatter in the calculations so far, but for rescaling of the picture, but it does\nin the study of the positivity properties below. Rescaling of Pi by ˆfp(q)\nimplies that each structural coefficient or constant c(q) computed so far\nhas to be multiplied by an appropriate power of ˆfp(q)m(c), where m(c) is a\nnonnegative integer depending on c. In what follows, it is implicitly assumed\nthat each c computed so far has been rescaled by a suitable power ˆfp(q)n(c),\nwhere n(c) ≤m(c) is the smallest nonnegative integer chosen so that the\n(nonstandard) positivity property of the coefficients of c becomes apparent\nThe picture remains the same even if we were to multiply by ˆfp(q)m(c), but\nwe choose n(c) as small as possible to keep the the degrees of the polynomials\nfrom blowing up.\nFigures 16-20 show the vectors associated with the nonzero coefficients\nof v0 in the {Pσ} basis. The vector (as defined in the begining of Section 5)\nfor the coefficient corresponding to each σ is obtained by concatenating the\nrows in front of that σ. Figure 21 shows the vectors associated with the\ntraces of the nonzero coefficients of v1\n1 and Figures 22-23 show their norms.\nHere the trace and norm of an element in the algerbraic extension K of\nQ(q) is defined in the usual fashion as the sum and product of its images\nunder the Frobenius automorphisms of K over Q(q); they are coefficients\nof the minimal polynomial of the element. It can be seen that all vectors\nin Figures 16-23 are positive and nonincreasing, except for the vectors as-\nsociated with v0 for σ = 212, 121, 1212, 2121, which are almost positive and\nnonincreasing. It was verified with the help of computer that the vectors\nassociated with the coefficients of other canonical basis elements are simi-\nlarly either positive and nonincreasing or almost positive and nonincreasing.\nThis is in accordance with Conjectures 3.5-3.6.\nFigures 24-35 show vectors associated with a few multiplicative con-\nstants, taking norms and traces whenever necessary; the coefficient of bb′\nwith respect to the basis element b′′ is denoted by c(b, b′; b′′). Again it can\nbe seen that these vectors are positive and nonincreasing, except a few,\nwhich are almost positive and nonincreasing. It was verified with the help\n48\n\nof computer that the picture is the same for other multiplicative constants\nas well. This too is in accordance with Conjectures 3.5-3.6.\n49\n\nσ\nvector\n∅\n34116640\n34028832\n33766665\n33333910\n32736719\n31983492\n31084702\n30052720\n28901524\n27646408\n26303647\n24890162\n23423208\n21920062\n20397697\n18872456\n17359791\n15874002\n14428069\n13033484\n11700135\n10436190\n9248100\n8140602\n7116788\n6178174\n5324820\n4555450\n3867635\n3257976\n2722277\n2255718\n1853025\n1508640\n1216881\n972102\n768790\n601662\n465735\n356396\n269443\n201126\n148131\n107564\n76935\n54142\n37439\n25404\n16892\n10988\n6976\n4308\n2576\n1484\n821\n434\n217\n100\n40\n12\n2\n2\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\nFigure 16: The vectors associated with the coefficients of v0\n50\n\nσ\nvector\n1\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\n12\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\nFigure 17: The vectors associated with the coefficients of v0 (continued)\n51\n\nσ\nvector\n21\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\n212\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n121\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n1212\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\n2121\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\nFigure 18: The vectors associated with the coefficients of v0 (continued)\n52\n\nσ\nvector\n21212\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n12121\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n121212\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n212121\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n2121212\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\nFigure 19: The vectors associated with the coefficients of v0 (continued)\n53\n\nσ\nvector\n1212121\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\n12121212\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n21212121\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n212121212\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\n121212121\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\nFigure 20: The vectors associated with the coefficients of v0 (continued)\n54\n\nσ\nvector\n1\n15864\n15864\n15680\n15496\n15160\n14824\n14334\n13844\n13228\n12612\n11932\n11252\n10505\n9758\n9008\n8258\n7534\n6810\n6126\n5442\n4834\n4226\n3702\n3178\n2738\n2298\n1948\n1598\n1333\n1068\n868\n668\n534\n400\n310\n220\n165\n110\n80\n50\n34\n18\n12\n6\n3\n121\n14690\n14690\n14484\n14278\n13930\n13582\n13065\n12548\n11880\n11212\n10509\n9806\n9032\n8258\n7498\n6738\n6031\n5324\n4684\n4044\n3498\n2952\n2515\n2078\n1729\n1380\n1124\n868\n698\n528\n405\n282\n212\n142\n101\n60\n42\n24\n15\n6\n3\n12121\n4974\n4974\n4886\n4798\n4683\n4568\n4380\n4192\n3914\n3636\n3367\n3098\n2803\n2508\n2217\n1926\n1669\n1412\n1205\n998\n825\n652\n529\n406\n316\n226\n172\n118\n89\n60\n41\n22\n14\n6\n3\n1212121\n708\n708\n694\n680\n673\n666\n642\n618\n565\n512\n464\n416\n365\n314\n262\n210\n170\n130\n106\n82\n62\n42\n30\n18\n11\n4\n2\nFigure 21: The vectors associated with the traces of the coefficients of v1\n1\n55\n\nσ\nvector\n1\n942062408\n940617136\n936306916\n929157344\n919242745\n906637444\n891460752\n873831980\n853909410\n831851324\n807845688\n782080468\n754764061\n726104864\n696321516\n665632656\n634256120\n602409744\n570301452\n538139168\n506112609\n474411492\n443199968\n412642188\n382872865\n354026712\n326205988\n299512952\n274016717\n249786396\n226859750\n205274540\n185039618\n166163836\n148631230\n132425836\n117511844\n103853444\n91399832\n80100204\n69893842\n60720028\n52513000\n45206996\n38734986\n33029940\n28027078\n23661620\n19873490\n16602612\n13795174\n11397364\n9362691\n7644664\n6204292\n5002584\n4007841\n3188364\n2519160\n1975236\n1537471\n1186744\n908860\n689624\n518854\n386368\n285084\n207920\n150131\n106972\n75388\n52324\n35881\n24160\n16048\n10432\n6681\n4164\n2550\n1508\n874\n484\n262\n132\n64\n28\n12\n4\n1\n121\n835471628\n833946584\n829404381\n821877948\n811459617\n798241720\n782371271\n763995284\n743308085\n720504000\n695809491\n669451020\n641674579\n612726160\n582858876\n552325840\n521370802\n490237512\n459149620\n428330776\n397976239\n368281268\n339402663\n311497224\n284682092\n259074408\n234750212\n211785544\n190216267\n170078244\n151372598\n134100452\n118233841\n103744800\n90582172\n78694800\n68016456\n58480912\n50013272\n42538640\n35978861\n30255780\n25293821\n21017408\n17356415\n14240716\n11608322\n9397244\n7554889\n6028664\n4775532\n3752456\n2925473\n2260620\n1732150\n1314316\n988221\n734968\n541214\n393616\n283165\n200852\n140775\n97032\n65989\n44012\n28926\n18556\n11701\n7160\n4299\n2484\n1405\n752\n391\n188\n88\n36\n14\n4\n1\nFigure 22: The vectors associated with the norms of the coefficients of v1\n1\n56\n\n12121\n100225178\n100001236\n99336179\n98236776\n96719406\n94800448\n92504960\n89858000\n86893481\n83645316\n80151751\n76451032\n72582611\n68585940\n64501436\n60369516\n56227388\n52112260\n48055958\n44090308\n40241933\n36537456\n32995876\n29636192\n26469934\n23508632\n20757113\n18220204\n15896382\n13784124\n11876674\n10167276\n8645972\n7302804\n6125315\n5101048\n4216788\n3459320\n2815838\n2273536\n1820593\n1445188\n1137127\n886216\n684162\n522672\n395267\n295468\n218449\n159384\n114887\n81572\n57144\n39308\n26622\n17644\n11492\n7284\n4521\n2704\n1573\n868\n461\n224\n102\n40\n15\n4\n1\n1212121\n2212002\n2205476\n2186350\n2155076\n2112378\n2058980\n1995675\n1923256\n1842975\n1756084\n1663849\n1567536\n1468102\n1366504\n1263855\n1161268\n1059834\n960644\n864459\n772040\n684050\n601152\n523831\n452572\n387494\n328716\n276117\n229576\n188839\n153652\n123592\n98236\n77108\n59732\n45640\n34364\n25488\n18596\n13346\n9396\n6493\n4384\n2894\n1848\n1142\n672\n377\n196\n95\n40\n15\n4\n1\nFigure 23: The vectors associated with the norms of the coefficients of v1\n1\n(continued)\n57\n\n9846\n9820\n9750\n9644\n9501\n9320\n9093\n8812\n8493\n8152\n7779\n7364\n6917\n6448\n5966\n5480\n4989\n4492\n4001\n3528\n3080\n2664\n2274\n1904\n1569\n1284\n1038\n820\n631\n472\n346\n256\n186\n120\n69\n44\n31\n16\n4\nFigure 24: The vector for the multiplicative constant c(v0, v1\n5; v0)\n13026\n12964\n12777\n12464\n12048\n11552\n10964\n10272\n9523\n8764\n7990\n7196\n6398\n5612\n4867\n4192\n3569\n2980\n2444\n1980\n1583\n1248\n967\n732\n539\n384\n267\n188\n131\n80\n42\n24\n16\n8\n2\nFigure 25: The vector for the multiplicative constant c(v0, v1\n4; v0)\n14180\n14088\n13832\n13432\n12894\n12224\n11448\n10592\n9682\n8744\n7800\n6872\n5976\n5128\n4342\n3632\n2998\n2440\n1956\n1544\n1202\n928\n706\n520\n374\n272\n198\n136\n88\n56\n36\n24\n16\n8\n2\nFigure 26: The vector for the multiplicative constant c(v0, v1\n3; v0)\n58\n\n1356922\n1356922\n1341857\n1326792\n1297628\n1268464\n1227083\n1185702\n1133960\n1082218\n1023821\n965424\n902616\n839808\n776306\n712804\n650849\n588894\n531320\n473746\n421861\n369976\n325182\n280388\n243022\n205656\n175677\n145698\n122582\n99466\n82260\n65054\n52971\n40888\n32575\n24262\n19002\n13742\n10480\n7218\n5402\n3586\n2573\n1560\n1107\n654\n431\n208\n135\n62\n35\n8\n4\nFigure 27: The vector for the trace of the multiplicative constant c(v0, v1\n1; v0)\n59\n\n128607887512140\n128408739182416\n127813071267725\n126826189536408\n125456853005552\n123717149426056\n121622324577366\n119190568251368\n116442761279740\n113402187959064\n110094219510476\n106545974210632\n102785960441146\n98843708903656\n94749400393129\n90533495522016\n86226372383558\n81857978142544\n77457499613617\n73053057887224\n68671430858636\n64337807515464\n60075576278933\n55906149694176\n51848826289840\n47920690427296\n44136549385454\n40508906927184\n37047971392480\n33761696363504\n30655850888930\n27734116255008\n24998205742188\n22448003806144\n20081720818165\n17896059499880\n15886389332178\n14046925218184\n12370907081114\n10850777077832\n9478351722036\n8244986211000\n7141729026165\n6159464877872\n5289044837698\n4521402501856\n3847655805468\n3259194107520\n2747750779948\n2305461534304\n1924909254250\nFigure 28:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0)\n60\n\n1599156102128\n1321763961652\n1086804278768\n888858554957\n723010747256\n584832826172\n470364742664\n376089969502\n298907782312\n236103258694\n185315973800\n144508153155\n111933043504\n86104032184\n65765045520\n49862540548\n37519404368\n28010898646\n20742786784\n15231640340\n11087321280\n7997549136\n5714462144\n4043023409\n2831123144\n1961213430\n1343311944\n909211769\n607734400\n400883734\n260758832\n167107927\n105406152\n65367982\n39805384\n23766532\n13889944\n7930388\n4412904\n2386644\n1250232\n631716\n306184\n141372\n61592\n25006\n9272\n3049\n848\n190\n32\n4\nFigure 29:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0) (continued)\n61\n\n612764\n609940\n601539\n587774\n568990\n545654\n518332\n487666\n454355\n419136\n382753\n345926\n309336\n273610\n239300\n206862\n176658\n148958\n123938\n101678\n82176\n65362\n51106\n39226\n29503\n21696\n15554\n10828\n7279\n4686\n2851\n1604\n800\n316\n52\n−68\n−101\n−90\n−64\n−38\n−18\n−6\n−1\nFigure 30: The vector for the multiplicative constant c(v1\n5, v1\n5; v1\n5)\n9738\n9694\n9563\n9348\n9052\n8676\n8228\n7732\n7209\n6658\n6077\n5484\n4900\n4332\n3784\n3268\n2794\n2360\n1962\n1604\n1292\n1028\n809\n626\n471\n344\n246\n172\n116\n76\n50\n32\n17\n6\n1\nFigure 31: The vector for the multiplicative constant c(v1\n4, v1\n4; v1\n4)\n5.5\nExperimental evidence for crystalization\nLet (Lρ\nα, Bρ\nα) be an upper crystal basis of W ρ\nq,α as in Conjecture 2.2. Since\nit is also a local crystal basis of W ρ\nq,α as an Hq-module, there is a crystal\ngraph over Bρ\nα whose connected components correspond to irreducible Hq-\nsubmodules of W ρ\nq,α. The elements of Bρ\nα that correspond to the highest\nweight nodes of these connected components are called the highest weight\ncrystal elements of the upper crystal base (Lρ\nα, Bρ\nα) with respect to Hq.\n5.6\nKronecker problem: n = 2, r = 3\nConsider again the special case of the Kronecker problem for n = 2 and\nr = 3 as in Section 5.1. Thus H = Gl2 × Gl2, G = Gl4 with H embedded\ndiagonally, X = Xq = Vq ⊗Wq is the standard four dimensional represen-\ntation of Hq, where Vq ∼= Wq is the standard two representation of GLq(2).\n62\n\n81298\n80886\n79660\n77650\n74908\n71510\n67546\n63110\n58306\n53254\n48074\n42870\n37740\n32786\n28097\n23732\n19734\n16144\n12987\n10258\n7938\n6010\n4449\n3212\n2251\n1526\n999\n628\n374\n208\n107\n50\n20\n6\n1\nFigure 32: The vector for the multiplicative constant c(v1\n3, v1\n3; v1\n3)\n976672\n974152\n971632\n954184\n936736\n915476\n894216\n860888\n827560\n791463\n755366\n713035\n670704\n627743\n584782\n540877\n496972\n454258\n411544\n373144\n334744\n298301\n261858\n232239\n202620\n175584\n148548\n128938\n109328\n91580\n73832\n62540\n51248\n41348\n31448\n25972\n20496\n15650\n10804\n8712\n6620\n4729\n2838\n2215\n1592\n986\n380\n312\n244\n113\n−18\n−9\n0\n−15\n−30\n−15\nFigure 33: The vector for the trace of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n63\n\n80255448992640\n80138274925167\n79787740098774\n79206792680861\n78400301248652\n77374989989221\n76139349486502\n74703523453335\n73079174468928\n71279332267508\n69318226392392\n67211105097154\n64974044241996\n62623750255631\n60177360010954\n57652240221271\n55065789702876\n52435247936361\n49777512793214\n47108969815061\n44445335040340\n41801513256271\n39191473712706\n36628144773627\n34123327701708\n31687629536553\n29330415837966\n27059783595895\n24882552858520\n22804275396566\n20829260005316\n18960613784310\n17200296928344\n15549188413159\n14007161465134\n12573167695619\n11245327257264\n10021022340390\n8896992772108\n7869432671924\n6934086054972\n6086339325247\n5321309766266\n4633929430149\n4019023248184\n3471380269320\n2985817677720\n2557237529092\n2180675978000\n1851344837246\n1564665615708\nFigure 34: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n64\n\n1316296386448\n1102151942468\n918417720699\n761557915658\n628318302423\n515724544156\n421076738262\n341940657408\n276136144552\n221723424848\n176988052223\n140424814566\n110720869883\n86738676456\n67499233418\n52165764220\n40027907728\n30486730148\n23040841355\n17273591498\n12841266667\n9462395940\n6908272529\n4994579014\n3573971135\n2529616304\n1769689525\n1222700770\n833511103\n559989972\n370279871\n240567010\n153252303\n95475892\n57962133\n34120478\n19338377\n10435196\n5256321\n2374094\n864025\n140844\n−155240\n−236220\n−220838\n−171968\n−120043\n−77166\n−46045\n−25460\n−12974\n−6040\n−2518\n−900\n−255\n−50\n−5\nFigure 35: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n(continued)\n65\n\nLet\nx1 = v1 ⊗w1, x2 = v1 ⊗w2, x3 = v2 ⊗w1, x4 = v2 ⊗w2,\nbe the standard basis of Xq. Let b1, . . . , b4 be the corresponding standard\ncrystal basis of L(Xq). The irreducible representations of GH\nq that occur in\nX⊗3\nq\nare:\n1. CH,3\nq\n(X), the 16-dimensional the degree-three component of the braided\nsymmetric algebra of GH\nq ,\n2. ∧H,3\nq\n(X), the four dimensional degree three component of the braided\nexterior algebra of GH\nq ,\n3. two copies of the 16-dimensional GH\nq -representation Wq,(2,1),1(X) de-\nfined in [GCT4] (it is denoted by Vq,(2,1),1(X) there), and\n4. two copies of the 4-dimensional representation Wq,(2,1),2(X) of GH\nq as\nalso defined there (it is called Vq,(2,1),2(X) there).\nEmbeddings of the braided symmetric and exterior algebra components in\nX⊗3\nq\nare uniquely defined. We denote their embedded images by CH,3\nq\n(X)\nand ∧H,3\nq\n(X) again.\nWe choose appropriate embeddings of Wq,(2,1),2(X)\nand Wq,(2,1),2(X) in X⊗3\nq\nand denote them by the same symbols again. As\nHq = GLq(V ) × GLq(W)-modules,\nCH,3\nq\n(X)\n=\nVq,(3)(V ) ⊗Vq,(3)(W) ⊕Wq,(2,1)(V ) ⊗Vq,(2,1)(W),\n∧H,3\nq\n(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W),\nWq,(2,1),1(X)\n=\nVq,(2,1)(V ) ⊗Vq,(3)(W) ⊕Vq,(3)(V ) ⊗Vq,(2,1)(W)\nWq,(2,1),2(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W).\nIt was verified by computer that they have upper crystal bases as per\nConjecture 2.2.\nThe highest weight crystal elements with respect to Hq\nfor these upper crystal bases, as shown separately for each module, are as\nfollows; we denote the monomial basis element bi1 ⊗bi2 ⊗bi3 of B(X⊗3\nq ) by\nbi1i2i3.\n66\n\nCH,3\nq\n(X) :\n{b114 + b141, b111}.\n∧H,3\nq\n(X) : {b123 + b132}.\nWq,(2,1),1(X) :\n{b121, b131}.\nWq,(2,1),2(X) :\n{b114 −b141}.\nThe highest weight crystal elements whose monomial support have size\ntwo correspond to the four dimensional Hq-module Vq,(2,1)(V ) ⊗Vq,(2,1)(W).\nThe element b111 corresponds to the Hq-module Vq,(3)(V ) ⊗Vq,(3)(W), the\nelement b121 to the Hq-module Vq,(3)(V ) ⊗Vq,(2,1)(W), and b131 to the Hq-\nmodule Vq,(2,1)(V ) ⊗Vq,(3)(W). Notice that not all highest weight crystal\nelements have monomial supports of size one as in the standard setting.\n5.6.1\nH = sl2, G = sl4\nNow we consider the case when Hq = Glq(2), Xq its four dimensional irre-\nducible representation, and GH\nq\nas in Section 5.3. Let W0, . . . , W5 be the\nirreducible representations of GH\nq occuring in X⊗3\nq\nas defined in Section 6.1.2\nof [GCT7], with W0 = CH,3\nq\n[X].\nAs Hq-modules,\nW0\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2),\nW1\n∼=\nVq,(6,3)(2),\nW2\n∼=\nVq,(6,3)(2),\nW3\n∼=\nVq,(8,1)(2),\nW4\n∼=\nVq,(5,4)(2),\nW5\n∼=\nVq,(7,2)(2),\n(19)\nwhere Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to the\npartition λ. Their dimensions are 16, 4, 4, 8, 2 and 6, respectively. Though\nW1 and W2 are isomorphic as Hq-modules, they are nonisomorphic as GH\nq -\nmodules.\nIt was verified by computer that they–or rather their embeddings in\nX⊗3\nq –have upper crystal bases as per Conjecture 2.2. The highest weight\ncrystal elements of of the embedding of W1, . . . , W5 have monomial supports\nof size one. Let bi1i2i3 = bi1 ⊗bi2 ⊗bi3 denote the monomial basis elements of\nB(X⊗3\nq ). Then the highest weight crystal elements of the uniquely defined\nembedding of W0 are b111 and b = b113 + b131. The latter element b here\nhas monomial support of size two, a phenomenon not seen in the standard\nsetting.\nNonzero coefficients of the element x in the lattice L(C3,H\nq\n(X))\n67\n\nwhose crystalization is b is shown in Figure 36, wherein x1, . . . , x4 are the\nstandard basis vectors of Xq.\n6\nComplexity theoretic properties of the canonical\nbasis\nIn the standard setting, elements of the canonical basis of Vq,λ are indexed\n(labelled) by semistandard tableau, for H = GL(V ), and by LS-paths [Li],\nfor general semisimple H. And combinatorial analogues of Kashiwara’s crys-\ntal operators [Li] on these labels can be computed efficiently [GCT6]. This\nis enough to imply a #P-formula for the generalized Littlewood-Richardson\ncoefficient though the canonical basis of Vq,λ is hard to compute.\nIn the same spirit, it may be conjectured that the canonical basis of\nO(MH\nq (X) (or rather the set of its labels) has additional complexity the-\noretic properties (to be described in the full version), based on its cellular\nand refined sub-cellular decomposition (Conjecture 2.6), that imply a pos-\nitive #P-formula for the multiplicity nα\nπ of the irreducible Hq-module Vq,π\nin Wq,α. This would solve the problem P1 in [GCT7].\nSimilarly, let mα\nλ denote the multiplicity of the Specht module Sλ of\nthe symmetric group Sr corresponding to the partition λ in Limq→1Tq,α,\nconsidered as an Sr-module. It may be conjectured that the canonical basis\nBH\nr (q) (or rather the set of its labels) has similar additional complexity\ntheoretic properties (to be described in the full version) based on its cellular\nand quasi-subcellular decompositions (Conjecture 3.3). This would imply a\npositive #P-formula for the multiplicity mπ\nλ, as needed in the problem P2\nin [GCT7].\nReferences\n[BBD]\nA.\nBeilinson,\nJ.\nBernstein,\nP.\nDeligne,\nFaisceaux\npervers,\nAst ́erisque 100, (1982), Soc. Math. France.\n[BZ]\nA. Berenstein, S. Zwicknagl, Braided symmetric and exterior alge-\nbras, arXiv:math/0504155v3, April, 2007.\n[DJM]\nM. Date, M. Jimbo, T. Miwa, Representations of Uq(ˆgl(n, C)) at\nq = 0 and the Robinson-Schensted correspondence, in Physics and\n68\n\nMonomial\nCoefficient\nx1 ⊗x1 ⊗x3\n−(q4 + 1)2(q2 + 1)4(q4 −q2 + 1)5(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q20 −2 q18 + q16 + q10 −q8 + q6 + q4 −q2 + 1)\n(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\nx1 ⊗x2 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q37\nx1 ⊗x3 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q16 + q14 −q12 + q10 + q4 −q2 + 1)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q46\nx2 ⊗x1 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q40\nx2 ⊗x2 ⊗x1\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q4 −q3 + q2 −q + 1)\n(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5(q4 −q2 + 1)5/q43\nx3 ⊗x1 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5\n+q4 + q3 + q2 + q + 1)(q12 −q10 + q8 + q6 + q4 −q2 + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q42\nFigure 36: Nonzero coefficients of x in the lattice L(C3,H\nq\n(X))\n69\n\nMathematics of Strings, World Scientific, Singapore, 1990, pp. 185-\n211.\n[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n[Dri]\nV. Drinfeld, Quantum groups, Proc. Int. Congr. Math. Berkeley,\n1986, vol. 1, Amer. Math. Soc. 1988, 798-820.\n[GCTflip1] K. Mulmuley, On P. vs. NP, geometric complexity theory, and\nthe flip I: a high-level view, Technical Report TR-2007-13, Com-\nputer Science Department, The University of Chicago, September\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu\n[GCT4] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\nthe-\nory\nIV:\nquantum\ngroup\nfor\nthe\nKronecker\nproblem,\ncs.\nArXiv preprint cs. CC/0703110,\nMarch,\n2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT6] K. Mulmuley, Geometric complexity theory VI: the flip via sat-\nurated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\nSci. Dept., The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: Nonstandard\nquantum group for the plethysm problem, technical report TR-\n2007-14, computer science dept., The university of Chicago, Sept.\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[Ji]\nM. Jimbo, A q-difference analogue of U(G) and the Yang-Baxter\nequation, Lett. Math. Phys. 10 (1985), 63-69.\n[Kas1]\nM. Kashiwara, Crystalizing the q-analogue of universal enveloping\nalgebras, Comm. Math. Phys. 133 (1990), 249-260.\n[Kas1]\nM. Kashiwara, On crystal bases of the q-analogue of universal en-\nveloping algebras, Duke Math. J. 63 (1991), 465-516.\n[Kas2]\nM. Kashiwara, Global crystal bases of quantum groups, Duke\nMathematical Journal, vol. 69, no.2, 455-485.\n[KL1]\nD. Kazhdan, G. Lusztig, Representations of Coxeter groups and\nHecke algebras, Invent. Math. 53 (1979), 165-184.\n70\n\n[KL2]\nD. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality,\nProc. Symp. Pure Math., AMS, 36 (1980), 185-203.\n[Kli]\nA. Klimyk, K. Schm ̈udgen, Quantum groups and their representa-\ntions, Springer, 1997.\n[Li]\nP. Littelmann, Paths and root operators in representation theory,\nAnn. of Math. 142 (1995), 499-525.\n[Lu1]\nG. Lusztig, Canonical bases arising from quantized enveloping al-\ngebras, J. Amer. Math. Soc. 3, (1990), 447-498.\n[Lu2]\nG. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[So]\nW. Soergel, Kazhdan-Lusztig polynomials and a combinatoric for\ntilting modules, Representation theory 1, (1997)\n[St]\nR. Stanley, Positivity problems and conjectures in algebraic com-\nbinatorics, In Mathematics: frontiers and perspectives, 295-319,\nAmer. Math. Soc. Providence, RI (2000).\n[W]\nS. Woronowicz: Compact matrix pseudogroups, Commun. Math.\nPhys. 111 (1987), 613-665.\n71","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.0751v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VIII: On canonical\nbases for the nonstandard quantum groups\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\n(Technical Report TR-2007-15\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nOctober 27, 2018\nAbstract\nThis article gives conjecturally correct algorithms to construct canon-\nical bases of the irreducible polynomial representations and the matrix\ncoordinate rings of the nonstandard quantum groups in GCT4 and\nGCT7, and canonical bases of the dually paired nonstandard deforma-\ntions of the symmetric group algebra therein. These are generalizations\nof the canonical bases of the irreducible polynomial representations\nand the matrix coordinate ring of the standard quantum group, as\nconstructed by Kashiwara and Lusztig, and the Kazhdan-Lusztig ba-\nsis of the Hecke algebra. A positive (#P-) formula for the well-known\nplethysm constants follows from their conjectural properties and the\nduality and reciprocity conjectures in [GCT7].\n1"},{"paragraph_id":"p2","order":2,"text":"1\nIntroduction\nLet H be a complex connected classical reductive group, X = Vμ(H) its\nirreducible polynomial representation with highest weight μ, G = GL(X),\nand ρ : H →G the representation map. Given a highest weight π of H\nand λ of G, the plethysm constant aπ\nλ,μ is defined to be the multiplicity of\nVπ(H) in Vλ(G), considered an H-module via ρ. A fundamental problem\nin representation theory is to find a positive (#P-) formula (rule) for the\nplethysm constant [GCT7, St] akin to the Littlewood-Richardson rule. Mo-\ntivated by this problem, the article [GCT7] constructs a quantization ρq of\nthe homomorphism ρ in the form\nρq : Hq →GH\nq ,\n(1)\nwhere Hq is the standard (Drinfeld-Jimbo) quantum group [Dri, Ji, RTF]\nassociated with H and GH\nq is the new (possibly singular) quantum group,\ncalled the nonstandard quantum group associated with ρ. In the standard\ncase, i.e., when H = G, this specializes to the standard quantum group, and\nin the Kronecker case, i.e., when H = GL(V ) × GL(W), X = V ⊗W with\nthe natural H action, this specializes to the nonstandard quantum group in\n[GCT4]. Also constructed in [GCT7] is a nonstandard quantization BH\nr (q)\nof the group algebra C[Sr], Sr the symmetric group, whose relationship with\nGH\nq is conjecturally similar to that of the Hecke algebra with the standard\nquantum group.\nThis article gives conjecturally correct algorithms for constructing canon-\nical bases of the irreducible polynomial representations and the matrix co-\nordinate ring of GH\nq (Section 2) and a canonical basis of BH\nr (q) (Section 3).\nWe call these nonstandard canonical bases. They are generalizations of the\ncanonical bases of the irreducible polynomial representations and the matrix\ncoordinate ring of the standard quantum group, as constructed by Kashi-\nwara and Lusztig [Kas1, Kas2, Lu1, Lu2], and the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. A positive (#P-) formula for the plethysm con-\nstant follows from their conjectural properties (Sections 2.4 and 3.4), which\nare akin to those of the standard canonical basis, and the conjectural duality\nand reciprocity between GH\nq and BH\nr (q); cf. [GCT7].\nExperimental evidence (Section 5) suggests that these algorithms should\nbe correct. But we can not prove this formally, nor the required properties of\nthe nonstandard canonical bases. Mainly because we are unable to deal with\nthe complexity of the minors of the nonstandard quantum group. Specifi-\ncally, in contrast to the elementary formula for the Laplace expansion of a\n2"},{"paragraph_id":"p3","order":3,"text":"minor of the standard quantum group–which is akin to the classical Laplace\nexpansion at q = 1–the Laplace expansion of a minor of a nonstandard\nquantum group is highly nonelementary; cf. [GCT7]. Its coefficients de-\npend on the multiplicative structural constants of a canonical basis akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nas per Kashiwara and Lusztig. In the Kronecker case, these constants are\nconjecturally polynomials in Q[q, q−1] with nonnegative coefficients, and in\ngeneral, polynomials with a conjectural relaxed form of this property. To\nprove this and to get explicit formulae for the minors in the nonstandard\nsetting, one needs explicit interpretations for these structural constants in\nthe spirit of the interpretation based on perverse sheaves for the Kazhdan-\nLusztig polynomials [KL2] and the multiplicative structural constants of the\ncanonical basis of the Drinfeld-Jimbo enveloping algebra [Lu2]. Thus even\nto get explicit formulae for the minors of the nonstandard quantum group\na (nonstandard) extension of the theory of perverse sheaves [BBD], and the\nunderlying Riemann hypothesis over finite fields [Dl2] seems necessary.\nMinors of the standard quantum group are in a sense the simplest (basic)\ncanonical basis elements in its matrix coordinate ring. That the simplest\ncanonical basis elements for the nonstandard quantum group–namely, its\nminors–are already so nonelementary in contrast to the standard case indi-\ncates the possible difficulties that may be encountered in proving correctness\nof the algorithms given here for constructing nonstandard canonical bases.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for the help in explicit\ncomputations in MATLAB.\nNotation: We use the symbols π and μ to denote labels of irreducible\nrepresentations of the standard quantum group and the symbols α, α0, α1, . . .\nto denote labels of irreducible representations of the nonstandard quantum\ngroup. Thus objects with subscripts π and μ are standard and the objects\nwith subscripts α, α0, . . . are nonstandard.\n2\nNonstandard canonical basis for GH\nq\nIn this section we describe a conjecturally correct algorithm for constructing\nthe canonical basis of the matrix coordinate ring of the nonstandard quan-\ntum group GH\nq . We follow the same terminology as in [Kli] for the basic\nquantum group notions.\nFor the sake of simplicity, let us assume that H = GL(V ). Let Hq =\n3"},{"paragraph_id":"p4","order":4,"text":"GLq(V ) denote its standard (Drinfeld-Jimbo) quantization [Dri, Ji, RTF],\nand Mq(V ) the standard quantization of the matrix space M(V ).\nLet\nO(Mq(V )) be the coordinate ring of Mq(V ). We call it the matrix coordinate\nring of GLq(V ). The coordinate ring O(GLq(V )) of GLq(V ) is obtained by\nlocalizing O(Mq(V )) at the quantum determinant of GLq(V ). Let H denote\nthe Lie algebra of H, and Uq(H) the Drinfeld-Jimbo universal enveloping\nalgebra of Hq = GLq(V ).\nTo quantize the homomorphism ρ : H →G = GL(X) as in (1), the arti-\ncle [GCT7] constructs a nonstandard matrix coordinate ring O(MH\nq (X)) of\na (virtual) nonstandard matrix space MH\nq (X), and then defines the nonstan-\ndard quantized universal enveloping algebra U H\nq (G) by dualization. The non-\nstandard quantum group GH\nq is the virtual object whose universal enveloping\nalgebra is U H\nq (G). The construction also yields natural bialgebra homomor-\nphisms from Uq(H) to U H\nq (G) and from O(MH\nq (X)) to O(Mq(V ), thereby\ngiving the desired quantizations of the homomorphisms U(H) →U(G)\nand O(M(X)) →O(M(V )). This is what is meant by the quantization\n(1) of the representation map ρ.\nThe determinant of GH\nq\nmay vanish,\nand hence, we cannot, in general, define its coordinate ring O(GH\nq ) by lo-\ncalizing O(MH\nq (X)). Fortunately, this does not matter since O(MH\nq (X))\nstill has properties similar to that of the standard matrix coordinate ring\nO(Mq(V )). Specifically, it is cosemisimple. This means all (finite dimen-\nsional) polynomial representations of GH\nq , by which we mean corepresen-\ntations of O(MH\nq (V )), are completely reducible. A nonstandard quantum\nanalogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\n(2)\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)). Fur-\nthermore, the nonstandard enveloping algebra U H\nq (G) is a bialgebra with a\ncompact real form (∗-structure).\nThe goal is to construct a canonical basis for the matrix coordinate ring\nO(MH\nq (X)) akin to the canonical basis of the standard matrix coordinate\nring O(Mq(V )) as per Kashiwara and Lusztig [Kas1, Kas2, Lu1, Lu2].\n2.1\nThe standard setting\nWe begin by reviewing the basic scheme of Kashiwara and Lusztig for con-\nstructing a canonical basis of the matrix coordinate ring O(Mq(V )). The\n4"},{"paragraph_id":"p5","order":5,"text":"canonical basis of the coordinate ring O(GLq(V )) is obtained by localizing\nat the determinant.\nFollowing Kashiwara, we first define a balanced triple. Let A and ̄A be\nthe ring of rational functions in q regular at q = 0 and q = ∞, respectively.\nLet V be a Q(q)-vector space, L0 a sub-A-module (A-lattice) of V , L∞a\nsub- ̄A-module ( ̄A-lattice) of V , and VQ a sub-Q[q, q−1]-module of V such\nthat\nV ∼= Q(q) ⊗Q[q,q−1] VQ ∼= Q(q) ⊗A L0 ∼= Q(q) ⊗ ̄\nA L∞.\nWe say that (VQ, L0, L∞) is a balanced triple if any of the following three\nequivalent conditions hold:\n(a) E = VQ ∩L0 ∩L∞→L0/qL0 is an isomorphism.\n(b) E →L∞/q−1L∞is an isomorphism.\n(c) Q[q, q−1] ⊗Q E →VQ, A ⊗Q E →L0, ̄A ⊗Q E →L∞are isomorphisms.\nLet R = O(Mq(V )).\nKashiwara constructs an A-submodule (lattice)\nL = L(R) ⊂R, an involution −of R, a Q[q, q−1]-submodule RQ ⊂R, and\na basis B of L/qL such that (RQ, L, ̄L) is a balanced triple and, letting G\ndenote the inverse of the isomorphism RQ ∩L ∩ ̄L →L/qL, {G(b) | b ∈B}\nis the canonical basis of R. The pair (L, B), called the (upper) crystal base\nof R, has the following form. By the q-analogue of the Peter-Weyl theorem\nfor the standard quantum group,\nR = ⊕πV ∗\nq,π ⊗Vq,π\n(3)\nas a bi-GLq(V )-module, where Vq,π = Vq,π(V ) is the irreducible polynomial\nrepresentation of GLq(V ) with highest weight π. Let (Lπ, Bπ) denote the\nupper crystal base of Vq,π. Then\n(L, B) = ⊕π(L∗\nπ, B∗\nπ) ⊗(Lπ, Bπ)\n(4)\nwith appropriate normalization.\nWe now describe a construction of the upper crystal base (Lπ, Bπ) that\ncan be generalized to the nonstandard setting.\nLet r be the size of the partition π. Choose any embedding ρ : Vq,π →\nV ⊗r such that the highest weight vector of the image V ρ\nq,π = ρ(Vq,π) be-\nlongs to the A-lattice L(V )⊗r of V ⊗r, where L(V ) denotes the lattice of\nV generated by its standard basis {vi}. We also assume that the highest\nweight vector does not belong to qL(V )⊗r. Choose a Hermitian form on\nV ⊗r so that its monomial basis {vi1 ⊗· · · vir} is orthonormal. Let V ρ,⊥\nq,π de-\nnote the orthogonal complement of V ρ\nq,π. Since GLq(V ) has a compact real\n5"},{"paragraph_id":"p6","order":6,"text":"form Uq(V )–i.e., the unitary compact subgroup in the sense of Woronowicz\n[W]–it follows that V ρ,⊥\nq,π is a GLq(V )-module. Thus V ⊗r = V ρ\nq,π ⊕V ρ,⊥\nq,π as a\nGLq(V )-module. Let\nLρ\nπ = L(V )⊗r ∩V ρ\nq,π\nand\nLρ,⊥\nπ\n= L(V )⊗r ∩V ρ,⊥\nq,π .\nIt follows from Kashiwara’s work [Kas1] that L(V )⊗r = Lρ\nπ ⊕Lρ,⊥\nπ\n.\nLet B(V ) = {bi = ψ(vi)} denote the basis of L(V )/qL(V ), where ψ :\nL(V ) →L(V )/qL(V ) is the natural projection. Let B(V )⊗r = {bi1 ⊗· · · ⊗\nbir} denote the monomial basis of B(V )⊗r. Given b ∈B(V )⊗r, let\nb =\nX\ni1,...,ir\nf(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients f(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\nIt follows from the works of Kashiwara [Kas1] and Date et al [DJM]\nthat Lρ\nπ/qLρ\nπ has a unique basis Bρ\nπ (up to scaling by constant multiples)\nsuch that the monomial supports of its elements are disjoint–in fact, one\ncan choose ρ so that the monomial support of each basis element consists of\njust one distinct monomial. This basis can be made completely unique by\nappropriate normalization. Then (Lρ\nπ, Bρ\nπ) coincides with the upper crystal\nbase of Vq,π as constructed by Kashiwara. Furthermore, this crystal base\ndoes not depend on the embedding ρ (up to isomorphism). Hence, we let\n(Lπ, Bπ) = (Lρ\nπ, Bρ\nπ) for any ρ as above. Kashiwara [Kas1] also shows that\nLπ and Bπ ∪{0} are invariant under certain crystal operators ̃ei and ̃fi\ncorresponding to the simple roots of Hq.\nThis scheme for constructing the upper crystal base (Lπ, Bπ) crucially\ndepends on the existence of a compact real form Uq(V ) ⊆GLq(V ) = Hq.\nEven the existence of a compact real form of the standard Drinfeld-Jimbo\nenveloping algebra Uq(H) suffices here.\n2.2\nNonstandard triple\nWe now generalize the preceding scheme to the nonstandard setting using\nthe compact real form of U H\nq (G), whose existence is proved in GCT7. The\ngoal is to construct an analogous triple for the matrix coordinate ring S =\nO(MH\nq (X)) of GH\nq . It will turn out that this triple need not be balanced as\n6"},{"paragraph_id":"p7","order":7,"text":"in the standard case. We shall describe in Section 2.3 how a canonical basis\ncan be constructed from such a triple despite the lack of balance.\nWe begin by recalling that the q-analogue of the Peter-Weyl theorem (2)\nin the nonstandard setting need not hold over Q(q) unlike in the standard\nsetting. It holds only over an appropriate algebraic extension K of Q(q)\n[GCT7]–thinking of q as a transcendental. It will be convenient to assume\nin what follows that K is actually an algebraic extension ̃Q(q), where ̃Q is\nthe algebraic closure of Q. We let AK and ̄AK be the subrings of algebraic\nfunctions in K that are regular at q = 0 and q = ∞, respectively. Let KQ\nbe the integral closure of Q[q, q−1] in K. Clearly, KQ ∩AK ∩ ̄AK = ˆQ, where\nˆQ denotes the integral closure of Q in ̃Q. In what follows, we let ˆQ, AK, ̄AK\nand KQ play the role of Q, A, ̄A and Q[q, q−1] in Section 2.1. Thus by a\nlattice at q = 0 we mean an AK-lattice, by a lattice at q = ∞a ̄AK-lattice,\nby a Q-form, a KQ-form (module).\nSimilarly, instead of Q-modules and\nQ(q)-modules, we will be considering ˆQ-modules and K-modules. That is,\nin what follows we shall assuming that the underlying base field is K, instead\nof Q(q).\nNow let us describe the construction of the upper crystal base of an\nirreducible polynomial representation Wq,α of GH\nq . Let Xq = Vq,μ denote the\nquantization of X = Vμ; i.e, the Hq-module with highest weight μ. Since\nthe underlying field is K, Xq is a K-module. Let {zi} denote its standard\nq-orthonormal Gelfand-Tsetlin basis. Let {xi} denote the rescaled version\nof Gelfand-Tsetlin basis (as described in Section 7.3.3 of [Kli]) so that there\nare no square roots in the explicit formulae for the action of the generators\nof the Drinfeld-Jimbo algebra UqH on this basis. Alternatively, we can let\n{xi} be the standard (upper) canonical basis [Kas1, Lu2] of Xq as an Hq-\nmodule. We assume that limq→0(zi −xi) = 0; this can always be arranged.\nIn what follows, we sometimes denote Xq by X. What is meant should be\nclear from the context. It is shown in [GCT7] that Wq,α can be embedded\nin X⊗r for an appropriate r. We choose an embedding ρ : Wq,α →X⊗r as\nfollows.\nIf r = 1, Wq,α = X, so this is trivial. Otherwise, choose any β of degree\nr−1 so that Wq,α occurs as a GH\nq -submodule of Wq,β⊗X. By semisimplicity,\nthe latter is completely reducible as a GH\nq -module:\nWq,β ⊗X = ⊕βjWq,βj,\n(5)\nwhere we assume that this is also a decomposition as a K-module. Choose\nany j such that Wq,βj ∼= Wq,α. This fixes an embedding of Wq,α in Wq,β ⊗X.\n7"},{"paragraph_id":"p8","order":8,"text":"(It is a plausible conjecture that the decomposition ( 5) is multiplicity free.\nThis would be a conjectural analogue of Pieri’s rule in the nonstandard\nsetting. It would imply that the embedding of Wq,α in Wq,β ⊗X is unique.\nBut this is not required here.). By induction, we have fixed an embedding\nof Wq,β in X⊗r−1. This fixes an embedding ρ of Wq,α in X⊗r (among many\npossible choices). Let W ρ\nq,α = ρ(Wq,α) be its image.\nChoose a Hermitian form on X⊗r so that its Gelfand-Tsetlin basis {zi1 ⊗\n· · · ⊗zir} is orthonormal. Let W ρ,⊥\nq,α denote the orthogonal complement of\nW ρ\nq,α. Since U H\nq (G) has a compact real form [GCT7] such that X⊗r\nq\nis its\nunitary representation with respect to this Hermitian form, it follows that\nW ρ,⊥\nq,α is a GH\nq -module. Thus X⊗r = W ρ\nq,α ⊕W ρ,⊥\nq,α as a GH\nq -module. Let\nLρ\nα = L(X)⊗r ∩W ρ\nq,α\nand\nLρ,⊥\nα\n= L(X)⊗r ∩W ρ,⊥\nq,α .\nThen, in analogy with Kashiwara’s work mentioned above:\nProposition 2.1 L(X)⊗r = Lρ\nα ⊕Lρ,⊥\nα .\nProof: The r.h.s. is clearly contained in the l.h.s. To show the converse it\nsuffices to show that Lρ\nα and Lρ,⊥\nα\nare projections of the lattice L(X)⊗r onto\nW ρ\nq,α and W ρ,⊥\nq,α respectively. Let us show this for Lρ\nα, the other case being\nsimilar. Clearly, the projection of L(X)⊗r onto W ρ\nq,α contains Lρ\nα. We only\nhave to show that the projection ˆy of any y ∈L(X)⊗r onto W ρ\nq,α also belongs\nto the lattice L(X)⊗r, and hence to Lρ\nα. Since y ∈L(X)⊗r, its length |y|\nw.r.t. the preceding Hermitian form tends to a well defined nonnegative real\nnumber as q →0. Since, the projection y →ˆy is orthonormal, the length\n|ˆy| of ˆy is at most |y|, and hence also tends to a well defined nonnegative\nreal number as q →0. This means ˆy is regular at q = 0 and hence belongs\nto L(X)⊗r. Q.E.D.\nLet B(X) = {bi = φ(xi)} denote the basis of the ̃Q-module L(X)/qL(X),\nwhere φ : L(X) →L(X)/qL(X) is the natural projection (the bi’s in this\nsection are different from the bi’s in Section 2.1). Let B(X)⊗r = {bi1 ⊗· · ·⊗\nbir} denote the monomial basis of L(X)⊗r. Given b ∈B(X)⊗r, let\nb =\nX\ni1,...,ir\ng(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients g(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\n8"},{"paragraph_id":"p9","order":9,"text":"In analogy with the work of Kashiwara and Date et al mentioned above,\nit may be conjectured that:\nConjecture 2.2 (Existence of (local) crystal basis)\nThe ̃Q-module Lρ\nα/qLρ\nα has a unique basis Bρ\nα (up to scaling by constant\nmultiples, and which can be made completely unique by appropriate normal-\nization) such that:\n1. The monomial supports of its elements are disjoint,\n2. Lρ\nα and Bρ\nα ∪{0} are invariant under Kashiwara’s crystal operators ̃ei\nand ̃fi for Hq (which are well defined since Uq(H) is a subalgebra of\nU H\nq (G)), and (Lρ\nα, Bρ\nα) is a local crystal basis, in the sense of Kashi-\nwara, of Wq,α as an Hq-module\nFurthermore, this crystal base does not depend on the embedding ρ (up to\nisomorphism).\nThis conjecture has been supported by experimental evidence; cf.\nSec-\ntion 5.5.\nAssuming this, we let (Lα, Bα) = (Lρ\nα, Bρ\nα) for any ρ as above.\nIt is\ncalled the upper crystal base of Wq,α.\nIn standard setting, the embedding ρ : Vq,π →V ⊗r in Section 2.1 can\nbe chosen so that the support of each basis element in Bρ\nπ consists of just\none monomial. That is, so that each b ∈Bρ\nπ is a monomial in B(V )⊗r. In\nthe nonstandard setting, it is not always possible to choose an embedding\nρ : Wq,α →X⊗r so that the support of each basis element in Bρ\nα in Conjec-\nture 2.1 consists of just one monomial; cf. Section 5.5 for a counterexample.\nIn view of the nonstandard q-analogue of the Peter-Weyl theorem (2),\nS = O(MH\nq (X)) has a natural upper crystal base\n(L(S), B(S)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(6)\nat q = 0 (with appropriate normalization). Let SQ be the KQ-forms (KQ-\nsubring) of S generated by the entries ui\nj of the generic nonstandard quantum\nmatrix u; recall (cf. [GCT7]) that S is the quotient of C⟨u⟩modulo appro-\npriate quadratic relations over the entries ui\nj’s of u. We define an involution\n−over S by a natural generalization of its definition in the standard setting.\nSpecifically, S is a O(MH\nq (X))-bi-comodule. Via the homomorphism from\n9"},{"paragraph_id":"p10","order":10,"text":"O(MH\nq (X)) to O(Mq(V )), S is also O(Mq(V ))-bi-comodule; i.e., an Hq-bi-\nmodule. In the spirit of [Kas2], for any u and v in S with Hq-bi-weights\n(λr, λl) and (μr, μl), let uv = q(λr,μr)−(λl,μl) ̄v ̄u, where ( , ) denotes the usual\ninner product in the Hq-weight space. We let ̄ui\nj = ui\nj, and ̄q = q−1. This\ndefines −on S completely.\nApplying −to (L(S), B(S)), we get an upper crystal base ( ̄L(S), ̄B(S))\nat q = ∞. In analogy with the standard setting, we can now ask:\nQuestion 2.3 Is the triple (SQ, L(S), ̄L(S)) balanced? In other words, is\nthe map ψ : E = SQ ∪L(S) ∪ ̄L(S) →L(S)/qL(S) of ˆQ-modules an isomor-\nphism.\nIf it were, we could have defined the canonical basis of S by a globalization\nprocedure very much as in the standard setting, i.e., as ψ−1(B(S)). But, as\nit turns out, this need not be so; cf. Section 5.3. Specifically, for a given\nb ∈B(S), the fibre ψ−1(b) need not be singleton. This is the major difference\nfrom the standard setting that makes construction of the canonical basis of\nS in the nonstandard setting much more complex. We turn to this in the\nnext section.\n2.3\nNonstandard globalization via minimization of degree\ncomplexity\nWe now give a conjectural procedure for choosing an unambiguously defined\ncanonical element yb ∈ψ−1(b), b ∈B(S). The set {yb |b ∈B(S)} will then\nbe the canonical basis of S.\nFix r. Let Sr denote the degree r component of S. Let A = AH\nr\n=\nAH\nr (q) be the nonstandard q-Schur algebra [GCT7], which is the dual S∗\nr =\nHom(Sr, K) of Sr. A polynomial irreducible GH\nq -module Wq,α of degree r\nis an irreducible A-module, and conversely every irreducible A-module is\nof this form. Furthermore, the q-analogue of the Peter-Weyl theorem also\nholds for A:\nA = ⊕αW ∗\nq,α ⊗Wq,α.\n(7)\nFor reasons given later (cf. Remark 1 below), it will be more convenient\nto construct the canonical basis of AH\nr first. The canonical basis of Sr will\nthen be defined to be its dual.\n10"},{"paragraph_id":"p11","order":11,"text":"Let\n⟨⟩: A ⊗Sr →K\nbe the natural pairing. The lattice L(A) is defined to be the dual lattice of\nL(Sr):\nL(A) = {a ∈A | ⟨a, L(Sr)⟩⊆AK}.\nThe automorphism −of A is defined by:\n⟨ ̄a, s⟩= ⟨a, ̄s⟩−.\nWe define ̄L(A) by applying −to L(A). We define the Q-form (i.e. KQ-\nform) AQ of A by:\nAQ = {a ∈A | ⟨a, Sr,Q⟩⊆KQ},\nwhere Sr,Q = Sr ∩SQ.\nWe define the basis B(A) of L(A)/qL(A) to be\nthe dual of B(S). Thus (L(A), B(A)) is the local crystal basis of A as per\nConjecture 2.2 and we have an analogue of (6):\n(L(A), B(A)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(8)\nThis defines the triple (AQ, L(A), ̄L(A)) for A. It need not be balanced\nin the standard sense, just like the triple (SQ, L(S), ̄L(S)) above. Now we\ndescribe the conjectural construction of a canonical basis of A.\nEach component B∗\nα ⊗Bα of B(A) has the left and right action of Kashi-\nwara’s crystal operators for Hq (cf. Conjecture 2.2). Thus, by [Kas1], we\nget a crystal graph on B∗\nα ⊗Bα, whose each connected component intutively\ncorresponds to an irreducible Hq-bi-submodule V ∗\nq,μ1 ⊗Vq,μ2 of the compo-\nnent W ∗\nq,α ⊗Wq,α in the Peter-Weyl decomposition of A (7). With each\nelement b ∈B(A) that occurs in such a connected component, we associate\nthe triple T(b) = (α, μ1, μ2). We call it the type of b. The types T(b)’s can\nbe partially ordered as follows.\nFirst, put a partial order ≤on the labels α of polynomial irreducible A-\nmodules Wq,α as follows. Consider Wq,α as an Hq-module. Let μ(α) denote\nthe highest weight in Wq,α as an Hq-module. There may be several highest\nweight vectors in Wq,α, since Wq,α need not be irreducible as an Hq-module.\nThat is fine. Let ≤denote the usual partial order on the highest weights of\nHq-modules: μ1 ≤μ2 iffμ2 −μ1 ∈P\nτ Nτ where τ ranges over the simple\npositive roots of H. We say α ≤α′ iffμ(α) ≤μ(α′).\n11"},{"paragraph_id":"p12","order":12,"text":"Now observe that each component W ∗\nq,α⊗Wq,α in the Peter-Weyl decom-\nposition of A is an Hq-bimodule; i.e., has a left and right action of Hq. With\neach irreducible Hq-bimodule in this component isomorphic to Vq,μ1 ⊗Vq,μ2,\nwhere μ1 and μ2 are highest left and right weights of Hq, we associate the\ntype T = (α, μ1, μ2). Put a partial order, which we shall again denote by\n≤, on the types T as per the partial order ≤on the individual components.\nThe type T(b) associated with each b ∈B(A) above is similar to this type.\nSo this also puts a partial order on the types T(b)’s.\nNext fix a b ∈B(A). Let T = T(b) be its type. We shall associate a\ncanonical basis element yb with each such b by induction on its type using\nthe preceding partial order. The set {yb} will then be the required canonical\nbasis of A.\nLet A≤T denote the span of all Hq-bimodules in A of types less than or\nequal to T as per ≤. Let\nL(A≤T) = L(A) ∩A≤T .\nWe define ̄L(A)≤T , and A≤T\nQ\nsimilarly. Consider the natural projection\nψT : A≤T\nQ\n∩L(A)≤T ∩ ̄L(A)≤T →L(A)≤T /qL(A)≤T .\nLet ψ−1\nT (b) be the fibre of b. If this fibre were to contain a unique element,\nthen we can simply let yb be this unique element. But this need not be\nso, because the triple (A≤T\nQ , L(A)≤T , ̄L(A)≤T ) need not be balanced. So\nwe have to resolve the ambiguity in some canonical way. Towards this end,\nwe shall associate with each element in AQ ∩L(A) a complexity measure,\ncalled its degree complexity. We shall then define yb to be the element in\nψ−1\nT (b) of minimum degree complexity–it would be conjecturally unique; cf.\nConjecture 2.5 below.\nThis scheme is in the spirit of [KL1] where each\nelement of the Kazhdan-Lusztig basis of the Hecke algebra is defined to be\nan element of minimum degree in a certain sense.\nSo let us define the degree complexity of an element y ∈AQ ∩L(A).\nSince X⊗r\nq\nis a represention of A = AH\nr (q), we have the injection\nη : A ֒→Z = End(X⊗r\nq ) = (X⊗r\nq )∗⊗X⊗r\nq .\nLet L(Z) = L(X⊗r\nq )∗⊗L(X⊗r\nq ) be the lattice associated with Z. The Q-form\n(or rather KQ-form) ZQ is defined similarly. Then\nProposition 2.4 The embedding η injects the Q-form AQ into ZQ. Fur-\nthermore, assuming Conjecture 2.2, η also injects the lattice L(A) into L(Z).\n12"},{"paragraph_id":"p13","order":13,"text":"The proof is easy. (To be filled in).\nFix the upper canonical basis {xi} of Xq as an Hq-module. Let {x∗\ni } be\nthe dual canonical basis of X∗\nq . This fixes the upper canonical basis CB(Z)\nof Z, whose each element is of the form\nzi1,...,ir;j1,...,jr = x∗\ni1 ⊗· · · ⊗x∗\nir ⊗xj1 ⊗· · · ⊗xjr.\nIt is also a basis of the Q-form ZQ and the lattice L(Z). Now given any\ny ∈A, let w = η(y). Express w in the canonical basis of Z:\nw = η(y) =\nX\nz\na(y, z)z,\n(9)\nwhere z ranges of the basis elements in CB(Z). Since w ∈L(Z) ∩ZQ, each\na(y, z) ∈AK ∩KQ. This means it is integral over Q[q] and hence has a well\ndefined degree d(y, z) at q = 0 (the same as the order of its pole at q = ∞);\nif a(y, z) = 0, we define d(y, z) = −∞. We define the degree complexity d(y)\nof y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We put a partial order\non the degree complexity as follows.\nLet U = Uq(H) be the Drinfeld-Jimbo enveloping algebra of Hq, U −the\nsubalgebra generated by its generators Fi’s. For any string ν = ν1, ν2, . . . of\npositive integers, let U −\nν be the subspace of U −spanned by the words in Fi’s\nin which each Fi occurs occurs νi times. Given a canonical basis element x\nof Xq, we define its length to be |x| = P\ni νi, where x ∈U −\nν x0, and x0 is\nthe highest weight vector of Xq. Order the canonical basis elements of Xq\nas per the reverse order on their lengths; so that x0 is the highest element\nin this order.\nPut a similar order on X∗\nq .\nThis puts an induced partial\norder on the elements of the canonical basis CB(Z) of Z. We let < denote\nthe strict less than relation as per this partial order. Given y and y′, and\nletting w = η(y), w′ = η(y′), we say that d(y) ≤d(y′) if for every z: either\nd(y, z) ≤d(y′, z), or for some ̄z < z d(y, ̄z) < d(y′, ̄z).\nConjecture 2.5 (Minimum degree) The fibre ψ−1\nT (b) contains a unique\nelement yb of minimum degree complexity. Minimum means d(yb) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (b).\nWe call yb the canonical basis element associated with b, and the set {yb}\nthe canonical basis CB(A) of A. The canonical basis CB(Sr) = {xb} of Sr\nis defined to be its dual. The canonical basis CB(S) of S is ∪rCB(Sr).\nRemark 1: The reader may wonder why we defined the canonical basis of A\nfirst, and that of Sr later, as its dual. Can we define the canonical basis of Sr\n13"},{"paragraph_id":"p14","order":14,"text":"directly? The algorithm for Sr would be similar as above. The main problem\nis to define the degree complexity of an element x ∈L(Sr) ∩SQ. We have\na natural projection from Z = End(X⊗d\nq ) to Sr, but not a natural injection\nthat injects Sr,Q = Sr ∩SQ into ZQ. So the analogue of Proposition 2.4 does\nnot hold.\nRemark 2: In the standard settting, Kashiwara and Lusztig give an efficient\nscheme for constructing each canonical basis element of the standard matrix\ncoordinate ring O(Mq(V )). We do not have here an analogous efficient algo-\nrithm for constructing the canonical basis element yb in Conjecture 2.5. In\nthe standard setting an efficient cosntruction was possible because the stan-\ndard Drinfeld-Jimbo universal enveloping algebra has an explicit presenta-\ntion in terms of generators and defining relations. This explicit presentation\nis crucially used in the construction of the standard canonical basis and also\nin proving correctness of the construction.\nThe nonstandard universal algebra does not have an analogous explicit\npresentation as yet; cf. [GCT7]. For this we need explicit formulae for the\ncoefficients of the Laplace relation in [GCT7] among the simplest nonstan-\ndard canonical basis elements in S, namely nonstandard minors, since as\ndiscussed in [GCT7], it is the mother relation in the representation theory\nof the nonstandard quantum group (just as in the standard setting). That\nis, we need explicit interpretation for these coefficients in the spirit of the ex-\nplicit interpretation for the coefficients of the Kazhdan-Lusztig polynomials\nin terms perverse sheaves. This is the basic core problem that needs to be\nsolved to prove that the preceding algorithm for constructing the nonstan-\ndard canonical basis is correct and to give an explicit, efficient construction\nof yb. Furthermore, if explicit presentation of the nonstandard universal al-\ngebra is so nonelementary, as against the elementary explicit presentation of\nthe standard (Drinfled-Jimbo) enveloping algebra, then the task of proving\ncorrectness may be formidable.\n2.4\nProperties of the nonstandard canonical basis\nIt may be conjectured that the nonstandard canonical bases CB(S) and\nCB(A) have properties akin to the standard canonical basis of O(Mq(V )):\n2.4.1\nCellular decomposition\nConjecture 2.6 (Cell decomposition) The refined Peter-Weyl theorem,\nakin to the one proved by Lusztig [Lu2] in the standard setting, holds for\n14"},{"paragraph_id":"p15","order":15,"text":"CB(S) and CB(A).\nThis means the left, right and two-sided cells in O(MH\nq (X)) with re-\nspect to CB(S) yield irreducible left, right, and two-sided (polynomial)\nrepresentations of GH\nq . And furthermore, the left sub-cells of each left cell\nwith respect to the restricted Hq-action yield irreducible Hq-representations.\nThe left cell of O(MH\nq (X)) is defined as follows.\nGiven b ∈CB(S), let\n∆(b) = P\nb′,b′′ cb′,b′′\nb\nb′ ⊗b′′, where ∆denotes comultiplication. Then we say\nthat b′′ ←L b if cb′,b′′\nb\nis nonzero for some b′. Let <L denote the transitive\nclosure of ←L. Using <L we define left cells in a natural way. The right\nand two-sided cells are defined similarly. The left, right and two-sided sub-\ncells with respect to the Hq-action are defined similarly. The definitions for\nCB(A) are similar.\nBy restricting the canonical basis to any left cell corresponding to an\nirreducible polynomial representation Wq,α of GH\nq , we get the canonical basis\nof Wq,α; here the choice of the left cell would conjecturally not matter up to\nscaling.\n2.4.2\nPositivity in the Kronecker case\nConjecture 2.7 (Positivity) In the Kronecker case–i.e. when H = GL(V )×\nGL(W), X = V ⊗W with the natural H-action–each coefficient g(q) of any\ncanonical basis element in CB(AH\nr ) in the basis CB(Z) (cf. eq.(9)) is a\npositive polynomial in q.\nSimilarly, each multiplicative or comultiplicative structural constant of\nCB(AH\nr ) is of the form\n+−(q −1\nq)af(q), where a is a nonnegative integer\nand f(q) is a −-invariant positive and unimodal polynomial in q and q−1.\nThe same also for CB(S).\nHere by a positive polynomial we mean a polynomial with nonnegative\nrational coefficients. By unimodality of the (−-invariant) polynomial f(q),\nwe mean its coefficients f−k, . . . , fk satisfy the condition\nf−k ≤f−k+1 ≤· · · ≤f−1 ≤f0 ≥f1 ≥· · · fk.\nBy multiplicative structural constants, we mean the coefficients mb′′\nb,b′ in the\nexpansion\nbb′ =\nX\nb′′\nmb′′\nb,b′b′′,\n15"},{"paragraph_id":"p16","order":16,"text":"for b, b′ ∈CB(S), and with b′′ ranging over the elements in CB(S). Comul-\ntiplicative structural constants are defined similarly.\nFor experimental evidence for the dual nonstandard algebra BH\nr (q), see\nSection 5.1.\nPresumably, the nonnegative coefficients of g(q) and f(q) may have a\ntopological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of f(q) may be a conse-\nquence of some result akin to the Hard Lefschetz theorem in the spirit of\nthe results on unimodal sequences in [St].\nThe Kronecker case is fundamental because O(MH\nq (X)) therein is a non-\nflat deformation of O(M(X)), though CH\nq [X] is a flat deformation of C[X].\nThus to prove the positivity Conjecture 2.7, some nonstandard extension\nof the theory of perverse sheaves and the work surrounding the Riemann\nhypothesis over finite fields [BBD, Dl2] that can deal with nonflat, noncom-\nmutative varieties like O(MH\nq (X)) may be neeeded.\n2.4.3\nNonstandard positivity and saturation\nIn the Kronecker case, the braided symmetric algebra CH\nq [X] is a flat de-\nformation of C[X]; i.e. dim(CH,r\nq\n[X]) = dim(Cr[X]). But when CH\nq [X] is a\nnonflat deformation the situation is much more complex. To see this, let c be\na coefficient of any canonical basis element in CB(AH\nr ) in the basis CB(Z).\nAs observed after eq.(9), it is integral over Q[q]. Hence every coefficient of\nits minimal polynomial fc is a polynomial in q with rational coefficients. In\nthe spirit of Conjecture 2.7, one may ask if the coefficients of this polynomial\nare always nonnegative. Unfortunately, this need not be so; cf. Section 5.3\nfor counterexamples in the dual setting of BH\nr (q). The following is a relaxed\nversion of Conjecture 2.7 for the general case.\nConjecture 2.8 (Saturation) Each coefficient a(q) of fc is a saturated\npolynomial (in the terminology of [GCT6]); this means a(1) is a positive\nrational if a(q) is not an identically zero polynomial.\nSimilarly, each coefficient b(q) of the minimal polynomial of a multipli-\nciative or comultiplicative structural constant of CB(AH\nr ) can be expressed\nin the form\nb(q) = (−1)e(q −1\nq )e′c(q)\n(10)\nfor some nonnegative integers e, e′, where\n16"},{"paragraph_id":"p17","order":17,"text":"1. e is chosen so that the middle term of c(q) is positive; here by the\nmiddle term we mean the coefficient of qi for the smallest i such that\nthis coefficient is nonzero, and\n2. c(q) is a saturated polynomial in q and q−1–this again means c(1) is a\npositive rational if c(q) is not an identically zero polynomial.\nIn the context of the plethysm problem, one is finally interested in the\nbehaviour of b(q) at q = 1 (cf. Section 6), so this relaxed saturation form\nof positivity should be sufficient; see also [GCT6] for the importance of\nsaturation in the context of the flip in GCT. A stronger positivity conjecture\nthat would specialize to Conjecture 2.7 in the Kronecker case is:\nConjecture 2.9 (Nonstandard Positivity) Each polynomial c(q) in Con-\njecture 2.8 is almost positive and unimodal.\nThat is, it is of the form\nc0(q) + c1(q), where, if c(q) is not identically zero,\n1. c0(1) >> |c1(1)|, where >> means much greater as r →∞, and more\ngenerally,\n2. c0(q) is a dominant positive unimodal polynomial, and c1(q) is a very\nsmall error-correction polynomial. Specifically,\n||c1(q)||/||c0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwhere μ and π are as in the plethysm problem (cf. begining of Sec-\ntion 1), ⟨\n⟩denotes the bitlength-of-specification function, and poly(\n)\nmeans polynomial of a fixed (constant) degree in the specified bitlengths,\nand ||\n|| denotes the L2-norm of the coefficient vector of the polyno-\nmial.\nSee Section 5.3 for experimental evidence for Conjectures 2.8-2.9 in the\ndual setting of BH\nr (q).\nPresumably, the nonnegative coefficients of such c0(q) may again have\na topological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of c0(q) may again be a con-\nsequence of some result akin to the Hard Lefschetz theorem. The correction\npolynomial c1(q) may also have a topological interpretation that depends on\na cohomological measure of nonflatness of CH\nq [X]. In the Kronecker case,\nwhen CH\nq [X] is a flat deformation of C[X], this correction would then van-\nish, and Conjecture 2.9 would reduce to Conjecture 2.7. Furthermore, the\n17"},{"paragraph_id":"p18","order":18,"text":"conjectural nonnegative value of c(1) may also have an interpretation akin\nto the representation-theoretic interpretation for the values of the Kazhdan-\nLusztig polynomials at q = 1.\nIf nonstandard extension of the work surronding the Riemann hypothe-\nsis over finite fields as needed to prove the positivity Conjecture refcposkro-\nneckerleft in the Kronecker case can be found, that may open the way for\ninvestigating the more complicated nonstandard form of positivity in Con-\njecture 2.9.\n3\nNonstandard canonical basis of BH\nr\nLet BH\nr = BH\nr (q) be the nonstandard quantization of the symmetric group\nring C[Sr] in [GCT7]. In this section, we describe an analogous conjecturally\ncorrect algorithm for constructing a nonstandard canonical basis E(r) of BH\nr .\nIn the standard setting–i.e., when H = G–this basis would conjecturally\nspecialize to the Kazhdan-Lusztig basis of the Hecke-algebra, though the\nspecialized algorithm here is different from the algorithm in [KL1].\nSince BH\nr (q) is semisimple [GCT4, GCT7], by the Wedderburn structure\ntheorem\nBH\nr (q) =\nM\nα\nT ∗\nq,α ⊗Tq,α,\n(11)\nwhere Tq,α ranges over the irreducible representations of BH\nr , assuming that\nthe underlying base field is a suitable algebraic extension of Q(q1/2). We\nshall denote this base field by K–it is the same as the base field K in\nSection 2.2, except that the role of q there is played by q1/2 here.\nLet\nAK, ̄AK, KQ, ˆQ be as in Section 2.2, with the role of q played by q1/2.\nWe assume that H is the general linear group GL(V ) or a product of\ngeneral linear groups. In this case (cf. [GCT7]), BH\nr is a −-invariant sub-\nalgebra of a suitable Hecke-algebra or a product of Hecke algebras, where\n−denotes the usual bar-automorphism on the Hecke algebra [KL1] (ana-\nlogue of −in Section 2). Let Pi’s and Qi’s denote the rescaled positive and\nnegative generators of BH\nr as defined in [GCT7] (denoted by p+,H\nX,i and q+,H\nX,i\ntherein) so that they belong to the usual Z[q1/2, q−1/2]-form on the ambient\nHecke algebra (or the ambient product of Hecke algebras). Let BH\nr,Q denote\nthe KQ-form of BH\nr generated ring-theoretically by Pi’s, or equivalently Qi’s.\nThe goal is to construct an AK-lattice R(r) ⊆BH\nr so that the canonical\nbasis E(r) of BH\nr\ncan then be constructed by a nonstandard globalization\n18"},{"paragraph_id":"p19","order":19,"text":"procedure on the triple (BH\nr,Q, R(r), ̄R(r)) analogous to the one Section 2.3.\nJust as in Section 2.2, it will turn out that this triple need not be balanced.\nWe will resolve the ambiguity caused by lack of balance using the notion of\nminimum degree complexity very much as in Section 2.3.\n3.1\nNonstandard Gelfand-Tsetlin basis of BH\nr (q)\nIn the construction of Kazhdan-Lusztig polynomials as described in [KL1,\nSo], the lattice in the Hecke algebra is constructed using its standard mono-\nmial basis. But in general BH\nr (q) does not have a naturally defined monomial\nbasis; see [GCT4] for an example. So we need a different way to construct\nthe lattice R(r). The construction here will be analogous to the construction\nof the lattice in the standard matrix coordinate ring O(Mq(V )) based on its\nGelfand-Tsetlin basis. This construction in O(Mq(V )) is different from the\none defined in Section 2.1. We shall recall it using the same notation as\nin Section 2.1. It is based on the observation that the Gelfand-Tsetlin ba-\nsis of an irreducible Hq-representation Vq,μ (after rescaling as described in\nsection 7.3.3 of [Kli]) is its local crystal basis: i.e., it is an A-basis of the\nlattice Lμ ⊆Vq,μ, and that its projection in Lμ/qLμ is equal to the basis\nBμ (whose elements have disjoint monomial supports as described before).\nThis observation was in fact the starting point for the theory of local crystal\nbasis [Kas1]. Let us denote the (rescaled) Gelfand-Tsetlin basis of Vq,μ by\nGTq,μ. The Gelfand-Tsetlin basis of O(Mq(V )) is defined as per the standard\nPeter-Weyl theorem ( 3):\nGT(O(Mq(V ))) =\n[\nμ\nGT ∗\nq,μ ⊗GTq,μ.\n(12)\nLet LGT be the lattice generated by this Gelfand-Tsetlin basis, and BGT the\nprojection of the Gelfand-Tsetlin basis on LGT /qLGT . Then (LGT , BGT )\ncoincides with the standard crystal base (L, B) of O(Mq(V )) in Section 2.1.\nThe algebra BH\nr (q) has a natural analgoue of the Gelfand-Tseltin basis,\nwhich can then be used to construct the lattice R(r). We begin by describing\nthis basis for an irreducible representation Tq,α of BH\nr (q).\nWe proceed by induction on r, the case r = 1 being easy. The following\nis a conjectural analogue of the standard Pieri’s rule in this setting:\nC1’: Tq,α has a multiplicity-free decomposition as a BH\nr−1-module.\n(This conjecture is not really necessary as long as there is a natural\nway to resolve the ambiguity caused by multiplicity).\nBy induction, we\n19"},{"paragraph_id":"p20","order":20,"text":"have defined a basis for each irreducible BH\nr−1(q)-submodule of Tq,α. Putting\nthese bases together, we get the sought nonstandard Gelfand-Tsetlin basis\nCα of Tq,α.\nAssuming multiplicity-free decomposition, such a basis is unique, up to\nscaling factors, which will be fixed in the course of the algorithm below.\nEach element x ∈Cα can be indexed by a nonstandard Gelfand-Tsetlin\ntableau, which is an analogue of the standard Gelfand-Tsetlin tableau [Kli]\nin this setting. It is defined to be the tuple (αr, αr−1, . . .), with αr = α, of the\nclassifying labels–which we shall call types–of the irreducible BH\ni -submodules\nTq,αi containing x, where Tq,αi ⊂Tq,αi+1.\nWe define the nonstandard Gelfand-Tsetlin basis C(r) of BH\nq (r) as per\nthe decomposition (11):\nC(r) =\n[\nα\nC∗\nα ⊗Cα.\nEach element of C(r) is indexed by a nonstandard Gelfand-Tsetlin bi-tableau\nas per this decomposition. We shall denote the element of C(r) indexed by\na nonstandard Gelfand-Tsetlin bitableau T by cT .\n3.2\nLocal crystal base\nThe sought lattice R(r) ⊆BH\nr (q) will be generated by the elements of C(r)\nafter scaling them appropriately in the course of the algorithm below. Let\nus assume at the moment that this scaling has already been given to us,\nand thus R(r) is fixed. Let bT denote the image of cT under the projection\nψ : R(r) →R(r)/q1/2R(r). Let B(r) = {bT } be the basis of R(r)/q1/2R(r).\nThen (R(r), B(r)) is the analgoue of the local crystal base in the standard\nsetting.\n3.3\nNonstandard globalization via minimization of degree\ncomplexity\nThe elements of C(r) need not be −-invariant.\nNext we globalize C(r)\nto get a −-invariant canonical basis E(r) of BH\nr\nin the spirit of Kazhdan\nand Lusztig [KL1], with the role of the standard basis in [KL1, So] played\nby the nonstandard Gelfand-Tsetlin basis of BH\nr (q) here. As already men-\ntioned, the main difference from the standard setting of Hecke algebra is\nthat (BH\nr (q), R(r), ̄R(r)) need not be balanced. This is the main problem\n20"},{"paragraph_id":"p21","order":21,"text":"that needs to be addresssed. The nonstandard globalization procedure here\nis analogous to the one in Section 2.3. It goes as follows.\n(1) In Section 2.3, we described a partial order ≤on the types (classifying\nlabels) of the irreducible modules Wq,α of AH\nr (q). By the nonstadard duality\nconjecture [GCT7], this induces a partial order ≤on the types (classifying\nlabels) of the (paired) irreducible modules Tq,α of BH\nr (q). (In the standard\nsetting, this procedure would yield a partial order on the partitions of size\nr, with the partition containing a single row of size r at the top of the order\nand the partition containing a single column of size r at the bottom of the\norder.)\n(2) This induces a lexicographic partial order ≤on the nonstandard Gelfand-\nTsetlin tableaux, since they are just tuples of types, and also on nonstandard\nGelfand-Tsetlin bitableau which index the basis elements C(r).\n(3) Let B≤T be the span of the basis elements cT ′ ∈C(r) such that T ′ ≤T.\nLet R≤T = R(r) ∩B≤T , ̄R≤T = ̄R(r) ∩B≤T, and B≤T\nQ\n= BH\nr,Q ∩B≤T . Then\nthe triple (B≤T\nQ , R≤T , ̄R≤T ) need not be balanced. To define a canonical\nbasis element eT associated with T, we associate a degree complexity with\neach element y ∈BH\nr,Q in the spirit of Section 2.3.\nThis is done as follows. Since we are assuming that H is GL(V ) or a\nproduct of general linear groups, BH\nr is a subalgebra of a product of Hecke\nalgebras, say Z = Hk1(q) ⊗· · · ⊗Hkl(q), where Hj(q) denotes the Hecke\nalgebra with rank j. Furthermore,\nBH\nr,Q ⊆ZQ = Hk1,Q × · · · × Hkl,Q,\nwhere Hj,Q denote the KQ-form of Hj(q) obtained by tensoring its usual\nQ[q1/2, q−1/2]-form with KQ. Consider the Kazhdan-Lusztig basis KL(Z)\nof Z formed by taking the product of the Kazhdan-Lusztig bases of its Hecke\nalgebra factors. Express y ∈BH\nr,Q in terms of KL(Z):\ny =\nX\nz\na(y, z)z,\n(13)\nwhere z ranges over the elements in KL(Z). Then each coefficient a(y, z) ∈\nKQ. Let d(y, z) denote the degree of a(y, z); i.e., the order of its pole at\nq = ∞.\nIf a(y, z) = 0, we define d(y, z) = −∞.\nWe define the degree\ncomplexity d(y) of y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We\nput a partial order on degree complexities as follows. Put a partial order\n≤on the Kazhdan-Lusztig basis of the Hecke algebra Hj(q) as per the\nreverse order on the (reduced) lengths of the permutation indices of the\n21"},{"paragraph_id":"p22","order":22,"text":"basis elements–so 1 is the highest element as per this order. This also puts\na partial order on KL(Z). Let < denote the strict less than relation as per\nthis partial order. Given y and y′, we say that d(y) ≤d(y′) if for every z:\neither d(y, z) ≤d(y′, z), or for some ̄z < z, d(y, ̄z) < d(y′, ̄z).\nConsider the natural projection\nψT : R≤T ∩ ̄R≤T ∩B≤T\nQ\n→R≤T /q1/2R≤T .\nLet ψ−1\nT (bT ) be the fibre of bT ∈B(r). The following is the analogue of\nConjecture 2.5 in this context (with different interpretation for b, y, ψ etc.\nfrom there):\nConjecture 3.1 (Minimum degree) The fibre ψ−1\nT (bT ) contains a unique\nelement eT of minimum degree complexity. Minimum means d(eT ) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (bT ).\nWe call eT the canonical basis element associated with T, and E(r) =\n{eT } the canonical basis BH\nr (q).\nSo far we have not discussed how to scale the nonstandard Gelfand-\nTsetlin basis of BH\nr (q) to get the lattice R(r). To complete the algorithm,\nit remains to fix this scaling.\nLet {c′\nT } denote the nonstandard Gelfand-Tsetlin basis of BH\nr (q) before\nscaling. The scaled cT will be of the form qaT c′\nT for some rational aT . We\nhave to determine all aT ’s. Assume that aT ′, T ′ < T, have been fixed. For\nany rational a, let ca,T = qac′\nT . Let Ra,≤T be the lattice generated by ca,T\nand cT ′, T < T, and ̄Ra,≤T obtained by applying −to it. Consider the\nprojection\nψa,T : Ra,≤T ∩ ̄Ra,≤T ∩B≤T\nQ\n→Ra,≤T /q1/2Ra,≤T .\nLet ba,T be the image of ca,T under the projection Ra,≤T /q1/2Ra,≤T . Let\nψ−1\na,T (ba,T ) be its fibre. The following is the strengthened form of Conjec-\nture 2.5.\nConjecture 3.2 (Minimum degree) There exists a unique aT and eT ∈\nψ−1\naT ,T(baT ,T) such that for any a and any y ∈ψ−1\na,T (ba,T ) d(eT ) ≤d(y). That\nis, eT is the unique element of minimum degree complexity over all choices\nof a.\n22"},{"paragraph_id":"p23","order":23,"text":"This fixes aT . Furthermore, ψT = ψaT ,T , bT = baT ,T , and eT in Conjec-\nture 3.1 is the same as here.\nIf instead of the order ≤among the classifying labels α’s of Tq,α, we\nuse its reverse order, and in the definition of degree complexity use the\nopposite of the Kazhdan-Lusztig basis (obtained by replacing Qi by Pi), we\nget another canonical basis Eopp(r) of BH\nr (q), which we shall call its opposite\ncanonical basis.\nTo prove Conjectures 3.1-3.2, we need to know relations among the gen-\nerators Pi’s of BH\nr explicitly, just as we know the relations among the gen-\nerators of the Hecke algebra explicitly. This is not known at present. See\n[GCT7] for the problems that arise in this context.\nEach element c of the Kazhdan-Lusztig basis of the Hecke algebra can\nbe expressed in the form\nc = c0 +\nX\nj>0\na(j)cj,\nwhere each cj is a monomial in the generators of the Hecke algebra, a(j) ∈\nQ[q1/2, q−1/2] and the length of each cj, j > 0, is smaller than that of\nc0. This need not be so in the nonstandard setting: there can be several\nmonomials of maximum length with nontrivial coefficients in any monomial\nrepresentation of a nonstandard canonical basis element; cf. Section 5.3 and\nFigure 11 therein for an example.\n3.4\nConjectural properties\nIt may be conjectured that the canonical bases E(r) and Eopp(r) have prop-\nerties akin to those of the Kazhdan-Lusztig basis of the Hecke algebra.\n3.4.1\nCellular decomposition\nConjecture 3.3 (Cell decomposition) Analogue of the cell decomposi-\ntion property of the Kazhdan-Lusztig basis also holds for E(r) and Eopp(r).\nSpecifically this means the following. Let us define the left, right and\ntwo-sided cells of BH\nr\nwith respect to the canonical basis E(r) very much\nas in Section 2.4. Then it may be conjectured that they yield irreducible\nleft, right and two-sided representations of BH\nr . The conjecture for Eopp(r)\nis similar.\n23"},{"paragraph_id":"p24","order":24,"text":"By restricting E(r) to any left cell corresponding to an irreducible BH\nr -\nmodule Wq,α, we get the canonical basis of Wq,α; here the choice of the left\ncell would conjecturally not matter (up to scaling).\n3.5\nPositivity in the Kronecker case\nThe following is an analogue of Conjecture 2.7 here.\nConjecture 3.4 (Positivity) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13).\nIn the Kronecker case, c is of the form\n+−(q1/2 −q−1/2)af(q), where a\nis a nonnegative integer and f(q) is a −-invariant positive and unimodal\npolynomial in q1/2 and q−1/2.\nThe same for Eopp(r).\nFor experimental evidence see Sections 5.1.2 and 5.2.\n3.5.1\nNonstandard positivity and saturation\nThe general case is much more complex as in Section 2.4.3. The following\nis an analogoue of Conjecture 2.8.\nConjecture 3.5 (Saturation) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13). Let fc be its minimal polynomial with\ncoefficients in Q(q1/2).\nThen each coefficient s(q) of fc can be expressed in the form\ns(q) = (−1)e(q1/2 −q−1/2)e′g(q)\n(14)\nfor some nonnegative integers e, e′, where\n1. e is chosen so that the middle term of g(q) is positive; and\n2. g(q) is a saturated polynomial in q1/2 and q−1/2.\n24"},{"paragraph_id":"p25","order":25,"text":"Analogue of the stronger Conjecture 2.9 in this case is:\nConjecture 3.6 (Nonstandard Positivity, informal) Each polynomial\ng(q) above is almost positive and unimodal; i.e. of the form g0(q) + g1(q),\nwhere, if g(q) is not identically zero,\n1. g0(1) >> |g1(1)|, and more generally,\n2. g0(q) is a dominant positive unimodal polynomial, and g1(q) is a very\nsmall error-correction polynomial. Specifically,\n||g1(q)||/||g0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwith the terminology as in Conjecture 2.9.\nFor experimental evidence, see Section 5.4.1.\nHere, g(1) and the conjectural nonnegative coefficients of g0(q) may have\na representation-theoretic/topological/cohomological interpretation akin to\nthat sought for the analogous quantities in Section 2.4.3.\n3.5.2\nQuasi-cellular decomposition\nConjecture 3.7 The opposite canonical basis Eopp also has the following\nquasi-cellular decomposition property.\nFor this we define a quasi-subcellular decomposition of each left or right\ncell with respect to Eopp(r). Specifically, given e′, e′′ belonging to the same\nleft cell, express\nee′ =\nX\ne′′\nǫ(e, e′, e′′)(q1/2 −q−1/2)δ(e,e′,e′′)de′′\ne,e′e′′,\n(15)\nwhere e′′ ∈E(r), the sign ǫ(e, e′, e′′) is either 1 or −1, δ(e, e′, e′′) is a non-\nnegative integer, and de′′\ne,e′ is a −-invariant saturated polynomial in q1/2 and\nq−1/2 as per the saturation Conjecture 3.5. We say that e′′ ∝L e′ if, for some\ne, de′′\ne,e′ in (15) is nonzero and δ(e, e′, e′′) is zero; i.e., if, for some e, e′′ occurs\nwith nonzero coefficient in the expansion of ee′ specialized at q = 1. Let ≺L\ndenote the transitive closure of ∝L. Using ≺L we define left quasi-subcells\nof a left-cell of Eopp(r). It may be conjectured that each left quasi-subcell\nof Eopp(r) yields an irreducible representation of the symmetric group Sr at\n25"},{"paragraph_id":"p26","order":26,"text":"q = 1. That is, when the BH\nr -representation Y corresponding this left cell\nis specialized at q = 1, so as to become a representation Yq=1 of C[Sr], the\npartial order on its left quasi-sub-cells induces a composition series of Yq=1\nwhose factors are irreducible representations of C[Sr].\nFix one such quasi-subcell C of Eopp(r). Let Sλ(C) be the irreducible\nrepresentation (Specht module) of C[Sr] that is isomorphic to the factor\nin correspondence with C in this composition series of Yq=1, where λ(C)\nis a partition depending on C. The canonical basis elements in C, after\nspecialization at q = 1 and projection, yield a basis of Sλ(C). It may be con-\njectured that this basis coincides with the Kazhdan-Lusztig basis of Sλ(C)\n(up to rescaling).\nBy the Kazhdan-Lusztig basis of Sλ(C), we mean spe-\ncialization at q = 1 of the Kazhdan-Lusztig basis of the quantized Specht\nmodule Sq,λ(C) of the Hecke algebra Hr(q).\nBut Conjecture 3.7 need not hold for E(r); cf. Section 5.3 for a coun-\nterexample. This is analgous to the fact that the refined Peter-Weyl theorem\nin [Lu2] for the coordinate ring of the standard quantum group Hq holds\nonly for the ordering ≤(as defined in Section 2.3) among the labels (high-\nest weights) of irreducible Hq-modules–there is no canonical basis of the\nstandard coordinate ring which admits refined Peter-Weyl theorem for the\nopposite of the order ≤.\n4\nInternal definition of degree complexity\nWe give here an internal definition of degree complexity which may be used\nin place of the definition in Section 3.3 during the construction of the canon-\nical basis. By internal, we mean it is based only on the structure of BH\nr (q)\nand does not depend on its embedding in the external ambient algebra Z\nthere. This notion of degree complexity does not coincide with the one in\nSection 3.3, but the canonical basis constructed using this definition may be\nconjectured to be the same as the one constructed therein.\nLet B[i] ⊆BH\nr (q) be the span of the monomials in Qj’s of length j, and\nB[< i] of length < i. We say that a given set of monomials in Qj’s of length\ni is independent if the images of these monomials in B[i]/B[< i] are linearly\nindependent. An expression\na =\nX\nm\namm,\n(16)\nwhere m ranges over monomials in Qj’s and am ∈KQ, is called valid\n26"},{"paragraph_id":"p27","order":27,"text":"if, for each i, the monomials m of length i with am ̸= 0 in this expres-\nsion are independent.\nAssume that a is −-invariant, so that each am is\n−-invariant.\nThe degree complexity ˆd(a) of a is defined to be the tuple\n⟨ˆdl(a), . . . , ˆdi(a), . . . , ˆd0(a)⟩where ˆdi(a) denotes the maximum degree (at\nq = 0) of am for any m of length i, and l is the maximum length of m\nwith am ̸= 0 in the expression (16); by definition, ˆdi(a) = −∞if there is\nno m in (16) of length i with am ̸= 0. We order these degree complexities\nlexicographically. The degree complexity ˆd(b) of an element b ∈BH\nr,Q(q) is\ndefined to be the minimum degree complexity of its any valid expression.\nIt may be conjectured that if this definition of degree complexity, with the\nlexicographic ordering as above, is used in place of the definition of degree\ncomplexity in Section 3.3, the algorithm therein still works correctly and\nconstructs the same canonical basis E(r).\nFor the opposite canonical basis Eopp(r), one can similarly use the inter-\nnal definition as above with Pi in place of Qi.\nThe definition of degree complexity in this section is not as satisfactory\nas in Section 3.3 because BH\nq (r) does not a natural monomial basis [GCT4].\nHence to find the degree complexity of an element, one has to consider all\nits monomial expressions, finite but huge in number. It will be interesting\nto know if there is a more efficient internal definition.\n5\nExperimental evidence for BH\nr (q)\nIn this section we shall verify the conjectures in this paper for two nontriv-\nial special cases of the nonstandard algebra B = BH\nr (q). The nonstandard\ncanonical bases of BH\nr (q) in these cases were computed with the help of a\ncomputer using the algorithm in Section 3 and the notion of degree com-\nplexity as in Section 3.3.\nFirst, some notation. Given a string σ = i1 · · · ik of positive integers,\nwe let Pσ denote the monomial Pi1 · · · Pik; Qσ is defined similarly. Given a\n−-invariant polynomial g(q) ∈Q[q, q−1], we define the vector ag associated\nwith g(q) as follows. Express g(q) in the form\n+−(q−1/q)eh(q), where e is the\nmaximum possible. Let h−l, . . . , h0, . . . , hl be the coefficients of h(q). Then\nag is defined to be [h0, . . . , hl]. In particular if h(q) is (positive) unimodal,\nthen aq is a (positive) nonincreasing sequence. The vector associated with\na −-invariant polynomial in q1/2 and q−1/2 is defined similarly.\n27"},{"paragraph_id":"p28","order":28,"text":"5.1\nKronecker problem: n = 2, r = 3\nConsider B = BH\n3 (q) in the special case of the Kronecker problem for n = 2\nand r = 3. Thus H = Gl2 ×Gl2, and G = Gl4 with H embedded diagonally.\nLet Pi, i = 1, 2, be as in Section 3 and [GCT4]. The nonstandard canonical\nbasis of B was computed in [GCT4] by an ad hoc method for r = 3, but it\ncoincides with the one computed by the algorithm here. It is as follows. Let\nc1 = q6 + 2q5 + 3q4 + 4q3 + 3q2 + 2q + 1\nq3\n,\nc2 = q4 + q3 + 4q2 + q + 1\nq2\n,\nb1 = −(q2 + 1)2/q2, and b2 = (q + 1)2/q.\nThen the opposite canonical basis Eopp(3) of B consists of the following\nten elements:\nΣ = c1P1 −c2P121 + P12121,\nγi\n1 = b1P1 + P121,\ni = 1, 2,\nγi\n12 = b1P12 + P1212,\ni = 1, 2,\nγi\n2 = b1P2 + P212,\ni = 1, 2,\nγi\n21 = b1P21 + P2121,\ni = 1, 2,\nμ = 1.\n(17)\nThe canonical basis E(3) is obtained by susbstituting Qi for Pi. In what\nfollows, we shall only consider Eopp(3).\n5.1.1\nCellular and quasi-cellular decomposition\nThe basis Eopp(3) has a cellular decomposition, in accordance with Conjec-\nture 3.3, with the following right cells:\nUσ\n=\n{Σ}\nV1\n=\n{γ1\n1, γ1\n12}\nV2\n=\n{γ2\n1, γ2\n12}\nW1\n=\n{γ1\n2, γ1\n21}\nW2\n=\n{γ2\n2, γ2\n21}\nUμ =\n=\n{μ}.\nThe left cell decomposition is similar. The representation of B supported\nby Uσ is the trivial one dimensional representation. The representation sup-\nported by V1 or W1 is isomorphic; let us call it χ1. Similarly, the representa-\ntion supported by V2 or W2 is isomorphic; let us call it χ2. Then χ1 and χ2\n28"},{"paragraph_id":"p29","order":29,"text":"are two nonisomorphic two-dimensional representations of B which special-\nize at q = 1 to the two-dimensional Specht module of the symmetric group\nS3 corresponding to the partition (2, 1). Thus quasi-cellular decomposition\n(Conjecture 3.7) holds trivially here.\n5.1.2\nPositivity\nCoefficients of the elements of W1 and W2 in the Kazhdan-Lusztig basis of\nH3(q) ⊗H3(q) ⊇BH\n3 (q) are shown in Figure 1 (with the Kazhdan-Lusztig\nbasis symmetrized and appropriately ordered as described in [GCT4]); the\nfirst column shows the coefficients of γ1\n1, the second of γ1\n12, and so on. It\ncan be observed that all coefficients are positive, and unimodal polynomials\nin Q[q, q−1]. The cofficients of other canonical basis elements can be found\nin [GCT4]; they too are positive, unimodal polynomials. This verifies the\npositivity Conjecture 3.4 for the structural coefficients of the canonical basis.\nA few typical nonzero multiplicative structural constants of the canonical\nbasis are shown in Figure 2, where the coefficient of bb′ with respect to the\nbasis element b′′ is denoted by c(b, b′; b′′). It can be seen that each constant\nis a polynomial of the form\n(−1)a(q1/2 −q−1/2)bf(q1/2, q−1/2),\nwhere f is a positive unimodal polynomial. It was verified with computer\nthat all multiplicative structural constants are of this form. This verifies the\npositivity Conjecture 3.4 for the multiplicative structural constants as well.\n5.2\nKronecker case, H = SL2, r = 4\nFor the Kronecker case, H = Gl2 × GL2, G = GL4, and r = 4, we could\ncompute just one canonical basis element Σ (akin to Σ in Section 5.1) corre-\nsponding to the trivial one dimensional representation of BH\n4 (q). Symbolic\ncomputations needed to compute other canonical basis elements turned out\nto be beyond the scope of MATLAB/Maple on an ordinary workstation.\nThe coefficients of Σ in the Kazhdan-Luztig basis of H4(q)⊗H4(q) ⊃BH\n4 (q)\nwere computed in MATLAB/Maple. There are 576 coefficients in total. Fig-\nures 3-5 show the vectors associated with distinct nonzero coefficients among\nthese. They can be seen to be positive and nonincreasing in accordance with\nConjecture 3.4.\n29"},{"paragraph_id":"p30","order":30,"text":"(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n8\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n8 (1+q)2\nq\n8\n8 (1+q)2\nq\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n(q2+6 q+1)(1+q)2\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2"},{"paragraph_id":"p31","order":31,"text":"Figure 1: Coefficients of the elements of W1 and W2 in the symmetrized\nKazhdan-Lusztig basis, as computed in [GCT4]\n30"},{"paragraph_id":"p32","order":32,"text":"5.3\nH = sl2, G = sl4\nNow we study the nonstandard algebra B = BH\n3 (q), when H = Gl2, X is its\nfour dimensional irreducible representation, and G = GL(X) = Gl4. It is\na 21-dimensional algebra whose explicit presentation is given in Section 7.1\nof [GCT7]. We follow the notation as therein. Let Pi and Qi be as defined\nin the begining of that section. The monomials Pσ, where σ ranges over\nstrings in 1 and 2 of length k ≤10 with no consecutive 1’s or 2’s, form a\nbasis of B. This algebra has one trivial one-dimensinal representation, and\nfive nonisomorphic two-dimensional representations, so that\n21 = 1 + 22 + 22 + 22 + 22 + 22.\nLet\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(18)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x]\nbe the algebraic extension of Q(q) obtained by adjoining x. It is shown in\n[GCT7] that B admits a complete Wederburn-structure decomposition over\nK, but not Q(q). In what follows, we assume that B is defined over this\nbase field K.\nThe nonstandard canonical bases E(3) and Eopp(3) of B were computed\nin MATLAB/Maple using the algorithm in Section 3. They are as follows.\nLet Ui, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the\nmatrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni"},{"paragraph_id":"p33","order":33,"text":",\nwhere u1\n1 is as specified in Figure 6, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 7-9. Elements\nare specified in these figures by giving their nonzero coefficients in the {Qσ}\nbasis; the coefficient for Qσ is shown in front of σ. Let u2\ni , 1 ≤i ≤5, be the\nelement obtained from u1\ni by interchanging Q1 and Q2. Let u12\ni\n= u1\ni Q2,\nand u21\ni\n= Q2u1\ni , for 1 ≤i ≤5. Let u0 = 1 (this definition of u0 is different\nfrom that in [GCT7]). Then u0 and the entries of ui form the canonical\nbasis E(3) of BH\n3 (q). The left cells of E(e) are {u0} and the columns of ui.\nThe right cells are {u0} and the rows of ui. The representation supported\n31"},{"paragraph_id":"p34","order":34,"text":"by {u0} is the trivial one-dimensional representation; let us denote it by\nΣ. The representations supported by the columns or rows of ui are two-\ndimensional representations of B, distinct for each i; let us denote them by\nχi.\nThe left cell {u0} is at the top of the ≤L partial order and the left\ncells corresponding to the columns Ui are at depth 1 from the top in this\npartial order (and mutually incomparable). The situation for the right cells\nis similar.\nLet\nvi =\n v1\ni\nv12\ni\nv21\ni\nv2\ni"},{"paragraph_id":"p35","order":35,"text":",\nwhere vα\ni is obtained from uα\ni by substituting Pσ for Qσ in the expression of\nuα\ni in the {Qσ} basis. Let v0 be the element whose coefficients in the {Pσ}\nbasis are as shown in Figures 10-11. Then v0 and the elements of vi form the\nopposite canonical basis Eopp(3) of BH\n3 (q). The left cells are {v0} and the\ncolumns of vi. The right cells are {v0} and the rows of vi. The left cell {v0}\nis at the bottom of the ≤L partial order and the left cells corresponding to\nthe columns of vi at height 1 from the bottom (and mutually incomparable).\nThe situation for the right cells is similar.\nLet Σ′ be the trivial one-dimensional representation of the subalgebra\nBH\n2 (q) ⊂B = BH\n3 (q) generated by P2. Let μ′ be the other one-dimensional\nrepresentation of BH\n2 (q) that specializes to the signed one-dimensional rep-\nresentation of the symmetric group S2 at q = 1. Then the nonstandard\nGelfand-Tsetlin tableau T(b)’s associated with the basis elements b’s of E(3)\nare as follows.\nIf b = u0, then T(b) = [Σ, Σ′]. If b = u1\ni , then T(b) = [χi, Σ′]. If b = u21\ni ,\nthen T(b) = [χi, μ′].\nIf b = u2\ni , then T(b) = [χi, Σ′].\nIf b = u12\ni , then\nT(b) = [χi, μ′].\nThe nonstandard Gelfand-Tsetlin tableau associated with the basis ele-\nments in Eopp(r) are similar.\n5.3.1\nViolation of standard balance\nLet R(3) be the AK-lattice generated by E(3), ̄R(3) = (R(3))−. Let Ropp(3)\nand ̄Ropp(3) be defined similarly. Then it turns out that the triple (BH\n3,Q, R(3), ̄R(3))\nassociated with the canonical basis E(3) is balanced, but the triple (BH\n3,Q, Ropp(3), ̄Ropp(3))\nassociated with the opposite canonical basis Eopp(3) is not balanced. Specif-\nically, the fibre ψ−1\nT (0) ̸= {0} when T = T(b) is the nonstandard tableau\nassociated with b = v2\ni , for any i (it is zero for all other b’s).\n32"},{"paragraph_id":"p36","order":36,"text":"For example, with the help of computer it was found that the Q-module\nψ−1\nT (0), for b = v2\n5, T = T(b), is generated by the two elements w and x\nspecified in Figures 12-15, which give their nonzero cofficients in the {Pσ}\nbasis. Clearly w belongs to the KQ form BH\n3,Q since the coefficients belong\nto Q[q, q−1]. It is −-invariant, since the coefficients are −-invariant. It can\nbe verified that w ∈qRopp(3). Specifically, it can be shown that\nw = a0v0 + c1v1\n5 + c12v12\n5 + c2v2\n5 + c21v21\n5 ,\nwhere the coefficient vector [a0, c1, c12, c2, c21] is the following"},{"paragraph_id":"p37","order":37,"text":"−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)"},{"paragraph_id":"p38","order":38,"text":"This means ψT (w) = 0. It can be verified that w belongs to B≤T . Similarly,\nx is a −-invariant element of BH\n3,Q ∩B≤T that belongs to qRopp(3).\n5.4\nNonstandard globalization via minimization of degree\ncomplexity\nThus each element in ψ−1\nT ( ̄b), b = v2\n5, ̄b = ψ(b), T = T(b), is a linear combi-\nnation of b, w and x. It easy to see from the explicit formulae in Figures 12-15\nand Figure 9 that b = v2\n5 is the element of minimum degree complexity in\nψ−1\nT ( ̄b), where the degree complexity is defined internally as in Section 4.\n(Remember that v2\n5 is obtained from u1\n5 in Figure 9 by substituting Pi for Qi\nand then interchanging P1 and P2.) The same is also true for all b’s. This\nverifies the minimum degree conjecture (Conjecture 3.1); Conjecture 3.2 can\nalso be verified similarly.\nWe could not use the external definition of degree complexity as in Sec-\ntion 3.3 here, since the smallest product of Hecke algebras containing BH\n3 (q)\nis Z = H3(q)⊗9 with dimension 10077696 = 69. It is impossible to carry out\nsymbolic computations in an algebra of this size in MATLAB/Maple.\n33"},{"paragraph_id":"p39","order":39,"text":"c(γ1\n1; γ1\n1; γ1\n1) = c(γ12\n1 ; γ1\n1; γ1\n1) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n2; γ1\n1; γ1\n21) = −(q2 + q + 1) ∗(q −1)2/q2;\nc(γ1\n21; γ1\n1; γ1\n21) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n1; γ1\n1; Σ) = c(γ2\n1; γ1\n1; Σ) = c(γ1\n2; γ1\n1; Σ) = c(γ2\n2; γ1\n1; Σ) = 1/q ∗(1 + q)2;\nc(γ1\n12; γ1\n12; Σ) = c(γ1\n21; γ1\n12; Σ) = (1 + q)2 ∗(6 ∗q + 5 ∗q2 + 5)/q2;\nc(γ2\n21; γ2\n21; γ2\n21) = (1 + q2)2 ∗(q2 + q + 1) ∗(q −1)2/q4;\nc(γ2\n21; γ2\n21; Σ) = c(γ2\n12; γ2\n21; Σ) = (1 + q)2 ∗(2 ∗q2 + q + 1) ∗(q2 + q + 2)/q3.\nFigure 2: Multiplicative structural constants of the canonical basis of BH\n3 (q)\nin the Kronecker case, n = 2, r = 3\n34"},{"paragraph_id":"p40","order":40,"text":"10\n4\n3\n44\n21\n7\n22\n17\n7\n1\n50\n30\n15\n2\n20\n12\n4\n14\n7\n3\n44\n31\n14\n5\n19\n12\n4\n1\n6\n5\n3\n1\n94\n64\n29\n4\n88\n65\n28\n7\n39\n24\n8\n1\n80\n45\n17\n2\n40\n32\n16\n4\n28\n21\n10\n3\n75\n45\n19\n5\n38\n31\n16\n5\n1\n11\n8\n4\n1\n122\n69\n23\n2\n62\n49\n23\n5\nFigure 3: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (ignoring a positive, unimodal factor)\n35"},{"paragraph_id":"p41","order":41,"text":"126\n92\n47\n12\n2\n158\n93\n33\n4\n153\n93\n35\n7\n78\n63\n32\n9\n1\n160\n125\n62\n19\n2\n72\n48\n20\n4\n150\n120\n64\n24\n5\n69\n47\n21\n6\n1\n22\n19\n12\n5\n1\n212\n163\n74\n17\n244\n191\n92\n25\n2\n111\n72\n28\n5\n102\n71\n33\n9\n1\n104\n88\n52\n20\n4\n100\n85\n52\n22\n6\n1\n30\n23\n13\n5\n1\n128\n106\n59\n20\n3\n316\n251\n126\n37\n4\n306\n246\n128\n42\n7\n141\n95\n41\n10\n1\nFigure 4: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q), continued\n36"},{"paragraph_id":"p42","order":42,"text":"144\n120\n68\n24\n4\n41\n31\n17\n6\n1\n344\n213\n79\n12\n375\n237\n91\n17\n162\n117\n59\n19\n3\n222\n183\n100\n33\n5\n204\n173\n104\n42\n10\n1\n192\n140\n72\n24\n4\n60\n53\n36\n18\n6\n1\n220\n191\n124\n58\n18\n3\n264\n188\n92\n28\n4\n82\n72\n48\n23\n7\n1\n280\n244\n160\n76\n24\n4\n83\n66\n41\n19\n6\n1\n750\n612\n328\n108\n17\n345\n237\n107\n28\n3\n324\n279\n176\n78\n22\n3\n113\n89\n54\n24\n7\n1\n106\n96\n71\n42\n19\n6\n1\n528\n452\n280\n120\n32\n4\nFigure 5: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (continued)\n37"},{"paragraph_id":"p43","order":43,"text":"σ\nCoefficient\n1\n1/2 (q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(q2 + 1)4\n×(x + 3 q28 + 4 q24 −2 q22 + 10 q20 −2 q18 + 4 q16 + 3 q12)/q40\n121\n−1/2 (q2 + 1)2(2 q18 −295 q28 −516 q36 + x + 210 q26 + 3 q56 −3 q54 + 47 q46 + 9 q52 −q48\n+2 q50 −84 q24 −295 q40 + 604 q34 + 462 q30 −xq2 + 47 q22 −9 q20x + 19 q10x −q26x + q28x −3 q14\n+q24x + 4 q22x + 30 q14x + 462 q38 −516 q32 + 19 q18x + 210 q42 −q20 + 9 q16 −24 q16x −24 q12x\n−9 q8x + 4 q6x + q4x −84 q44 + 3 q12)/q36\n12121\n1/2 (q18 −2 q28 + 22 q36 + x + 45 q26 + 2 q46 + 3 q48 + 22 q24 + 24 q40 + 45 q34 + 92 q30 + 18 q22 + q20x\n+6 q10x + 2 q14 + q14x + 18 q38 −2 q32 + q42 + 24 q20 + 9 q16 + q16x + q6x + q4x + 9 q44 + 3 q12)/q30\n1212121\n−1/2 (22 q20 + 6 q16 + 6 q24 + 2 q26 + 2 q14 + 2 q30 + 2 q10 + 3 q28 −2 q22 −2 q18 + 3 q12 + x)/q20\n121212121\n1\nFigure 6: Coefficients of u1\n1\n38"},{"paragraph_id":"p44","order":44,"text":"σ\nCoefficient\n1\n(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4/q32\n121\n−(1 −4 q10 + 14 q8 −30 q14 + 44 q28 + 73 q16 + 3 q2 + 14 q32 −30 q26 + 73 q24 + 3 q38\n+102 q20 −53 q18 + q40 + 44 q12 −53 q22 −4 q30 + 5 q4 + 5 q36)(q2 + 1)2(q4 −q2 + 1)2/q26\n12121\n(3 + 72 q18 + 14 q28 + 3 q36 + 20 q26 + 10 q24 + 2 q34 + 2 q30 + 36 q22 + 14 q8 + 7 q4\n+2 q2 + 7 q32 + 2 q6 −10 q20 −10 q16 + 10 q12 + 20 q10 + 36 q14)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + q4 + 3 q2 + 4 q6 + q20 + 4 q14 + 3 q18 + q16)/q10\n121212121\n1\nFigure 7: Coefficients of u1\n3\n39"},{"paragraph_id":"p45","order":45,"text":"σ\nCoefficient\n1\n(q2 + 1)2(q4 −q2 + 1)2(q4 + 1)2(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2/q30\n121\n−(1 + 75 q18 −49 q28 + 42 q36 + 206 q26 −q46 + q52 + 7 q48 −49 q24 + 40 q40 + 75 q34 + 158 q30 + 158 q22\n+22 q8 + 7 q4 + 17 q14 + 17 q38 + q32 −q6 + q20 + 42 q16 + 22 q44 + 40 q12)q26\n12121\n(3 + 80 q18 + 10 q28 + 3 q36 + 26 q26 −3 q24 + q34 + 5 q30 + 52 q22 + 10 q8 + 5 q4 + 52 q14 + q2 + 5 q32\n+5 q6 −19 q20 −19 q16 −3 q12 + 26 q10)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + 3 q2 + 5 q6 + q20 + 5 q14 + 3 q18)/q10\n121212121\n1\nFigure 8: Coefficients of u1\n4\n40"},{"paragraph_id":"p46","order":46,"text":"σ\nCoefficient\n1\n(q2 + 1)2(q4 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q4 −q2 + 1)4/q26\n121\n−(q36 + 3 q34 + 10 q32 + 19 q30 + 33 q28 + 53 q26 + 64 q24 + 91 q22 + 84 q20 + 116 q18 + 84 q16 + 91 q14\n+64 q12 + 53 q10 + 33 q8 + 19 q6 + 10 q4 + 3 q2 + 1)(q4 −q2 + 1)2/q22\n12121\n(80 q16 + 3 q26 + 26 q24 + 4 q22 + q32 + 7 q28 + 50 q20 + 3 q6 + 3 q2 + 50 q12 + 1 + 3 q30 + 7 q4 −14 q18\n−14 q14 + 4 q10 + 26 q8)/q16\n1212121\n−(3 + 5 q12 −2 q10 + 14 q8 + 5 q4 + q2 −2 q6 + q14 + 3 q16)/q8\n121212121\n1\nFigure 9: Coefficients of u1\n5\n41"},{"paragraph_id":"p47","order":47,"text":"σ\nCoefficient\n∅\n(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n×(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2\n×(q4 + 1)2(q −1)4(q + 1)4(q2 + 1)4(q4 −q2 + 1)5/q46\n2\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n1\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n12\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n21\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30\n+2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q31\n121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52\n−77 q22 + 2 q48 −2 q46 −2 q44)/q31\n1212\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18\n+q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\n2121\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16\n−110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\nFigure 10: First nine coefficients of v0 in {Pσ} basis\n42"},{"paragraph_id":"p48","order":48,"text":"σ\nCoefficient\n21212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n12121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n212121\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n2121212\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n1212121\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n12121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n21212121\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n212121212\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n121212121\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n1212121212\n1\n2121212121\n1\nFigure 11: Last twelve coefficients of v0 in {Pσ} basis\n43"},{"paragraph_id":"p49","order":49,"text":"σ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(q4 + 1)2(q2 −q + 1)2(1 + q2 + q)2(q −1)4(q + 1)4(q2 + 1)2\n(q4 −q2 + 1)5)/q40\n2\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40\n−14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n1\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10\n+q40 −14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n12\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34\n−q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n21\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34 −q32\n−3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n1212\n((q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q20\nFigure 12: A −-invariant element w in qRopp(3): the first eight coefficients\n44"},{"paragraph_id":"p50","order":50,"text":"σ\nCoefficient\n2121\n(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)/q20\n21212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n12121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n121212\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n212121\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n2121212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n1212121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n12121212\n1\n21212121\n1\nFigure 13: A −-invariant element w in qRopp(3): the last nine coefficients\n45"},{"paragraph_id":"p51","order":51,"text":"σ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q2 + 1)3(q4 −q2 + 1)5)/q45\n2\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56\n−4 q58 + 12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n1\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56 −4 q58\n+12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n12\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8\n−q46 + q12 −2 q10 −3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50\n−3 q28 −8 q34 −10 q22 −q32 −3 q38 + q40))/q35\n21\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8 −q46 + q12 −2 q10\n−3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50 −3 q28 −8 q34 −10 q22\n−q32 −3 q38 + q40))/q35\n212\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n121\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n1212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26\n+25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\nFigure 14: A −-invariant element x in qRopp(3): the first eight coefficients\n46"},{"paragraph_id":"p52","order":52,"text":"σ\nCoefficient\n2121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\n21212\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n12121\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))/q15\n212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))//q15\n2121212\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n1212121\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n12121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n21212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n212121212\n−((q4 + 1)(q4 −q2 + 1))/q4\n121212121\n−((q4 + 1)(q4 −q2 + 1))/q4\nFigure 15: A −-invariant element x in qRopp(3): the last eleven coefficients\n47"},{"paragraph_id":"p53","order":53,"text":"5.4.1\nNonstandard positivity\nNow we shall describe evidence for Conjecture 3.6 in the case under consid-\neration.\nSo far we are assuming that Pi’s are as defined in the begining of Section\n7.1 in [GCT7]. But as explained towards the end of that section, the actual\nPi’s differ from these (chosen for convenience and simplicity) by a positive,\nunimodal factor ˆfp(q) ∈Q[q, q−1] given there. As it turns this does not\nmatter in the calculations so far, but for rescaling of the picture, but it does\nin the study of the positivity properties below. Rescaling of Pi by ˆfp(q)\nimplies that each structural coefficient or constant c(q) computed so far\nhas to be multiplied by an appropriate power of ˆfp(q)m(c), where m(c) is a\nnonnegative integer depending on c. In what follows, it is implicitly assumed\nthat each c computed so far has been rescaled by a suitable power ˆfp(q)n(c),\nwhere n(c) ≤m(c) is the smallest nonnegative integer chosen so that the\n(nonstandard) positivity property of the coefficients of c becomes apparent\nThe picture remains the same even if we were to multiply by ˆfp(q)m(c), but\nwe choose n(c) as small as possible to keep the the degrees of the polynomials\nfrom blowing up.\nFigures 16-20 show the vectors associated with the nonzero coefficients\nof v0 in the {Pσ} basis. The vector (as defined in the begining of Section 5)\nfor the coefficient corresponding to each σ is obtained by concatenating the\nrows in front of that σ. Figure 21 shows the vectors associated with the\ntraces of the nonzero coefficients of v1\n1 and Figures 22-23 show their norms.\nHere the trace and norm of an element in the algerbraic extension K of\nQ(q) is defined in the usual fashion as the sum and product of its images\nunder the Frobenius automorphisms of K over Q(q); they are coefficients\nof the minimal polynomial of the element. It can be seen that all vectors\nin Figures 16-23 are positive and nonincreasing, except for the vectors as-\nsociated with v0 for σ = 212, 121, 1212, 2121, which are almost positive and\nnonincreasing. It was verified with the help of computer that the vectors\nassociated with the coefficients of other canonical basis elements are simi-\nlarly either positive and nonincreasing or almost positive and nonincreasing.\nThis is in accordance with Conjectures 3.5-3.6.\nFigures 24-35 show vectors associated with a few multiplicative con-\nstants, taking norms and traces whenever necessary; the coefficient of bb′\nwith respect to the basis element b′′ is denoted by c(b, b′; b′′). Again it can\nbe seen that these vectors are positive and nonincreasing, except a few,\nwhich are almost positive and nonincreasing. It was verified with the help\n48"},{"paragraph_id":"p54","order":54,"text":"of computer that the picture is the same for other multiplicative constants\nas well. This too is in accordance with Conjectures 3.5-3.6.\n49"},{"paragraph_id":"p55","order":55,"text":"σ\nvector\n∅\n34116640\n34028832\n33766665\n33333910\n32736719\n31983492\n31084702\n30052720\n28901524\n27646408\n26303647\n24890162\n23423208\n21920062\n20397697\n18872456\n17359791\n15874002\n14428069\n13033484\n11700135\n10436190\n9248100\n8140602\n7116788\n6178174\n5324820\n4555450\n3867635\n3257976\n2722277\n2255718\n1853025\n1508640\n1216881\n972102\n768790\n601662\n465735\n356396\n269443\n201126\n148131\n107564\n76935\n54142\n37439\n25404\n16892\n10988\n6976\n4308\n2576\n1484\n821\n434\n217\n100\n40\n12\n2\n2\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\nFigure 16: The vectors associated with the coefficients of v0\n50"},{"paragraph_id":"p56","order":56,"text":"σ\nvector\n1\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\n12\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\nFigure 17: The vectors associated with the coefficients of v0 (continued)\n51"},{"paragraph_id":"p57","order":57,"text":"σ\nvector\n21\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\n212\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n121\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n1212\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\n2121\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\nFigure 18: The vectors associated with the coefficients of v0 (continued)\n52"},{"paragraph_id":"p58","order":58,"text":"σ\nvector\n21212\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n12121\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n121212\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n212121\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n2121212\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\nFigure 19: The vectors associated with the coefficients of v0 (continued)\n53"},{"paragraph_id":"p59","order":59,"text":"σ\nvector\n1212121\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\n12121212\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n21212121\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n212121212\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\n121212121\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\nFigure 20: The vectors associated with the coefficients of v0 (continued)\n54"},{"paragraph_id":"p60","order":60,"text":"σ\nvector\n1\n15864\n15864\n15680\n15496\n15160\n14824\n14334\n13844\n13228\n12612\n11932\n11252\n10505\n9758\n9008\n8258\n7534\n6810\n6126\n5442\n4834\n4226\n3702\n3178\n2738\n2298\n1948\n1598\n1333\n1068\n868\n668\n534\n400\n310\n220\n165\n110\n80\n50\n34\n18\n12\n6\n3\n121\n14690\n14690\n14484\n14278\n13930\n13582\n13065\n12548\n11880\n11212\n10509\n9806\n9032\n8258\n7498\n6738\n6031\n5324\n4684\n4044\n3498\n2952\n2515\n2078\n1729\n1380\n1124\n868\n698\n528\n405\n282\n212\n142\n101\n60\n42\n24\n15\n6\n3\n12121\n4974\n4974\n4886\n4798\n4683\n4568\n4380\n4192\n3914\n3636\n3367\n3098\n2803\n2508\n2217\n1926\n1669\n1412\n1205\n998\n825\n652\n529\n406\n316\n226\n172\n118\n89\n60\n41\n22\n14\n6\n3\n1212121\n708\n708\n694\n680\n673\n666\n642\n618\n565\n512\n464\n416\n365\n314\n262\n210\n170\n130\n106\n82\n62\n42\n30\n18\n11\n4\n2\nFigure 21: The vectors associated with the traces of the coefficients of v1\n1\n55"},{"paragraph_id":"p61","order":61,"text":"σ\nvector\n1\n942062408\n940617136\n936306916\n929157344\n919242745\n906637444\n891460752\n873831980\n853909410\n831851324\n807845688\n782080468\n754764061\n726104864\n696321516\n665632656\n634256120\n602409744\n570301452\n538139168\n506112609\n474411492\n443199968\n412642188\n382872865\n354026712\n326205988\n299512952\n274016717\n249786396\n226859750\n205274540\n185039618\n166163836\n148631230\n132425836\n117511844\n103853444\n91399832\n80100204\n69893842\n60720028\n52513000\n45206996\n38734986\n33029940\n28027078\n23661620\n19873490\n16602612\n13795174\n11397364\n9362691\n7644664\n6204292\n5002584\n4007841\n3188364\n2519160\n1975236\n1537471\n1186744\n908860\n689624\n518854\n386368\n285084\n207920\n150131\n106972\n75388\n52324\n35881\n24160\n16048\n10432\n6681\n4164\n2550\n1508\n874\n484\n262\n132\n64\n28\n12\n4\n1\n121\n835471628\n833946584\n829404381\n821877948\n811459617\n798241720\n782371271\n763995284\n743308085\n720504000\n695809491\n669451020\n641674579\n612726160\n582858876\n552325840\n521370802\n490237512\n459149620\n428330776\n397976239\n368281268\n339402663\n311497224\n284682092\n259074408\n234750212\n211785544\n190216267\n170078244\n151372598\n134100452\n118233841\n103744800\n90582172\n78694800\n68016456\n58480912\n50013272\n42538640\n35978861\n30255780\n25293821\n21017408\n17356415\n14240716\n11608322\n9397244\n7554889\n6028664\n4775532\n3752456\n2925473\n2260620\n1732150\n1314316\n988221\n734968\n541214\n393616\n283165\n200852\n140775\n97032\n65989\n44012\n28926\n18556\n11701\n7160\n4299\n2484\n1405\n752\n391\n188\n88\n36\n14\n4\n1\nFigure 22: The vectors associated with the norms of the coefficients of v1\n1\n56"},{"paragraph_id":"p62","order":62,"text":"12121\n100225178\n100001236\n99336179\n98236776\n96719406\n94800448\n92504960\n89858000\n86893481\n83645316\n80151751\n76451032\n72582611\n68585940\n64501436\n60369516\n56227388\n52112260\n48055958\n44090308\n40241933\n36537456\n32995876\n29636192\n26469934\n23508632\n20757113\n18220204\n15896382\n13784124\n11876674\n10167276\n8645972\n7302804\n6125315\n5101048\n4216788\n3459320\n2815838\n2273536\n1820593\n1445188\n1137127\n886216\n684162\n522672\n395267\n295468\n218449\n159384\n114887\n81572\n57144\n39308\n26622\n17644\n11492\n7284\n4521\n2704\n1573\n868\n461\n224\n102\n40\n15\n4\n1\n1212121\n2212002\n2205476\n2186350\n2155076\n2112378\n2058980\n1995675\n1923256\n1842975\n1756084\n1663849\n1567536\n1468102\n1366504\n1263855\n1161268\n1059834\n960644\n864459\n772040\n684050\n601152\n523831\n452572\n387494\n328716\n276117\n229576\n188839\n153652\n123592\n98236\n77108\n59732\n45640\n34364\n25488\n18596\n13346\n9396\n6493\n4384\n2894\n1848\n1142\n672\n377\n196\n95\n40\n15\n4\n1\nFigure 23: The vectors associated with the norms of the coefficients of v1\n1\n(continued)\n57"},{"paragraph_id":"p63","order":63,"text":"9846\n9820\n9750\n9644\n9501\n9320\n9093\n8812\n8493\n8152\n7779\n7364\n6917\n6448\n5966\n5480\n4989\n4492\n4001\n3528\n3080\n2664\n2274\n1904\n1569\n1284\n1038\n820\n631\n472\n346\n256\n186\n120\n69\n44\n31\n16\n4\nFigure 24: The vector for the multiplicative constant c(v0, v1\n5; v0)\n13026\n12964\n12777\n12464\n12048\n11552\n10964\n10272\n9523\n8764\n7990\n7196\n6398\n5612\n4867\n4192\n3569\n2980\n2444\n1980\n1583\n1248\n967\n732\n539\n384\n267\n188\n131\n80\n42\n24\n16\n8\n2\nFigure 25: The vector for the multiplicative constant c(v0, v1\n4; v0)\n14180\n14088\n13832\n13432\n12894\n12224\n11448\n10592\n9682\n8744\n7800\n6872\n5976\n5128\n4342\n3632\n2998\n2440\n1956\n1544\n1202\n928\n706\n520\n374\n272\n198\n136\n88\n56\n36\n24\n16\n8\n2\nFigure 26: The vector for the multiplicative constant c(v0, v1\n3; v0)\n58"},{"paragraph_id":"p64","order":64,"text":"1356922\n1356922\n1341857\n1326792\n1297628\n1268464\n1227083\n1185702\n1133960\n1082218\n1023821\n965424\n902616\n839808\n776306\n712804\n650849\n588894\n531320\n473746\n421861\n369976\n325182\n280388\n243022\n205656\n175677\n145698\n122582\n99466\n82260\n65054\n52971\n40888\n32575\n24262\n19002\n13742\n10480\n7218\n5402\n3586\n2573\n1560\n1107\n654\n431\n208\n135\n62\n35\n8\n4\nFigure 27: The vector for the trace of the multiplicative constant c(v0, v1\n1; v0)\n59"},{"paragraph_id":"p65","order":65,"text":"128607887512140\n128408739182416\n127813071267725\n126826189536408\n125456853005552\n123717149426056\n121622324577366\n119190568251368\n116442761279740\n113402187959064\n110094219510476\n106545974210632\n102785960441146\n98843708903656\n94749400393129\n90533495522016\n86226372383558\n81857978142544\n77457499613617\n73053057887224\n68671430858636\n64337807515464\n60075576278933\n55906149694176\n51848826289840\n47920690427296\n44136549385454\n40508906927184\n37047971392480\n33761696363504\n30655850888930\n27734116255008\n24998205742188\n22448003806144\n20081720818165\n17896059499880\n15886389332178\n14046925218184\n12370907081114\n10850777077832\n9478351722036\n8244986211000\n7141729026165\n6159464877872\n5289044837698\n4521402501856\n3847655805468\n3259194107520\n2747750779948\n2305461534304\n1924909254250\nFigure 28:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0)\n60"},{"paragraph_id":"p66","order":66,"text":"1599156102128\n1321763961652\n1086804278768\n888858554957\n723010747256\n584832826172\n470364742664\n376089969502\n298907782312\n236103258694\n185315973800\n144508153155\n111933043504\n86104032184\n65765045520\n49862540548\n37519404368\n28010898646\n20742786784\n15231640340\n11087321280\n7997549136\n5714462144\n4043023409\n2831123144\n1961213430\n1343311944\n909211769\n607734400\n400883734\n260758832\n167107927\n105406152\n65367982\n39805384\n23766532\n13889944\n7930388\n4412904\n2386644\n1250232\n631716\n306184\n141372\n61592\n25006\n9272\n3049\n848\n190\n32\n4\nFigure 29:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0) (continued)\n61"},{"paragraph_id":"p67","order":67,"text":"612764\n609940\n601539\n587774\n568990\n545654\n518332\n487666\n454355\n419136\n382753\n345926\n309336\n273610\n239300\n206862\n176658\n148958\n123938\n101678\n82176\n65362\n51106\n39226\n29503\n21696\n15554\n10828\n7279\n4686\n2851\n1604\n800\n316\n52\n−68\n−101\n−90\n−64\n−38\n−18\n−6\n−1\nFigure 30: The vector for the multiplicative constant c(v1\n5, v1\n5; v1\n5)\n9738\n9694\n9563\n9348\n9052\n8676\n8228\n7732\n7209\n6658\n6077\n5484\n4900\n4332\n3784\n3268\n2794\n2360\n1962\n1604\n1292\n1028\n809\n626\n471\n344\n246\n172\n116\n76\n50\n32\n17\n6\n1\nFigure 31: The vector for the multiplicative constant c(v1\n4, v1\n4; v1\n4)\n5.5\nExperimental evidence for crystalization\nLet (Lρ\nα, Bρ\nα) be an upper crystal basis of W ρ\nq,α as in Conjecture 2.2. Since\nit is also a local crystal basis of W ρ\nq,α as an Hq-module, there is a crystal\ngraph over Bρ\nα whose connected components correspond to irreducible Hq-\nsubmodules of W ρ\nq,α. The elements of Bρ\nα that correspond to the highest\nweight nodes of these connected components are called the highest weight\ncrystal elements of the upper crystal base (Lρ\nα, Bρ\nα) with respect to Hq.\n5.6\nKronecker problem: n = 2, r = 3\nConsider again the special case of the Kronecker problem for n = 2 and\nr = 3 as in Section 5.1. Thus H = Gl2 × Gl2, G = Gl4 with H embedded\ndiagonally, X = Xq = Vq ⊗Wq is the standard four dimensional represen-\ntation of Hq, where Vq ∼= Wq is the standard two representation of GLq(2).\n62"},{"paragraph_id":"p68","order":68,"text":"81298\n80886\n79660\n77650\n74908\n71510\n67546\n63110\n58306\n53254\n48074\n42870\n37740\n32786\n28097\n23732\n19734\n16144\n12987\n10258\n7938\n6010\n4449\n3212\n2251\n1526\n999\n628\n374\n208\n107\n50\n20\n6\n1\nFigure 32: The vector for the multiplicative constant c(v1\n3, v1\n3; v1\n3)\n976672\n974152\n971632\n954184\n936736\n915476\n894216\n860888\n827560\n791463\n755366\n713035\n670704\n627743\n584782\n540877\n496972\n454258\n411544\n373144\n334744\n298301\n261858\n232239\n202620\n175584\n148548\n128938\n109328\n91580\n73832\n62540\n51248\n41348\n31448\n25972\n20496\n15650\n10804\n8712\n6620\n4729\n2838\n2215\n1592\n986\n380\n312\n244\n113\n−18\n−9\n0\n−15\n−30\n−15\nFigure 33: The vector for the trace of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n63"},{"paragraph_id":"p69","order":69,"text":"80255448992640\n80138274925167\n79787740098774\n79206792680861\n78400301248652\n77374989989221\n76139349486502\n74703523453335\n73079174468928\n71279332267508\n69318226392392\n67211105097154\n64974044241996\n62623750255631\n60177360010954\n57652240221271\n55065789702876\n52435247936361\n49777512793214\n47108969815061\n44445335040340\n41801513256271\n39191473712706\n36628144773627\n34123327701708\n31687629536553\n29330415837966\n27059783595895\n24882552858520\n22804275396566\n20829260005316\n18960613784310\n17200296928344\n15549188413159\n14007161465134\n12573167695619\n11245327257264\n10021022340390\n8896992772108\n7869432671924\n6934086054972\n6086339325247\n5321309766266\n4633929430149\n4019023248184\n3471380269320\n2985817677720\n2557237529092\n2180675978000\n1851344837246\n1564665615708\nFigure 34: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n64"},{"paragraph_id":"p70","order":70,"text":"1316296386448\n1102151942468\n918417720699\n761557915658\n628318302423\n515724544156\n421076738262\n341940657408\n276136144552\n221723424848\n176988052223\n140424814566\n110720869883\n86738676456\n67499233418\n52165764220\n40027907728\n30486730148\n23040841355\n17273591498\n12841266667\n9462395940\n6908272529\n4994579014\n3573971135\n2529616304\n1769689525\n1222700770\n833511103\n559989972\n370279871\n240567010\n153252303\n95475892\n57962133\n34120478\n19338377\n10435196\n5256321\n2374094\n864025\n140844\n−155240\n−236220\n−220838\n−171968\n−120043\n−77166\n−46045\n−25460\n−12974\n−6040\n−2518\n−900\n−255\n−50\n−5\nFigure 35: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n(continued)\n65"},{"paragraph_id":"p71","order":71,"text":"Let\nx1 = v1 ⊗w1, x2 = v1 ⊗w2, x3 = v2 ⊗w1, x4 = v2 ⊗w2,\nbe the standard basis of Xq. Let b1, . . . , b4 be the corresponding standard\ncrystal basis of L(Xq). The irreducible representations of GH\nq that occur in\nX⊗3\nq\nare:\n1. CH,3\nq\n(X), the 16-dimensional the degree-three component of the braided\nsymmetric algebra of GH\nq ,\n2. ∧H,3\nq\n(X), the four dimensional degree three component of the braided\nexterior algebra of GH\nq ,\n3. two copies of the 16-dimensional GH\nq -representation Wq,(2,1),1(X) de-\nfined in [GCT4] (it is denoted by Vq,(2,1),1(X) there), and\n4. two copies of the 4-dimensional representation Wq,(2,1),2(X) of GH\nq as\nalso defined there (it is called Vq,(2,1),2(X) there).\nEmbeddings of the braided symmetric and exterior algebra components in\nX⊗3\nq\nare uniquely defined. We denote their embedded images by CH,3\nq\n(X)\nand ∧H,3\nq\n(X) again.\nWe choose appropriate embeddings of Wq,(2,1),2(X)\nand Wq,(2,1),2(X) in X⊗3\nq\nand denote them by the same symbols again. As\nHq = GLq(V ) × GLq(W)-modules,\nCH,3\nq\n(X)\n=\nVq,(3)(V ) ⊗Vq,(3)(W) ⊕Wq,(2,1)(V ) ⊗Vq,(2,1)(W),\n∧H,3\nq\n(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W),\nWq,(2,1),1(X)\n=\nVq,(2,1)(V ) ⊗Vq,(3)(W) ⊕Vq,(3)(V ) ⊗Vq,(2,1)(W)\nWq,(2,1),2(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W).\nIt was verified by computer that they have upper crystal bases as per\nConjecture 2.2.\nThe highest weight crystal elements with respect to Hq\nfor these upper crystal bases, as shown separately for each module, are as\nfollows; we denote the monomial basis element bi1 ⊗bi2 ⊗bi3 of B(X⊗3\nq ) by\nbi1i2i3.\n66"},{"paragraph_id":"p72","order":72,"text":"CH,3\nq\n(X) :\n{b114 + b141, b111}.\n∧H,3\nq\n(X) : {b123 + b132}.\nWq,(2,1),1(X) :\n{b121, b131}.\nWq,(2,1),2(X) :\n{b114 −b141}.\nThe highest weight crystal elements whose monomial support have size\ntwo correspond to the four dimensional Hq-module Vq,(2,1)(V ) ⊗Vq,(2,1)(W).\nThe element b111 corresponds to the Hq-module Vq,(3)(V ) ⊗Vq,(3)(W), the\nelement b121 to the Hq-module Vq,(3)(V ) ⊗Vq,(2,1)(W), and b131 to the Hq-\nmodule Vq,(2,1)(V ) ⊗Vq,(3)(W). Notice that not all highest weight crystal\nelements have monomial supports of size one as in the standard setting.\n5.6.1\nH = sl2, G = sl4\nNow we consider the case when Hq = Glq(2), Xq its four dimensional irre-\nducible representation, and GH\nq\nas in Section 5.3. Let W0, . . . , W5 be the\nirreducible representations of GH\nq occuring in X⊗3\nq\nas defined in Section 6.1.2\nof [GCT7], with W0 = CH,3\nq\n[X].\nAs Hq-modules,\nW0\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2),\nW1\n∼=\nVq,(6,3)(2),\nW2\n∼=\nVq,(6,3)(2),\nW3\n∼=\nVq,(8,1)(2),\nW4\n∼=\nVq,(5,4)(2),\nW5\n∼=\nVq,(7,2)(2),\n(19)\nwhere Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to the\npartition λ. Their dimensions are 16, 4, 4, 8, 2 and 6, respectively. Though\nW1 and W2 are isomorphic as Hq-modules, they are nonisomorphic as GH\nq -\nmodules.\nIt was verified by computer that they–or rather their embeddings in\nX⊗3\nq –have upper crystal bases as per Conjecture 2.2. The highest weight\ncrystal elements of of the embedding of W1, . . . , W5 have monomial supports\nof size one. Let bi1i2i3 = bi1 ⊗bi2 ⊗bi3 denote the monomial basis elements of\nB(X⊗3\nq ). Then the highest weight crystal elements of the uniquely defined\nembedding of W0 are b111 and b = b113 + b131. The latter element b here\nhas monomial support of size two, a phenomenon not seen in the standard\nsetting.\nNonzero coefficients of the element x in the lattice L(C3,H\nq\n(X))\n67"},{"paragraph_id":"p73","order":73,"text":"whose crystalization is b is shown in Figure 36, wherein x1, . . . , x4 are the\nstandard basis vectors of Xq.\n6\nComplexity theoretic properties of the canonical\nbasis\nIn the standard setting, elements of the canonical basis of Vq,λ are indexed\n(labelled) by semistandard tableau, for H = GL(V ), and by LS-paths [Li],\nfor general semisimple H. And combinatorial analogues of Kashiwara’s crys-\ntal operators [Li] on these labels can be computed efficiently [GCT6]. This\nis enough to imply a #P-formula for the generalized Littlewood-Richardson\ncoefficient though the canonical basis of Vq,λ is hard to compute.\nIn the same spirit, it may be conjectured that the canonical basis of\nO(MH\nq (X) (or rather the set of its labels) has additional complexity the-\noretic properties (to be described in the full version), based on its cellular\nand refined sub-cellular decomposition (Conjecture 2.6), that imply a pos-\nitive #P-formula for the multiplicity nα\nπ of the irreducible Hq-module Vq,π\nin Wq,α. This would solve the problem P1 in [GCT7].\nSimilarly, let mα\nλ denote the multiplicity of the Specht module Sλ of\nthe symmetric group Sr corresponding to the partition λ in Limq→1Tq,α,\nconsidered as an Sr-module. It may be conjectured that the canonical basis\nBH\nr (q) (or rather the set of its labels) has similar additional complexity\ntheoretic properties (to be described in the full version) based on its cellular\nand quasi-subcellular decompositions (Conjecture 3.3). This would imply a\npositive #P-formula for the multiplicity mπ\nλ, as needed in the problem P2\nin [GCT7].\nReferences\n[BBD]\nA.\nBeilinson,\nJ.\nBernstein,\nP.\nDeligne,\nFaisceaux\npervers,\nAst ́erisque 100, (1982), Soc. Math. France.\n[BZ]\nA. Berenstein, S. Zwicknagl, Braided symmetric and exterior alge-\nbras, arXiv:math/0504155v3, April, 2007.\n[DJM]\nM. Date, M. Jimbo, T. Miwa, Representations of Uq(ˆgl(n, C)) at\nq = 0 and the Robinson-Schensted correspondence, in Physics and\n68"},{"paragraph_id":"p74","order":74,"text":"Monomial\nCoefficient\nx1 ⊗x1 ⊗x3\n−(q4 + 1)2(q2 + 1)4(q4 −q2 + 1)5(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q20 −2 q18 + q16 + q10 −q8 + q6 + q4 −q2 + 1)\n(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\nx1 ⊗x2 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q37\nx1 ⊗x3 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q16 + q14 −q12 + q10 + q4 −q2 + 1)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q46\nx2 ⊗x1 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q40\nx2 ⊗x2 ⊗x1\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q4 −q3 + q2 −q + 1)\n(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5(q4 −q2 + 1)5/q43\nx3 ⊗x1 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5\n+q4 + q3 + q2 + q + 1)(q12 −q10 + q8 + q6 + q4 −q2 + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q42\nFigure 36: Nonzero coefficients of x in the lattice L(C3,H\nq\n(X))\n69"},{"paragraph_id":"p75","order":75,"text":"Mathematics of Strings, World Scientific, Singapore, 1990, pp. 185-\n211.\n[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n[Dri]\nV. Drinfeld, Quantum groups, Proc. Int. Congr. Math. Berkeley,\n1986, vol. 1, Amer. Math. Soc. 1988, 798-820.\n[GCTflip1] K. Mulmuley, On P. vs. NP, geometric complexity theory, and\nthe flip I: a high-level view, Technical Report TR-2007-13, Com-\nputer Science Department, The University of Chicago, September\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu\n[GCT4] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\nthe-\nory\nIV:\nquantum\ngroup\nfor\nthe\nKronecker\nproblem,\ncs.\nArXiv preprint cs. CC/0703110,\nMarch,\n2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT6] K. Mulmuley, Geometric complexity theory VI: the flip via sat-\nurated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\nSci. Dept., The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: Nonstandard\nquantum group for the plethysm problem, technical report TR-\n2007-14, computer science dept., The university of Chicago, Sept.\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[Ji]\nM. Jimbo, A q-difference analogue of U(G) and the Yang-Baxter\nequation, Lett. Math. Phys. 10 (1985), 63-69.\n[Kas1]\nM. Kashiwara, Crystalizing the q-analogue of universal enveloping\nalgebras, Comm. Math. Phys. 133 (1990), 249-260.\n[Kas1]\nM. Kashiwara, On crystal bases of the q-analogue of universal en-\nveloping algebras, Duke Math. J. 63 (1991), 465-516.\n[Kas2]\nM. Kashiwara, Global crystal bases of quantum groups, Duke\nMathematical Journal, vol. 69, no.2, 455-485.\n[KL1]\nD. Kazhdan, G. Lusztig, Representations of Coxeter groups and\nHecke algebras, Invent. Math. 53 (1979), 165-184.\n70"},{"paragraph_id":"p76","order":76,"text":"[KL2]\nD. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality,\nProc. Symp. Pure Math., AMS, 36 (1980), 185-203.\n[Kli]\nA. Klimyk, K. Schm ̈udgen, Quantum groups and their representa-\ntions, Springer, 1997.\n[Li]\nP. Littelmann, Paths and root operators in representation theory,\nAnn. of Math. 142 (1995), 499-525.\n[Lu1]\nG. Lusztig, Canonical bases arising from quantized enveloping al-\ngebras, J. Amer. Math. Soc. 3, (1990), 447-498.\n[Lu2]\nG. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[So]\nW. Soergel, Kazhdan-Lusztig polynomials and a combinatoric for\ntilting modules, Representation theory 1, (1997)\n[St]\nR. Stanley, Positivity problems and conjectures in algebraic com-\nbinatorics, In Mathematics: frontiers and perspectives, 295-319,\nAmer. Math. Soc. Providence, RI (2000).\n[W]\nS. Woronowicz: Compact matrix pseudogroups, Commun. Math.\nPhys. 111 (1987), 613-665.\n71"}],"pages":[{"page":1,"text":"arXiv:0709.0751v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VIII: On canonical\nbases for the nonstandard quantum groups\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley\nThe University of Chicago\n(Technical Report TR-2007-15\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nOctober 27, 2018\nAbstract\nThis article gives conjecturally correct algorithms to construct canon-\nical bases of the irreducible polynomial representations and the matrix\ncoordinate rings of the nonstandard quantum groups in GCT4 and\nGCT7, and canonical bases of the dually paired nonstandard deforma-\ntions of the symmetric group algebra therein. These are generalizations\nof the canonical bases of the irreducible polynomial representations\nand the matrix coordinate ring of the standard quantum group, as\nconstructed by Kashiwara and Lusztig, and the Kazhdan-Lusztig ba-\nsis of the Hecke algebra. A positive (#P-) formula for the well-known\nplethysm constants follows from their conjectural properties and the\nduality and reciprocity conjectures in [GCT7].\n1"},{"page":2,"text":"1\nIntroduction\nLet H be a complex connected classical reductive group, X = Vμ(H) its\nirreducible polynomial representation with highest weight μ, G = GL(X),\nand ρ : H →G the representation map. Given a highest weight π of H\nand λ of G, the plethysm constant aπ\nλ,μ is defined to be the multiplicity of\nVπ(H) in Vλ(G), considered an H-module via ρ. A fundamental problem\nin representation theory is to find a positive (#P-) formula (rule) for the\nplethysm constant [GCT7, St] akin to the Littlewood-Richardson rule. Mo-\ntivated by this problem, the article [GCT7] constructs a quantization ρq of\nthe homomorphism ρ in the form\nρq : Hq →GH\nq ,\n(1)\nwhere Hq is the standard (Drinfeld-Jimbo) quantum group [Dri, Ji, RTF]\nassociated with H and GH\nq is the new (possibly singular) quantum group,\ncalled the nonstandard quantum group associated with ρ. In the standard\ncase, i.e., when H = G, this specializes to the standard quantum group, and\nin the Kronecker case, i.e., when H = GL(V ) × GL(W), X = V ⊗W with\nthe natural H action, this specializes to the nonstandard quantum group in\n[GCT4]. Also constructed in [GCT7] is a nonstandard quantization BH\nr (q)\nof the group algebra C[Sr], Sr the symmetric group, whose relationship with\nGH\nq is conjecturally similar to that of the Hecke algebra with the standard\nquantum group.\nThis article gives conjecturally correct algorithms for constructing canon-\nical bases of the irreducible polynomial representations and the matrix co-\nordinate ring of GH\nq (Section 2) and a canonical basis of BH\nr (q) (Section 3).\nWe call these nonstandard canonical bases. They are generalizations of the\ncanonical bases of the irreducible polynomial representations and the matrix\ncoordinate ring of the standard quantum group, as constructed by Kashi-\nwara and Lusztig [Kas1, Kas2, Lu1, Lu2], and the Kazhdan-Lusztig basis\n[KL1] of the Hecke algebra. A positive (#P-) formula for the plethysm con-\nstant follows from their conjectural properties (Sections 2.4 and 3.4), which\nare akin to those of the standard canonical basis, and the conjectural duality\nand reciprocity between GH\nq and BH\nr (q); cf. [GCT7].\nExperimental evidence (Section 5) suggests that these algorithms should\nbe correct. But we can not prove this formally, nor the required properties of\nthe nonstandard canonical bases. Mainly because we are unable to deal with\nthe complexity of the minors of the nonstandard quantum group. Specifi-\ncally, in contrast to the elementary formula for the Laplace expansion of a\n2"},{"page":3,"text":"minor of the standard quantum group–which is akin to the classical Laplace\nexpansion at q = 1–the Laplace expansion of a minor of a nonstandard\nquantum group is highly nonelementary; cf. [GCT7]. Its coefficients de-\npend on the multiplicative structural constants of a canonical basis akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nas per Kashiwara and Lusztig. In the Kronecker case, these constants are\nconjecturally polynomials in Q[q, q−1] with nonnegative coefficients, and in\ngeneral, polynomials with a conjectural relaxed form of this property. To\nprove this and to get explicit formulae for the minors in the nonstandard\nsetting, one needs explicit interpretations for these structural constants in\nthe spirit of the interpretation based on perverse sheaves for the Kazhdan-\nLusztig polynomials [KL2] and the multiplicative structural constants of the\ncanonical basis of the Drinfeld-Jimbo enveloping algebra [Lu2]. Thus even\nto get explicit formulae for the minors of the nonstandard quantum group\na (nonstandard) extension of the theory of perverse sheaves [BBD], and the\nunderlying Riemann hypothesis over finite fields [Dl2] seems necessary.\nMinors of the standard quantum group are in a sense the simplest (basic)\ncanonical basis elements in its matrix coordinate ring. That the simplest\ncanonical basis elements for the nonstandard quantum group–namely, its\nminors–are already so nonelementary in contrast to the standard case indi-\ncates the possible difficulties that may be encountered in proving correctness\nof the algorithms given here for constructing nonstandard canonical bases.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for the help in explicit\ncomputations in MATLAB.\nNotation: We use the symbols π and μ to denote labels of irreducible\nrepresentations of the standard quantum group and the symbols α, α0, α1, . . .\nto denote labels of irreducible representations of the nonstandard quantum\ngroup. Thus objects with subscripts π and μ are standard and the objects\nwith subscripts α, α0, . . . are nonstandard.\n2\nNonstandard canonical basis for GH\nq\nIn this section we describe a conjecturally correct algorithm for constructing\nthe canonical basis of the matrix coordinate ring of the nonstandard quan-\ntum group GH\nq . We follow the same terminology as in [Kli] for the basic\nquantum group notions.\nFor the sake of simplicity, let us assume that H = GL(V ). Let Hq =\n3"},{"page":4,"text":"GLq(V ) denote its standard (Drinfeld-Jimbo) quantization [Dri, Ji, RTF],\nand Mq(V ) the standard quantization of the matrix space M(V ).\nLet\nO(Mq(V )) be the coordinate ring of Mq(V ). We call it the matrix coordinate\nring of GLq(V ). The coordinate ring O(GLq(V )) of GLq(V ) is obtained by\nlocalizing O(Mq(V )) at the quantum determinant of GLq(V ). Let H denote\nthe Lie algebra of H, and Uq(H) the Drinfeld-Jimbo universal enveloping\nalgebra of Hq = GLq(V ).\nTo quantize the homomorphism ρ : H →G = GL(X) as in (1), the arti-\ncle [GCT7] constructs a nonstandard matrix coordinate ring O(MH\nq (X)) of\na (virtual) nonstandard matrix space MH\nq (X), and then defines the nonstan-\ndard quantized universal enveloping algebra U H\nq (G) by dualization. The non-\nstandard quantum group GH\nq is the virtual object whose universal enveloping\nalgebra is U H\nq (G). The construction also yields natural bialgebra homomor-\nphisms from Uq(H) to U H\nq (G) and from O(MH\nq (X)) to O(Mq(V ), thereby\ngiving the desired quantizations of the homomorphisms U(H) →U(G)\nand O(M(X)) →O(M(V )). This is what is meant by the quantization\n(1) of the representation map ρ.\nThe determinant of GH\nq\nmay vanish,\nand hence, we cannot, in general, define its coordinate ring O(GH\nq ) by lo-\ncalizing O(MH\nq (X)). Fortunately, this does not matter since O(MH\nq (X))\nstill has properties similar to that of the standard matrix coordinate ring\nO(Mq(V )). Specifically, it is cosemisimple. This means all (finite dimen-\nsional) polynomial representations of GH\nq , by which we mean corepresen-\ntations of O(MH\nq (V )), are completely reducible. A nonstandard quantum\nanalogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\n(2)\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)). Fur-\nthermore, the nonstandard enveloping algebra U H\nq (G) is a bialgebra with a\ncompact real form (∗-structure).\nThe goal is to construct a canonical basis for the matrix coordinate ring\nO(MH\nq (X)) akin to the canonical basis of the standard matrix coordinate\nring O(Mq(V )) as per Kashiwara and Lusztig [Kas1, Kas2, Lu1, Lu2].\n2.1\nThe standard setting\nWe begin by reviewing the basic scheme of Kashiwara and Lusztig for con-\nstructing a canonical basis of the matrix coordinate ring O(Mq(V )). The\n4"},{"page":5,"text":"canonical basis of the coordinate ring O(GLq(V )) is obtained by localizing\nat the determinant.\nFollowing Kashiwara, we first define a balanced triple. Let A and ̄A be\nthe ring of rational functions in q regular at q = 0 and q = ∞, respectively.\nLet V be a Q(q)-vector space, L0 a sub-A-module (A-lattice) of V , L∞a\nsub- ̄A-module ( ̄A-lattice) of V , and VQ a sub-Q[q, q−1]-module of V such\nthat\nV ∼= Q(q) ⊗Q[q,q−1] VQ ∼= Q(q) ⊗A L0 ∼= Q(q) ⊗ ̄\nA L∞.\nWe say that (VQ, L0, L∞) is a balanced triple if any of the following three\nequivalent conditions hold:\n(a) E = VQ ∩L0 ∩L∞→L0/qL0 is an isomorphism.\n(b) E →L∞/q−1L∞is an isomorphism.\n(c) Q[q, q−1] ⊗Q E →VQ, A ⊗Q E →L0, ̄A ⊗Q E →L∞are isomorphisms.\nLet R = O(Mq(V )).\nKashiwara constructs an A-submodule (lattice)\nL = L(R) ⊂R, an involution −of R, a Q[q, q−1]-submodule RQ ⊂R, and\na basis B of L/qL such that (RQ, L, ̄L) is a balanced triple and, letting G\ndenote the inverse of the isomorphism RQ ∩L ∩ ̄L →L/qL, {G(b) | b ∈B}\nis the canonical basis of R. The pair (L, B), called the (upper) crystal base\nof R, has the following form. By the q-analogue of the Peter-Weyl theorem\nfor the standard quantum group,\nR = ⊕πV ∗\nq,π ⊗Vq,π\n(3)\nas a bi-GLq(V )-module, where Vq,π = Vq,π(V ) is the irreducible polynomial\nrepresentation of GLq(V ) with highest weight π. Let (Lπ, Bπ) denote the\nupper crystal base of Vq,π. Then\n(L, B) = ⊕π(L∗\nπ, B∗\nπ) ⊗(Lπ, Bπ)\n(4)\nwith appropriate normalization.\nWe now describe a construction of the upper crystal base (Lπ, Bπ) that\ncan be generalized to the nonstandard setting.\nLet r be the size of the partition π. Choose any embedding ρ : Vq,π →\nV ⊗r such that the highest weight vector of the image V ρ\nq,π = ρ(Vq,π) be-\nlongs to the A-lattice L(V )⊗r of V ⊗r, where L(V ) denotes the lattice of\nV generated by its standard basis {vi}. We also assume that the highest\nweight vector does not belong to qL(V )⊗r. Choose a Hermitian form on\nV ⊗r so that its monomial basis {vi1 ⊗· · · vir} is orthonormal. Let V ρ,⊥\nq,π de-\nnote the orthogonal complement of V ρ\nq,π. Since GLq(V ) has a compact real\n5"},{"page":6,"text":"form Uq(V )–i.e., the unitary compact subgroup in the sense of Woronowicz\n[W]–it follows that V ρ,⊥\nq,π is a GLq(V )-module. Thus V ⊗r = V ρ\nq,π ⊕V ρ,⊥\nq,π as a\nGLq(V )-module. Let\nLρ\nπ = L(V )⊗r ∩V ρ\nq,π\nand\nLρ,⊥\nπ\n= L(V )⊗r ∩V ρ,⊥\nq,π .\nIt follows from Kashiwara’s work [Kas1] that L(V )⊗r = Lρ\nπ ⊕Lρ,⊥\nπ\n.\nLet B(V ) = {bi = ψ(vi)} denote the basis of L(V )/qL(V ), where ψ :\nL(V ) →L(V )/qL(V ) is the natural projection. Let B(V )⊗r = {bi1 ⊗· · · ⊗\nbir} denote the monomial basis of B(V )⊗r. Given b ∈B(V )⊗r, let\nb =\nX\ni1,...,ir\nf(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients f(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\nIt follows from the works of Kashiwara [Kas1] and Date et al [DJM]\nthat Lρ\nπ/qLρ\nπ has a unique basis Bρ\nπ (up to scaling by constant multiples)\nsuch that the monomial supports of its elements are disjoint–in fact, one\ncan choose ρ so that the monomial support of each basis element consists of\njust one distinct monomial. This basis can be made completely unique by\nappropriate normalization. Then (Lρ\nπ, Bρ\nπ) coincides with the upper crystal\nbase of Vq,π as constructed by Kashiwara. Furthermore, this crystal base\ndoes not depend on the embedding ρ (up to isomorphism). Hence, we let\n(Lπ, Bπ) = (Lρ\nπ, Bρ\nπ) for any ρ as above. Kashiwara [Kas1] also shows that\nLπ and Bπ ∪{0} are invariant under certain crystal operators ̃ei and ̃fi\ncorresponding to the simple roots of Hq.\nThis scheme for constructing the upper crystal base (Lπ, Bπ) crucially\ndepends on the existence of a compact real form Uq(V ) ⊆GLq(V ) = Hq.\nEven the existence of a compact real form of the standard Drinfeld-Jimbo\nenveloping algebra Uq(H) suffices here.\n2.2\nNonstandard triple\nWe now generalize the preceding scheme to the nonstandard setting using\nthe compact real form of U H\nq (G), whose existence is proved in GCT7. The\ngoal is to construct an analogous triple for the matrix coordinate ring S =\nO(MH\nq (X)) of GH\nq . It will turn out that this triple need not be balanced as\n6"},{"page":7,"text":"in the standard case. We shall describe in Section 2.3 how a canonical basis\ncan be constructed from such a triple despite the lack of balance.\nWe begin by recalling that the q-analogue of the Peter-Weyl theorem (2)\nin the nonstandard setting need not hold over Q(q) unlike in the standard\nsetting. It holds only over an appropriate algebraic extension K of Q(q)\n[GCT7]–thinking of q as a transcendental. It will be convenient to assume\nin what follows that K is actually an algebraic extension ̃Q(q), where ̃Q is\nthe algebraic closure of Q. We let AK and ̄AK be the subrings of algebraic\nfunctions in K that are regular at q = 0 and q = ∞, respectively. Let KQ\nbe the integral closure of Q[q, q−1] in K. Clearly, KQ ∩AK ∩ ̄AK = ˆQ, where\nˆQ denotes the integral closure of Q in ̃Q. In what follows, we let ˆQ, AK, ̄AK\nand KQ play the role of Q, A, ̄A and Q[q, q−1] in Section 2.1. Thus by a\nlattice at q = 0 we mean an AK-lattice, by a lattice at q = ∞a ̄AK-lattice,\nby a Q-form, a KQ-form (module).\nSimilarly, instead of Q-modules and\nQ(q)-modules, we will be considering ˆQ-modules and K-modules. That is,\nin what follows we shall assuming that the underlying base field is K, instead\nof Q(q).\nNow let us describe the construction of the upper crystal base of an\nirreducible polynomial representation Wq,α of GH\nq . Let Xq = Vq,μ denote the\nquantization of X = Vμ; i.e, the Hq-module with highest weight μ. Since\nthe underlying field is K, Xq is a K-module. Let {zi} denote its standard\nq-orthonormal Gelfand-Tsetlin basis. Let {xi} denote the rescaled version\nof Gelfand-Tsetlin basis (as described in Section 7.3.3 of [Kli]) so that there\nare no square roots in the explicit formulae for the action of the generators\nof the Drinfeld-Jimbo algebra UqH on this basis. Alternatively, we can let\n{xi} be the standard (upper) canonical basis [Kas1, Lu2] of Xq as an Hq-\nmodule. We assume that limq→0(zi −xi) = 0; this can always be arranged.\nIn what follows, we sometimes denote Xq by X. What is meant should be\nclear from the context. It is shown in [GCT7] that Wq,α can be embedded\nin X⊗r for an appropriate r. We choose an embedding ρ : Wq,α →X⊗r as\nfollows.\nIf r = 1, Wq,α = X, so this is trivial. Otherwise, choose any β of degree\nr−1 so that Wq,α occurs as a GH\nq -submodule of Wq,β⊗X. By semisimplicity,\nthe latter is completely reducible as a GH\nq -module:\nWq,β ⊗X = ⊕βjWq,βj,\n(5)\nwhere we assume that this is also a decomposition as a K-module. Choose\nany j such that Wq,βj ∼= Wq,α. This fixes an embedding of Wq,α in Wq,β ⊗X.\n7"},{"page":8,"text":"(It is a plausible conjecture that the decomposition ( 5) is multiplicity free.\nThis would be a conjectural analogue of Pieri’s rule in the nonstandard\nsetting. It would imply that the embedding of Wq,α in Wq,β ⊗X is unique.\nBut this is not required here.). By induction, we have fixed an embedding\nof Wq,β in X⊗r−1. This fixes an embedding ρ of Wq,α in X⊗r (among many\npossible choices). Let W ρ\nq,α = ρ(Wq,α) be its image.\nChoose a Hermitian form on X⊗r so that its Gelfand-Tsetlin basis {zi1 ⊗\n· · · ⊗zir} is orthonormal. Let W ρ,⊥\nq,α denote the orthogonal complement of\nW ρ\nq,α. Since U H\nq (G) has a compact real form [GCT7] such that X⊗r\nq\nis its\nunitary representation with respect to this Hermitian form, it follows that\nW ρ,⊥\nq,α is a GH\nq -module. Thus X⊗r = W ρ\nq,α ⊕W ρ,⊥\nq,α as a GH\nq -module. Let\nLρ\nα = L(X)⊗r ∩W ρ\nq,α\nand\nLρ,⊥\nα\n= L(X)⊗r ∩W ρ,⊥\nq,α .\nThen, in analogy with Kashiwara’s work mentioned above:\nProposition 2.1 L(X)⊗r = Lρ\nα ⊕Lρ,⊥\nα .\nProof: The r.h.s. is clearly contained in the l.h.s. To show the converse it\nsuffices to show that Lρ\nα and Lρ,⊥\nα\nare projections of the lattice L(X)⊗r onto\nW ρ\nq,α and W ρ,⊥\nq,α respectively. Let us show this for Lρ\nα, the other case being\nsimilar. Clearly, the projection of L(X)⊗r onto W ρ\nq,α contains Lρ\nα. We only\nhave to show that the projection ˆy of any y ∈L(X)⊗r onto W ρ\nq,α also belongs\nto the lattice L(X)⊗r, and hence to Lρ\nα. Since y ∈L(X)⊗r, its length |y|\nw.r.t. the preceding Hermitian form tends to a well defined nonnegative real\nnumber as q →0. Since, the projection y →ˆy is orthonormal, the length\n|ˆy| of ˆy is at most |y|, and hence also tends to a well defined nonnegative\nreal number as q →0. This means ˆy is regular at q = 0 and hence belongs\nto L(X)⊗r. Q.E.D.\nLet B(X) = {bi = φ(xi)} denote the basis of the ̃Q-module L(X)/qL(X),\nwhere φ : L(X) →L(X)/qL(X) is the natural projection (the bi’s in this\nsection are different from the bi’s in Section 2.1). Let B(X)⊗r = {bi1 ⊗· · ·⊗\nbir} denote the monomial basis of L(X)⊗r. Given b ∈B(X)⊗r, let\nb =\nX\ni1,...,ir\ng(b; i1, . . . , ir)bi1 ⊗· · · ⊗bir,\nbe its expansion in the monomial basis. The set of monomials bi1 ⊗· · · ⊗bir\nsuch that the coefficients g(b; i1, . . . , ir) are nonzero is called the monomial\nsupport of b.\n8"},{"page":9,"text":"In analogy with the work of Kashiwara and Date et al mentioned above,\nit may be conjectured that:\nConjecture 2.2 (Existence of (local) crystal basis)\nThe ̃Q-module Lρ\nα/qLρ\nα has a unique basis Bρ\nα (up to scaling by constant\nmultiples, and which can be made completely unique by appropriate normal-\nization) such that:\n1. The monomial supports of its elements are disjoint,\n2. Lρ\nα and Bρ\nα ∪{0} are invariant under Kashiwara’s crystal operators ̃ei\nand ̃fi for Hq (which are well defined since Uq(H) is a subalgebra of\nU H\nq (G)), and (Lρ\nα, Bρ\nα) is a local crystal basis, in the sense of Kashi-\nwara, of Wq,α as an Hq-module\nFurthermore, this crystal base does not depend on the embedding ρ (up to\nisomorphism).\nThis conjecture has been supported by experimental evidence; cf.\nSec-\ntion 5.5.\nAssuming this, we let (Lα, Bα) = (Lρ\nα, Bρ\nα) for any ρ as above.\nIt is\ncalled the upper crystal base of Wq,α.\nIn standard setting, the embedding ρ : Vq,π →V ⊗r in Section 2.1 can\nbe chosen so that the support of each basis element in Bρ\nπ consists of just\none monomial. That is, so that each b ∈Bρ\nπ is a monomial in B(V )⊗r. In\nthe nonstandard setting, it is not always possible to choose an embedding\nρ : Wq,α →X⊗r so that the support of each basis element in Bρ\nα in Conjec-\nture 2.1 consists of just one monomial; cf. Section 5.5 for a counterexample.\nIn view of the nonstandard q-analogue of the Peter-Weyl theorem (2),\nS = O(MH\nq (X)) has a natural upper crystal base\n(L(S), B(S)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(6)\nat q = 0 (with appropriate normalization). Let SQ be the KQ-forms (KQ-\nsubring) of S generated by the entries ui\nj of the generic nonstandard quantum\nmatrix u; recall (cf. [GCT7]) that S is the quotient of C⟨u⟩modulo appro-\npriate quadratic relations over the entries ui\nj’s of u. We define an involution\n−over S by a natural generalization of its definition in the standard setting.\nSpecifically, S is a O(MH\nq (X))-bi-comodule. Via the homomorphism from\n9"},{"page":10,"text":"O(MH\nq (X)) to O(Mq(V )), S is also O(Mq(V ))-bi-comodule; i.e., an Hq-bi-\nmodule. In the spirit of [Kas2], for any u and v in S with Hq-bi-weights\n(λr, λl) and (μr, μl), let uv = q(λr,μr)−(λl,μl) ̄v ̄u, where ( , ) denotes the usual\ninner product in the Hq-weight space. We let ̄ui\nj = ui\nj, and ̄q = q−1. This\ndefines −on S completely.\nApplying −to (L(S), B(S)), we get an upper crystal base ( ̄L(S), ̄B(S))\nat q = ∞. In analogy with the standard setting, we can now ask:\nQuestion 2.3 Is the triple (SQ, L(S), ̄L(S)) balanced? In other words, is\nthe map ψ : E = SQ ∪L(S) ∪ ̄L(S) →L(S)/qL(S) of ˆQ-modules an isomor-\nphism.\nIf it were, we could have defined the canonical basis of S by a globalization\nprocedure very much as in the standard setting, i.e., as ψ−1(B(S)). But, as\nit turns out, this need not be so; cf. Section 5.3. Specifically, for a given\nb ∈B(S), the fibre ψ−1(b) need not be singleton. This is the major difference\nfrom the standard setting that makes construction of the canonical basis of\nS in the nonstandard setting much more complex. We turn to this in the\nnext section.\n2.3\nNonstandard globalization via minimization of degree\ncomplexity\nWe now give a conjectural procedure for choosing an unambiguously defined\ncanonical element yb ∈ψ−1(b), b ∈B(S). The set {yb |b ∈B(S)} will then\nbe the canonical basis of S.\nFix r. Let Sr denote the degree r component of S. Let A = AH\nr\n=\nAH\nr (q) be the nonstandard q-Schur algebra [GCT7], which is the dual S∗\nr =\nHom(Sr, K) of Sr. A polynomial irreducible GH\nq -module Wq,α of degree r\nis an irreducible A-module, and conversely every irreducible A-module is\nof this form. Furthermore, the q-analogue of the Peter-Weyl theorem also\nholds for A:\nA = ⊕αW ∗\nq,α ⊗Wq,α.\n(7)\nFor reasons given later (cf. Remark 1 below), it will be more convenient\nto construct the canonical basis of AH\nr first. The canonical basis of Sr will\nthen be defined to be its dual.\n10"},{"page":11,"text":"Let\n⟨⟩: A ⊗Sr →K\nbe the natural pairing. The lattice L(A) is defined to be the dual lattice of\nL(Sr):\nL(A) = {a ∈A | ⟨a, L(Sr)⟩⊆AK}.\nThe automorphism −of A is defined by:\n⟨ ̄a, s⟩= ⟨a, ̄s⟩−.\nWe define ̄L(A) by applying −to L(A). We define the Q-form (i.e. KQ-\nform) AQ of A by:\nAQ = {a ∈A | ⟨a, Sr,Q⟩⊆KQ},\nwhere Sr,Q = Sr ∩SQ.\nWe define the basis B(A) of L(A)/qL(A) to be\nthe dual of B(S). Thus (L(A), B(A)) is the local crystal basis of A as per\nConjecture 2.2 and we have an analogue of (6):\n(L(A), B(A)) =\nM\nα\n(L∗\nα, B∗\nα) ⊗(Lα, Bα)\n(8)\nThis defines the triple (AQ, L(A), ̄L(A)) for A. It need not be balanced\nin the standard sense, just like the triple (SQ, L(S), ̄L(S)) above. Now we\ndescribe the conjectural construction of a canonical basis of A.\nEach component B∗\nα ⊗Bα of B(A) has the left and right action of Kashi-\nwara’s crystal operators for Hq (cf. Conjecture 2.2). Thus, by [Kas1], we\nget a crystal graph on B∗\nα ⊗Bα, whose each connected component intutively\ncorresponds to an irreducible Hq-bi-submodule V ∗\nq,μ1 ⊗Vq,μ2 of the compo-\nnent W ∗\nq,α ⊗Wq,α in the Peter-Weyl decomposition of A (7). With each\nelement b ∈B(A) that occurs in such a connected component, we associate\nthe triple T(b) = (α, μ1, μ2). We call it the type of b. The types T(b)’s can\nbe partially ordered as follows.\nFirst, put a partial order ≤on the labels α of polynomial irreducible A-\nmodules Wq,α as follows. Consider Wq,α as an Hq-module. Let μ(α) denote\nthe highest weight in Wq,α as an Hq-module. There may be several highest\nweight vectors in Wq,α, since Wq,α need not be irreducible as an Hq-module.\nThat is fine. Let ≤denote the usual partial order on the highest weights of\nHq-modules: μ1 ≤μ2 iffμ2 −μ1 ∈P\nτ Nτ where τ ranges over the simple\npositive roots of H. We say α ≤α′ iffμ(α) ≤μ(α′).\n11"},{"page":12,"text":"Now observe that each component W ∗\nq,α⊗Wq,α in the Peter-Weyl decom-\nposition of A is an Hq-bimodule; i.e., has a left and right action of Hq. With\neach irreducible Hq-bimodule in this component isomorphic to Vq,μ1 ⊗Vq,μ2,\nwhere μ1 and μ2 are highest left and right weights of Hq, we associate the\ntype T = (α, μ1, μ2). Put a partial order, which we shall again denote by\n≤, on the types T as per the partial order ≤on the individual components.\nThe type T(b) associated with each b ∈B(A) above is similar to this type.\nSo this also puts a partial order on the types T(b)’s.\nNext fix a b ∈B(A). Let T = T(b) be its type. We shall associate a\ncanonical basis element yb with each such b by induction on its type using\nthe preceding partial order. The set {yb} will then be the required canonical\nbasis of A.\nLet A≤T denote the span of all Hq-bimodules in A of types less than or\nequal to T as per ≤. Let\nL(A≤T) = L(A) ∩A≤T .\nWe define ̄L(A)≤T , and A≤T\nQ\nsimilarly. Consider the natural projection\nψT : A≤T\nQ\n∩L(A)≤T ∩ ̄L(A)≤T →L(A)≤T /qL(A)≤T .\nLet ψ−1\nT (b) be the fibre of b. If this fibre were to contain a unique element,\nthen we can simply let yb be this unique element. But this need not be\nso, because the triple (A≤T\nQ , L(A)≤T , ̄L(A)≤T ) need not be balanced. So\nwe have to resolve the ambiguity in some canonical way. Towards this end,\nwe shall associate with each element in AQ ∩L(A) a complexity measure,\ncalled its degree complexity. We shall then define yb to be the element in\nψ−1\nT (b) of minimum degree complexity–it would be conjecturally unique; cf.\nConjecture 2.5 below.\nThis scheme is in the spirit of [KL1] where each\nelement of the Kazhdan-Lusztig basis of the Hecke algebra is defined to be\nan element of minimum degree in a certain sense.\nSo let us define the degree complexity of an element y ∈AQ ∩L(A).\nSince X⊗r\nq\nis a represention of A = AH\nr (q), we have the injection\nη : A ֒→Z = End(X⊗r\nq ) = (X⊗r\nq )∗⊗X⊗r\nq .\nLet L(Z) = L(X⊗r\nq )∗⊗L(X⊗r\nq ) be the lattice associated with Z. The Q-form\n(or rather KQ-form) ZQ is defined similarly. Then\nProposition 2.4 The embedding η injects the Q-form AQ into ZQ. Fur-\nthermore, assuming Conjecture 2.2, η also injects the lattice L(A) into L(Z).\n12"},{"page":13,"text":"The proof is easy. (To be filled in).\nFix the upper canonical basis {xi} of Xq as an Hq-module. Let {x∗\ni } be\nthe dual canonical basis of X∗\nq . This fixes the upper canonical basis CB(Z)\nof Z, whose each element is of the form\nzi1,...,ir;j1,...,jr = x∗\ni1 ⊗· · · ⊗x∗\nir ⊗xj1 ⊗· · · ⊗xjr.\nIt is also a basis of the Q-form ZQ and the lattice L(Z). Now given any\ny ∈A, let w = η(y). Express w in the canonical basis of Z:\nw = η(y) =\nX\nz\na(y, z)z,\n(9)\nwhere z ranges of the basis elements in CB(Z). Since w ∈L(Z) ∩ZQ, each\na(y, z) ∈AK ∩KQ. This means it is integral over Q[q] and hence has a well\ndefined degree d(y, z) at q = 0 (the same as the order of its pole at q = ∞);\nif a(y, z) = 0, we define d(y, z) = −∞. We define the degree complexity d(y)\nof y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We put a partial order\non the degree complexity as follows.\nLet U = Uq(H) be the Drinfeld-Jimbo enveloping algebra of Hq, U −the\nsubalgebra generated by its generators Fi’s. For any string ν = ν1, ν2, . . . of\npositive integers, let U −\nν be the subspace of U −spanned by the words in Fi’s\nin which each Fi occurs occurs νi times. Given a canonical basis element x\nof Xq, we define its length to be |x| = P\ni νi, where x ∈U −\nν x0, and x0 is\nthe highest weight vector of Xq. Order the canonical basis elements of Xq\nas per the reverse order on their lengths; so that x0 is the highest element\nin this order.\nPut a similar order on X∗\nq .\nThis puts an induced partial\norder on the elements of the canonical basis CB(Z) of Z. We let < denote\nthe strict less than relation as per this partial order. Given y and y′, and\nletting w = η(y), w′ = η(y′), we say that d(y) ≤d(y′) if for every z: either\nd(y, z) ≤d(y′, z), or for some ̄z < z d(y, ̄z) < d(y′, ̄z).\nConjecture 2.5 (Minimum degree) The fibre ψ−1\nT (b) contains a unique\nelement yb of minimum degree complexity. Minimum means d(yb) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (b).\nWe call yb the canonical basis element associated with b, and the set {yb}\nthe canonical basis CB(A) of A. The canonical basis CB(Sr) = {xb} of Sr\nis defined to be its dual. The canonical basis CB(S) of S is ∪rCB(Sr).\nRemark 1: The reader may wonder why we defined the canonical basis of A\nfirst, and that of Sr later, as its dual. Can we define the canonical basis of Sr\n13"},{"page":14,"text":"directly? The algorithm for Sr would be similar as above. The main problem\nis to define the degree complexity of an element x ∈L(Sr) ∩SQ. We have\na natural projection from Z = End(X⊗d\nq ) to Sr, but not a natural injection\nthat injects Sr,Q = Sr ∩SQ into ZQ. So the analogue of Proposition 2.4 does\nnot hold.\nRemark 2: In the standard settting, Kashiwara and Lusztig give an efficient\nscheme for constructing each canonical basis element of the standard matrix\ncoordinate ring O(Mq(V )). We do not have here an analogous efficient algo-\nrithm for constructing the canonical basis element yb in Conjecture 2.5. In\nthe standard setting an efficient cosntruction was possible because the stan-\ndard Drinfeld-Jimbo universal enveloping algebra has an explicit presenta-\ntion in terms of generators and defining relations. This explicit presentation\nis crucially used in the construction of the standard canonical basis and also\nin proving correctness of the construction.\nThe nonstandard universal algebra does not have an analogous explicit\npresentation as yet; cf. [GCT7]. For this we need explicit formulae for the\ncoefficients of the Laplace relation in [GCT7] among the simplest nonstan-\ndard canonical basis elements in S, namely nonstandard minors, since as\ndiscussed in [GCT7], it is the mother relation in the representation theory\nof the nonstandard quantum group (just as in the standard setting). That\nis, we need explicit interpretation for these coefficients in the spirit of the ex-\nplicit interpretation for the coefficients of the Kazhdan-Lusztig polynomials\nin terms perverse sheaves. This is the basic core problem that needs to be\nsolved to prove that the preceding algorithm for constructing the nonstan-\ndard canonical basis is correct and to give an explicit, efficient construction\nof yb. Furthermore, if explicit presentation of the nonstandard universal al-\ngebra is so nonelementary, as against the elementary explicit presentation of\nthe standard (Drinfled-Jimbo) enveloping algebra, then the task of proving\ncorrectness may be formidable.\n2.4\nProperties of the nonstandard canonical basis\nIt may be conjectured that the nonstandard canonical bases CB(S) and\nCB(A) have properties akin to the standard canonical basis of O(Mq(V )):\n2.4.1\nCellular decomposition\nConjecture 2.6 (Cell decomposition) The refined Peter-Weyl theorem,\nakin to the one proved by Lusztig [Lu2] in the standard setting, holds for\n14"},{"page":15,"text":"CB(S) and CB(A).\nThis means the left, right and two-sided cells in O(MH\nq (X)) with re-\nspect to CB(S) yield irreducible left, right, and two-sided (polynomial)\nrepresentations of GH\nq . And furthermore, the left sub-cells of each left cell\nwith respect to the restricted Hq-action yield irreducible Hq-representations.\nThe left cell of O(MH\nq (X)) is defined as follows.\nGiven b ∈CB(S), let\n∆(b) = P\nb′,b′′ cb′,b′′\nb\nb′ ⊗b′′, where ∆denotes comultiplication. Then we say\nthat b′′ ←L b if cb′,b′′\nb\nis nonzero for some b′. Let <L denote the transitive\nclosure of ←L. Using <L we define left cells in a natural way. The right\nand two-sided cells are defined similarly. The left, right and two-sided sub-\ncells with respect to the Hq-action are defined similarly. The definitions for\nCB(A) are similar.\nBy restricting the canonical basis to any left cell corresponding to an\nirreducible polynomial representation Wq,α of GH\nq , we get the canonical basis\nof Wq,α; here the choice of the left cell would conjecturally not matter up to\nscaling.\n2.4.2\nPositivity in the Kronecker case\nConjecture 2.7 (Positivity) In the Kronecker case–i.e. when H = GL(V )×\nGL(W), X = V ⊗W with the natural H-action–each coefficient g(q) of any\ncanonical basis element in CB(AH\nr ) in the basis CB(Z) (cf. eq.(9)) is a\npositive polynomial in q.\nSimilarly, each multiplicative or comultiplicative structural constant of\nCB(AH\nr ) is of the form\n+−(q −1\nq)af(q), where a is a nonnegative integer\nand f(q) is a −-invariant positive and unimodal polynomial in q and q−1.\nThe same also for CB(S).\nHere by a positive polynomial we mean a polynomial with nonnegative\nrational coefficients. By unimodality of the (−-invariant) polynomial f(q),\nwe mean its coefficients f−k, . . . , fk satisfy the condition\nf−k ≤f−k+1 ≤· · · ≤f−1 ≤f0 ≥f1 ≥· · · fk.\nBy multiplicative structural constants, we mean the coefficients mb′′\nb,b′ in the\nexpansion\nbb′ =\nX\nb′′\nmb′′\nb,b′b′′,\n15"},{"page":16,"text":"for b, b′ ∈CB(S), and with b′′ ranging over the elements in CB(S). Comul-\ntiplicative structural constants are defined similarly.\nFor experimental evidence for the dual nonstandard algebra BH\nr (q), see\nSection 5.1.\nPresumably, the nonnegative coefficients of g(q) and f(q) may have a\ntopological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of f(q) may be a conse-\nquence of some result akin to the Hard Lefschetz theorem in the spirit of\nthe results on unimodal sequences in [St].\nThe Kronecker case is fundamental because O(MH\nq (X)) therein is a non-\nflat deformation of O(M(X)), though CH\nq [X] is a flat deformation of C[X].\nThus to prove the positivity Conjecture 2.7, some nonstandard extension\nof the theory of perverse sheaves and the work surrounding the Riemann\nhypothesis over finite fields [BBD, Dl2] that can deal with nonflat, noncom-\nmutative varieties like O(MH\nq (X)) may be neeeded.\n2.4.3\nNonstandard positivity and saturation\nIn the Kronecker case, the braided symmetric algebra CH\nq [X] is a flat de-\nformation of C[X]; i.e. dim(CH,r\nq\n[X]) = dim(Cr[X]). But when CH\nq [X] is a\nnonflat deformation the situation is much more complex. To see this, let c be\na coefficient of any canonical basis element in CB(AH\nr ) in the basis CB(Z).\nAs observed after eq.(9), it is integral over Q[q]. Hence every coefficient of\nits minimal polynomial fc is a polynomial in q with rational coefficients. In\nthe spirit of Conjecture 2.7, one may ask if the coefficients of this polynomial\nare always nonnegative. Unfortunately, this need not be so; cf. Section 5.3\nfor counterexamples in the dual setting of BH\nr (q). The following is a relaxed\nversion of Conjecture 2.7 for the general case.\nConjecture 2.8 (Saturation) Each coefficient a(q) of fc is a saturated\npolynomial (in the terminology of [GCT6]); this means a(1) is a positive\nrational if a(q) is not an identically zero polynomial.\nSimilarly, each coefficient b(q) of the minimal polynomial of a multipli-\nciative or comultiplicative structural constant of CB(AH\nr ) can be expressed\nin the form\nb(q) = (−1)e(q −1\nq )e′c(q)\n(10)\nfor some nonnegative integers e, e′, where\n16"},{"page":17,"text":"1. e is chosen so that the middle term of c(q) is positive; here by the\nmiddle term we mean the coefficient of qi for the smallest i such that\nthis coefficient is nonzero, and\n2. c(q) is a saturated polynomial in q and q−1–this again means c(1) is a\npositive rational if c(q) is not an identically zero polynomial.\nIn the context of the plethysm problem, one is finally interested in the\nbehaviour of b(q) at q = 1 (cf. Section 6), so this relaxed saturation form\nof positivity should be sufficient; see also [GCT6] for the importance of\nsaturation in the context of the flip in GCT. A stronger positivity conjecture\nthat would specialize to Conjecture 2.7 in the Kronecker case is:\nConjecture 2.9 (Nonstandard Positivity) Each polynomial c(q) in Con-\njecture 2.8 is almost positive and unimodal.\nThat is, it is of the form\nc0(q) + c1(q), where, if c(q) is not identically zero,\n1. c0(1) >> |c1(1)|, where >> means much greater as r →∞, and more\ngenerally,\n2. c0(q) is a dominant positive unimodal polynomial, and c1(q) is a very\nsmall error-correction polynomial. Specifically,\n||c1(q)||/||c0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwhere μ and π are as in the plethysm problem (cf. begining of Sec-\ntion 1), ⟨\n⟩denotes the bitlength-of-specification function, and poly(\n)\nmeans polynomial of a fixed (constant) degree in the specified bitlengths,\nand ||\n|| denotes the L2-norm of the coefficient vector of the polyno-\nmial.\nSee Section 5.3 for experimental evidence for Conjectures 2.8-2.9 in the\ndual setting of BH\nr (q).\nPresumably, the nonnegative coefficients of such c0(q) may again have\na topological interpretation in the spirit of that for the coefficients of the\nKazhdan-Lusztig polynomials, and unimodality of c0(q) may again be a con-\nsequence of some result akin to the Hard Lefschetz theorem. The correction\npolynomial c1(q) may also have a topological interpretation that depends on\na cohomological measure of nonflatness of CH\nq [X]. In the Kronecker case,\nwhen CH\nq [X] is a flat deformation of C[X], this correction would then van-\nish, and Conjecture 2.9 would reduce to Conjecture 2.7. Furthermore, the\n17"},{"page":18,"text":"conjectural nonnegative value of c(1) may also have an interpretation akin\nto the representation-theoretic interpretation for the values of the Kazhdan-\nLusztig polynomials at q = 1.\nIf nonstandard extension of the work surronding the Riemann hypothe-\nsis over finite fields as needed to prove the positivity Conjecture refcposkro-\nneckerleft in the Kronecker case can be found, that may open the way for\ninvestigating the more complicated nonstandard form of positivity in Con-\njecture 2.9.\n3\nNonstandard canonical basis of BH\nr\nLet BH\nr = BH\nr (q) be the nonstandard quantization of the symmetric group\nring C[Sr] in [GCT7]. In this section, we describe an analogous conjecturally\ncorrect algorithm for constructing a nonstandard canonical basis E(r) of BH\nr .\nIn the standard setting–i.e., when H = G–this basis would conjecturally\nspecialize to the Kazhdan-Lusztig basis of the Hecke-algebra, though the\nspecialized algorithm here is different from the algorithm in [KL1].\nSince BH\nr (q) is semisimple [GCT4, GCT7], by the Wedderburn structure\ntheorem\nBH\nr (q) =\nM\nα\nT ∗\nq,α ⊗Tq,α,\n(11)\nwhere Tq,α ranges over the irreducible representations of BH\nr , assuming that\nthe underlying base field is a suitable algebraic extension of Q(q1/2). We\nshall denote this base field by K–it is the same as the base field K in\nSection 2.2, except that the role of q there is played by q1/2 here.\nLet\nAK, ̄AK, KQ, ˆQ be as in Section 2.2, with the role of q played by q1/2.\nWe assume that H is the general linear group GL(V ) or a product of\ngeneral linear groups. In this case (cf. [GCT7]), BH\nr is a −-invariant sub-\nalgebra of a suitable Hecke-algebra or a product of Hecke algebras, where\n−denotes the usual bar-automorphism on the Hecke algebra [KL1] (ana-\nlogue of −in Section 2). Let Pi’s and Qi’s denote the rescaled positive and\nnegative generators of BH\nr as defined in [GCT7] (denoted by p+,H\nX,i and q+,H\nX,i\ntherein) so that they belong to the usual Z[q1/2, q−1/2]-form on the ambient\nHecke algebra (or the ambient product of Hecke algebras). Let BH\nr,Q denote\nthe KQ-form of BH\nr generated ring-theoretically by Pi’s, or equivalently Qi’s.\nThe goal is to construct an AK-lattice R(r) ⊆BH\nr so that the canonical\nbasis E(r) of BH\nr\ncan then be constructed by a nonstandard globalization\n18"},{"page":19,"text":"procedure on the triple (BH\nr,Q, R(r), ̄R(r)) analogous to the one Section 2.3.\nJust as in Section 2.2, it will turn out that this triple need not be balanced.\nWe will resolve the ambiguity caused by lack of balance using the notion of\nminimum degree complexity very much as in Section 2.3.\n3.1\nNonstandard Gelfand-Tsetlin basis of BH\nr (q)\nIn the construction of Kazhdan-Lusztig polynomials as described in [KL1,\nSo], the lattice in the Hecke algebra is constructed using its standard mono-\nmial basis. But in general BH\nr (q) does not have a naturally defined monomial\nbasis; see [GCT4] for an example. So we need a different way to construct\nthe lattice R(r). The construction here will be analogous to the construction\nof the lattice in the standard matrix coordinate ring O(Mq(V )) based on its\nGelfand-Tsetlin basis. This construction in O(Mq(V )) is different from the\none defined in Section 2.1. We shall recall it using the same notation as\nin Section 2.1. It is based on the observation that the Gelfand-Tsetlin ba-\nsis of an irreducible Hq-representation Vq,μ (after rescaling as described in\nsection 7.3.3 of [Kli]) is its local crystal basis: i.e., it is an A-basis of the\nlattice Lμ ⊆Vq,μ, and that its projection in Lμ/qLμ is equal to the basis\nBμ (whose elements have disjoint monomial supports as described before).\nThis observation was in fact the starting point for the theory of local crystal\nbasis [Kas1]. Let us denote the (rescaled) Gelfand-Tsetlin basis of Vq,μ by\nGTq,μ. The Gelfand-Tsetlin basis of O(Mq(V )) is defined as per the standard\nPeter-Weyl theorem ( 3):\nGT(O(Mq(V ))) =\n[\nμ\nGT ∗\nq,μ ⊗GTq,μ.\n(12)\nLet LGT be the lattice generated by this Gelfand-Tsetlin basis, and BGT the\nprojection of the Gelfand-Tsetlin basis on LGT /qLGT . Then (LGT , BGT )\ncoincides with the standard crystal base (L, B) of O(Mq(V )) in Section 2.1.\nThe algebra BH\nr (q) has a natural analgoue of the Gelfand-Tseltin basis,\nwhich can then be used to construct the lattice R(r). We begin by describing\nthis basis for an irreducible representation Tq,α of BH\nr (q).\nWe proceed by induction on r, the case r = 1 being easy. The following\nis a conjectural analogue of the standard Pieri’s rule in this setting:\nC1’: Tq,α has a multiplicity-free decomposition as a BH\nr−1-module.\n(This conjecture is not really necessary as long as there is a natural\nway to resolve the ambiguity caused by multiplicity).\nBy induction, we\n19"},{"page":20,"text":"have defined a basis for each irreducible BH\nr−1(q)-submodule of Tq,α. Putting\nthese bases together, we get the sought nonstandard Gelfand-Tsetlin basis\nCα of Tq,α.\nAssuming multiplicity-free decomposition, such a basis is unique, up to\nscaling factors, which will be fixed in the course of the algorithm below.\nEach element x ∈Cα can be indexed by a nonstandard Gelfand-Tsetlin\ntableau, which is an analogue of the standard Gelfand-Tsetlin tableau [Kli]\nin this setting. It is defined to be the tuple (αr, αr−1, . . .), with αr = α, of the\nclassifying labels–which we shall call types–of the irreducible BH\ni -submodules\nTq,αi containing x, where Tq,αi ⊂Tq,αi+1.\nWe define the nonstandard Gelfand-Tsetlin basis C(r) of BH\nq (r) as per\nthe decomposition (11):\nC(r) =\n[\nα\nC∗\nα ⊗Cα.\nEach element of C(r) is indexed by a nonstandard Gelfand-Tsetlin bi-tableau\nas per this decomposition. We shall denote the element of C(r) indexed by\na nonstandard Gelfand-Tsetlin bitableau T by cT .\n3.2\nLocal crystal base\nThe sought lattice R(r) ⊆BH\nr (q) will be generated by the elements of C(r)\nafter scaling them appropriately in the course of the algorithm below. Let\nus assume at the moment that this scaling has already been given to us,\nand thus R(r) is fixed. Let bT denote the image of cT under the projection\nψ : R(r) →R(r)/q1/2R(r). Let B(r) = {bT } be the basis of R(r)/q1/2R(r).\nThen (R(r), B(r)) is the analgoue of the local crystal base in the standard\nsetting.\n3.3\nNonstandard globalization via minimization of degree\ncomplexity\nThe elements of C(r) need not be −-invariant.\nNext we globalize C(r)\nto get a −-invariant canonical basis E(r) of BH\nr\nin the spirit of Kazhdan\nand Lusztig [KL1], with the role of the standard basis in [KL1, So] played\nby the nonstandard Gelfand-Tsetlin basis of BH\nr (q) here. As already men-\ntioned, the main difference from the standard setting of Hecke algebra is\nthat (BH\nr (q), R(r), ̄R(r)) need not be balanced. This is the main problem\n20"},{"page":21,"text":"that needs to be addresssed. The nonstandard globalization procedure here\nis analogous to the one in Section 2.3. It goes as follows.\n(1) In Section 2.3, we described a partial order ≤on the types (classifying\nlabels) of the irreducible modules Wq,α of AH\nr (q). By the nonstadard duality\nconjecture [GCT7], this induces a partial order ≤on the types (classifying\nlabels) of the (paired) irreducible modules Tq,α of BH\nr (q). (In the standard\nsetting, this procedure would yield a partial order on the partitions of size\nr, with the partition containing a single row of size r at the top of the order\nand the partition containing a single column of size r at the bottom of the\norder.)\n(2) This induces a lexicographic partial order ≤on the nonstandard Gelfand-\nTsetlin tableaux, since they are just tuples of types, and also on nonstandard\nGelfand-Tsetlin bitableau which index the basis elements C(r).\n(3) Let B≤T be the span of the basis elements cT ′ ∈C(r) such that T ′ ≤T.\nLet R≤T = R(r) ∩B≤T , ̄R≤T = ̄R(r) ∩B≤T, and B≤T\nQ\n= BH\nr,Q ∩B≤T . Then\nthe triple (B≤T\nQ , R≤T , ̄R≤T ) need not be balanced. To define a canonical\nbasis element eT associated with T, we associate a degree complexity with\neach element y ∈BH\nr,Q in the spirit of Section 2.3.\nThis is done as follows. Since we are assuming that H is GL(V ) or a\nproduct of general linear groups, BH\nr is a subalgebra of a product of Hecke\nalgebras, say Z = Hk1(q) ⊗· · · ⊗Hkl(q), where Hj(q) denotes the Hecke\nalgebra with rank j. Furthermore,\nBH\nr,Q ⊆ZQ = Hk1,Q × · · · × Hkl,Q,\nwhere Hj,Q denote the KQ-form of Hj(q) obtained by tensoring its usual\nQ[q1/2, q−1/2]-form with KQ. Consider the Kazhdan-Lusztig basis KL(Z)\nof Z formed by taking the product of the Kazhdan-Lusztig bases of its Hecke\nalgebra factors. Express y ∈BH\nr,Q in terms of KL(Z):\ny =\nX\nz\na(y, z)z,\n(13)\nwhere z ranges over the elements in KL(Z). Then each coefficient a(y, z) ∈\nKQ. Let d(y, z) denote the degree of a(y, z); i.e., the order of its pole at\nq = ∞.\nIf a(y, z) = 0, we define d(y, z) = −∞.\nWe define the degree\ncomplexity d(y) of y to be the tuple ⟨. . . , d(y, z), . . .⟩of these degrees. We\nput a partial order on degree complexities as follows. Put a partial order\n≤on the Kazhdan-Lusztig basis of the Hecke algebra Hj(q) as per the\nreverse order on the (reduced) lengths of the permutation indices of the\n21"},{"page":22,"text":"basis elements–so 1 is the highest element as per this order. This also puts\na partial order on KL(Z). Let < denote the strict less than relation as per\nthis partial order. Given y and y′, we say that d(y) ≤d(y′) if for every z:\neither d(y, z) ≤d(y′, z), or for some ̄z < z, d(y, ̄z) < d(y′, ̄z).\nConsider the natural projection\nψT : R≤T ∩ ̄R≤T ∩B≤T\nQ\n→R≤T /q1/2R≤T .\nLet ψ−1\nT (bT ) be the fibre of bT ∈B(r). The following is the analogue of\nConjecture 2.5 in this context (with different interpretation for b, y, ψ etc.\nfrom there):\nConjecture 3.1 (Minimum degree) The fibre ψ−1\nT (bT ) contains a unique\nelement eT of minimum degree complexity. Minimum means d(eT ) ≤d(y),\nas per the ordering on the degree tuples above, for any y ∈ψ−1\nT (bT ).\nWe call eT the canonical basis element associated with T, and E(r) =\n{eT } the canonical basis BH\nr (q).\nSo far we have not discussed how to scale the nonstandard Gelfand-\nTsetlin basis of BH\nr (q) to get the lattice R(r). To complete the algorithm,\nit remains to fix this scaling.\nLet {c′\nT } denote the nonstandard Gelfand-Tsetlin basis of BH\nr (q) before\nscaling. The scaled cT will be of the form qaT c′\nT for some rational aT . We\nhave to determine all aT ’s. Assume that aT ′, T ′ < T, have been fixed. For\nany rational a, let ca,T = qac′\nT . Let Ra,≤T be the lattice generated by ca,T\nand cT ′, T < T, and ̄Ra,≤T obtained by applying −to it. Consider the\nprojection\nψa,T : Ra,≤T ∩ ̄Ra,≤T ∩B≤T\nQ\n→Ra,≤T /q1/2Ra,≤T .\nLet ba,T be the image of ca,T under the projection Ra,≤T /q1/2Ra,≤T . Let\nψ−1\na,T (ba,T ) be its fibre. The following is the strengthened form of Conjec-\nture 2.5.\nConjecture 3.2 (Minimum degree) There exists a unique aT and eT ∈\nψ−1\naT ,T(baT ,T) such that for any a and any y ∈ψ−1\na,T (ba,T ) d(eT ) ≤d(y). That\nis, eT is the unique element of minimum degree complexity over all choices\nof a.\n22"},{"page":23,"text":"This fixes aT . Furthermore, ψT = ψaT ,T , bT = baT ,T , and eT in Conjec-\nture 3.1 is the same as here.\nIf instead of the order ≤among the classifying labels α’s of Tq,α, we\nuse its reverse order, and in the definition of degree complexity use the\nopposite of the Kazhdan-Lusztig basis (obtained by replacing Qi by Pi), we\nget another canonical basis Eopp(r) of BH\nr (q), which we shall call its opposite\ncanonical basis.\nTo prove Conjectures 3.1-3.2, we need to know relations among the gen-\nerators Pi’s of BH\nr explicitly, just as we know the relations among the gen-\nerators of the Hecke algebra explicitly. This is not known at present. See\n[GCT7] for the problems that arise in this context.\nEach element c of the Kazhdan-Lusztig basis of the Hecke algebra can\nbe expressed in the form\nc = c0 +\nX\nj>0\na(j)cj,\nwhere each cj is a monomial in the generators of the Hecke algebra, a(j) ∈\nQ[q1/2, q−1/2] and the length of each cj, j > 0, is smaller than that of\nc0. This need not be so in the nonstandard setting: there can be several\nmonomials of maximum length with nontrivial coefficients in any monomial\nrepresentation of a nonstandard canonical basis element; cf. Section 5.3 and\nFigure 11 therein for an example.\n3.4\nConjectural properties\nIt may be conjectured that the canonical bases E(r) and Eopp(r) have prop-\nerties akin to those of the Kazhdan-Lusztig basis of the Hecke algebra.\n3.4.1\nCellular decomposition\nConjecture 3.3 (Cell decomposition) Analogue of the cell decomposi-\ntion property of the Kazhdan-Lusztig basis also holds for E(r) and Eopp(r).\nSpecifically this means the following. Let us define the left, right and\ntwo-sided cells of BH\nr\nwith respect to the canonical basis E(r) very much\nas in Section 2.4. Then it may be conjectured that they yield irreducible\nleft, right and two-sided representations of BH\nr . The conjecture for Eopp(r)\nis similar.\n23"},{"page":24,"text":"By restricting E(r) to any left cell corresponding to an irreducible BH\nr -\nmodule Wq,α, we get the canonical basis of Wq,α; here the choice of the left\ncell would conjecturally not matter (up to scaling).\n3.5\nPositivity in the Kronecker case\nThe following is an analogue of Conjecture 2.7 here.\nConjecture 3.4 (Positivity) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13).\nIn the Kronecker case, c is of the form\n+−(q1/2 −q−1/2)af(q), where a\nis a nonnegative integer and f(q) is a −-invariant positive and unimodal\npolynomial in q1/2 and q−1/2.\nThe same for Eopp(r).\nFor experimental evidence see Sections 5.1.2 and 5.2.\n3.5.1\nNonstandard positivity and saturation\nThe general case is much more complex as in Section 2.4.3. The following\nis an analogoue of Conjecture 2.8.\nConjecture 3.5 (Saturation) Let c ∈KQ be a multiplicative or comulti-\nplicative structural constant of E(r) or a structural coefficient of a canoni-\ncal basis element in E(r)–i.e., a coefficient of its expression in terms of the\ncanonical basis of Z as in eq.(13). Let fc be its minimal polynomial with\ncoefficients in Q(q1/2).\nThen each coefficient s(q) of fc can be expressed in the form\ns(q) = (−1)e(q1/2 −q−1/2)e′g(q)\n(14)\nfor some nonnegative integers e, e′, where\n1. e is chosen so that the middle term of g(q) is positive; and\n2. g(q) is a saturated polynomial in q1/2 and q−1/2.\n24"},{"page":25,"text":"Analogue of the stronger Conjecture 2.9 in this case is:\nConjecture 3.6 (Nonstandard Positivity, informal) Each polynomial\ng(q) above is almost positive and unimodal; i.e. of the form g0(q) + g1(q),\nwhere, if g(q) is not identically zero,\n1. g0(1) >> |g1(1)|, and more generally,\n2. g0(q) is a dominant positive unimodal polynomial, and g1(q) is a very\nsmall error-correction polynomial. Specifically,\n||g1(q)||/||g0(q)|| ≤1/poly(⟨μ⟩, ⟨π⟩, ⟨r⟩),\nwith the terminology as in Conjecture 2.9.\nFor experimental evidence, see Section 5.4.1.\nHere, g(1) and the conjectural nonnegative coefficients of g0(q) may have\na representation-theoretic/topological/cohomological interpretation akin to\nthat sought for the analogous quantities in Section 2.4.3.\n3.5.2\nQuasi-cellular decomposition\nConjecture 3.7 The opposite canonical basis Eopp also has the following\nquasi-cellular decomposition property.\nFor this we define a quasi-subcellular decomposition of each left or right\ncell with respect to Eopp(r). Specifically, given e′, e′′ belonging to the same\nleft cell, express\nee′ =\nX\ne′′\nǫ(e, e′, e′′)(q1/2 −q−1/2)δ(e,e′,e′′)de′′\ne,e′e′′,\n(15)\nwhere e′′ ∈E(r), the sign ǫ(e, e′, e′′) is either 1 or −1, δ(e, e′, e′′) is a non-\nnegative integer, and de′′\ne,e′ is a −-invariant saturated polynomial in q1/2 and\nq−1/2 as per the saturation Conjecture 3.5. We say that e′′ ∝L e′ if, for some\ne, de′′\ne,e′ in (15) is nonzero and δ(e, e′, e′′) is zero; i.e., if, for some e, e′′ occurs\nwith nonzero coefficient in the expansion of ee′ specialized at q = 1. Let ≺L\ndenote the transitive closure of ∝L. Using ≺L we define left quasi-subcells\nof a left-cell of Eopp(r). It may be conjectured that each left quasi-subcell\nof Eopp(r) yields an irreducible representation of the symmetric group Sr at\n25"},{"page":26,"text":"q = 1. That is, when the BH\nr -representation Y corresponding this left cell\nis specialized at q = 1, so as to become a representation Yq=1 of C[Sr], the\npartial order on its left quasi-sub-cells induces a composition series of Yq=1\nwhose factors are irreducible representations of C[Sr].\nFix one such quasi-subcell C of Eopp(r). Let Sλ(C) be the irreducible\nrepresentation (Specht module) of C[Sr] that is isomorphic to the factor\nin correspondence with C in this composition series of Yq=1, where λ(C)\nis a partition depending on C. The canonical basis elements in C, after\nspecialization at q = 1 and projection, yield a basis of Sλ(C). It may be con-\njectured that this basis coincides with the Kazhdan-Lusztig basis of Sλ(C)\n(up to rescaling).\nBy the Kazhdan-Lusztig basis of Sλ(C), we mean spe-\ncialization at q = 1 of the Kazhdan-Lusztig basis of the quantized Specht\nmodule Sq,λ(C) of the Hecke algebra Hr(q).\nBut Conjecture 3.7 need not hold for E(r); cf. Section 5.3 for a coun-\nterexample. This is analgous to the fact that the refined Peter-Weyl theorem\nin [Lu2] for the coordinate ring of the standard quantum group Hq holds\nonly for the ordering ≤(as defined in Section 2.3) among the labels (high-\nest weights) of irreducible Hq-modules–there is no canonical basis of the\nstandard coordinate ring which admits refined Peter-Weyl theorem for the\nopposite of the order ≤.\n4\nInternal definition of degree complexity\nWe give here an internal definition of degree complexity which may be used\nin place of the definition in Section 3.3 during the construction of the canon-\nical basis. By internal, we mean it is based only on the structure of BH\nr (q)\nand does not depend on its embedding in the external ambient algebra Z\nthere. This notion of degree complexity does not coincide with the one in\nSection 3.3, but the canonical basis constructed using this definition may be\nconjectured to be the same as the one constructed therein.\nLet B[i] ⊆BH\nr (q) be the span of the monomials in Qj’s of length j, and\nB[< i] of length < i. We say that a given set of monomials in Qj’s of length\ni is independent if the images of these monomials in B[i]/B[< i] are linearly\nindependent. An expression\na =\nX\nm\namm,\n(16)\nwhere m ranges over monomials in Qj’s and am ∈KQ, is called valid\n26"},{"page":27,"text":"if, for each i, the monomials m of length i with am ̸= 0 in this expres-\nsion are independent.\nAssume that a is −-invariant, so that each am is\n−-invariant.\nThe degree complexity ˆd(a) of a is defined to be the tuple\n⟨ˆdl(a), . . . , ˆdi(a), . . . , ˆd0(a)⟩where ˆdi(a) denotes the maximum degree (at\nq = 0) of am for any m of length i, and l is the maximum length of m\nwith am ̸= 0 in the expression (16); by definition, ˆdi(a) = −∞if there is\nno m in (16) of length i with am ̸= 0. We order these degree complexities\nlexicographically. The degree complexity ˆd(b) of an element b ∈BH\nr,Q(q) is\ndefined to be the minimum degree complexity of its any valid expression.\nIt may be conjectured that if this definition of degree complexity, with the\nlexicographic ordering as above, is used in place of the definition of degree\ncomplexity in Section 3.3, the algorithm therein still works correctly and\nconstructs the same canonical basis E(r).\nFor the opposite canonical basis Eopp(r), one can similarly use the inter-\nnal definition as above with Pi in place of Qi.\nThe definition of degree complexity in this section is not as satisfactory\nas in Section 3.3 because BH\nq (r) does not a natural monomial basis [GCT4].\nHence to find the degree complexity of an element, one has to consider all\nits monomial expressions, finite but huge in number. It will be interesting\nto know if there is a more efficient internal definition.\n5\nExperimental evidence for BH\nr (q)\nIn this section we shall verify the conjectures in this paper for two nontriv-\nial special cases of the nonstandard algebra B = BH\nr (q). The nonstandard\ncanonical bases of BH\nr (q) in these cases were computed with the help of a\ncomputer using the algorithm in Section 3 and the notion of degree com-\nplexity as in Section 3.3.\nFirst, some notation. Given a string σ = i1 · · · ik of positive integers,\nwe let Pσ denote the monomial Pi1 · · · Pik; Qσ is defined similarly. Given a\n−-invariant polynomial g(q) ∈Q[q, q−1], we define the vector ag associated\nwith g(q) as follows. Express g(q) in the form\n+−(q−1/q)eh(q), where e is the\nmaximum possible. Let h−l, . . . , h0, . . . , hl be the coefficients of h(q). Then\nag is defined to be [h0, . . . , hl]. In particular if h(q) is (positive) unimodal,\nthen aq is a (positive) nonincreasing sequence. The vector associated with\na −-invariant polynomial in q1/2 and q−1/2 is defined similarly.\n27"},{"page":28,"text":"5.1\nKronecker problem: n = 2, r = 3\nConsider B = BH\n3 (q) in the special case of the Kronecker problem for n = 2\nand r = 3. Thus H = Gl2 ×Gl2, and G = Gl4 with H embedded diagonally.\nLet Pi, i = 1, 2, be as in Section 3 and [GCT4]. The nonstandard canonical\nbasis of B was computed in [GCT4] by an ad hoc method for r = 3, but it\ncoincides with the one computed by the algorithm here. It is as follows. Let\nc1 = q6 + 2q5 + 3q4 + 4q3 + 3q2 + 2q + 1\nq3\n,\nc2 = q4 + q3 + 4q2 + q + 1\nq2\n,\nb1 = −(q2 + 1)2/q2, and b2 = (q + 1)2/q.\nThen the opposite canonical basis Eopp(3) of B consists of the following\nten elements:\nΣ = c1P1 −c2P121 + P12121,\nγi\n1 = b1P1 + P121,\ni = 1, 2,\nγi\n12 = b1P12 + P1212,\ni = 1, 2,\nγi\n2 = b1P2 + P212,\ni = 1, 2,\nγi\n21 = b1P21 + P2121,\ni = 1, 2,\nμ = 1.\n(17)\nThe canonical basis E(3) is obtained by susbstituting Qi for Pi. In what\nfollows, we shall only consider Eopp(3).\n5.1.1\nCellular and quasi-cellular decomposition\nThe basis Eopp(3) has a cellular decomposition, in accordance with Conjec-\nture 3.3, with the following right cells:\nUσ\n=\n{Σ}\nV1\n=\n{γ1\n1, γ1\n12}\nV2\n=\n{γ2\n1, γ2\n12}\nW1\n=\n{γ1\n2, γ1\n21}\nW2\n=\n{γ2\n2, γ2\n21}\nUμ =\n=\n{μ}.\nThe left cell decomposition is similar. The representation of B supported\nby Uσ is the trivial one dimensional representation. The representation sup-\nported by V1 or W1 is isomorphic; let us call it χ1. Similarly, the representa-\ntion supported by V2 or W2 is isomorphic; let us call it χ2. Then χ1 and χ2\n28"},{"page":29,"text":"are two nonisomorphic two-dimensional representations of B which special-\nize at q = 1 to the two-dimensional Specht module of the symmetric group\nS3 corresponding to the partition (2, 1). Thus quasi-cellular decomposition\n(Conjecture 3.7) holds trivially here.\n5.1.2\nPositivity\nCoefficients of the elements of W1 and W2 in the Kazhdan-Lusztig basis of\nH3(q) ⊗H3(q) ⊇BH\n3 (q) are shown in Figure 1 (with the Kazhdan-Lusztig\nbasis symmetrized and appropriately ordered as described in [GCT4]); the\nfirst column shows the coefficients of γ1\n1, the second of γ1\n12, and so on. It\ncan be observed that all coefficients are positive, and unimodal polynomials\nin Q[q, q−1]. The cofficients of other canonical basis elements can be found\nin [GCT4]; they too are positive, unimodal polynomials. This verifies the\npositivity Conjecture 3.4 for the structural coefficients of the canonical basis.\nA few typical nonzero multiplicative structural constants of the canonical\nbasis are shown in Figure 2, where the coefficient of bb′ with respect to the\nbasis element b′′ is denoted by c(b, b′; b′′). It can be seen that each constant\nis a polynomial of the form\n(−1)a(q1/2 −q−1/2)bf(q1/2, q−1/2),\nwhere f is a positive unimodal polynomial. It was verified with computer\nthat all multiplicative structural constants are of this form. This verifies the\npositivity Conjecture 3.4 for the multiplicative structural constants as well.\n5.2\nKronecker case, H = SL2, r = 4\nFor the Kronecker case, H = Gl2 × GL2, G = GL4, and r = 4, we could\ncompute just one canonical basis element Σ (akin to Σ in Section 5.1) corre-\nsponding to the trivial one dimensional representation of BH\n4 (q). Symbolic\ncomputations needed to compute other canonical basis elements turned out\nto be beyond the scope of MATLAB/Maple on an ordinary workstation.\nThe coefficients of Σ in the Kazhdan-Luztig basis of H4(q)⊗H4(q) ⊃BH\n4 (q)\nwere computed in MATLAB/Maple. There are 576 coefficients in total. Fig-\nures 3-5 show the vectors associated with distinct nonzero coefficients among\nthese. They can be seen to be positive and nonincreasing in accordance with\nConjecture 3.4.\n29"},{"page":30,"text":"(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(q2+q+1)(1+q)4\nq3\n(q2+q+1)(1+q)6\nq4\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)3\nq3/2\n(1+q)5\nq5/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)5\nq5/2\n(1+q)7\nq7/2\n(1+q)4\nq2\n(1+q)6\nq3\n(1+q)4\nq2\n(1+q)6\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n8\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(q2+4 q+1)(1+q)2\nq2\n(q4+5 q3+12 q2+5 q+1)(1+q)2\nq3\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)3\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n8 (1+q)2\nq\n8\n8 (1+q)2\nq\n2 (1+q)2\nq\n(q2+6 q+1)(1+q)2\nq2\n8\n(q2+6 q+1)(1+q)2\nq2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n4 1+q\n√q\n4 (1+q)3\nq3/2\n2 (1+q)4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 3 q3+1+3 q+8 q2+q4\nq2\n2 (3 q3+1+3 q+8 q2+q4)(1+q)2\nq3\n2 (1+q)3\nq3/2\n(q2+6 q+1)(1+q)3\nq5/2\n(1+q)(q2+6 q+1)\nq3/2\n2\n(1+q)(3 q3+1+3 q+8 q2+q4)\nq5/2\n4 (1+q)2\nq\n4 (1+q)4\nq2\n4 (1+q)2\nq\n2 3 q4+6 q3+14 q2+6 q+3\nq2\n \n \nFigure 1: Coefficients of the elements of W1 and W2 in the symmetrized\nKazhdan-Lusztig basis, as computed in [GCT4]\n30"},{"page":31,"text":"5.3\nH = sl2, G = sl4\nNow we study the nonstandard algebra B = BH\n3 (q), when H = Gl2, X is its\nfour dimensional irreducible representation, and G = GL(X) = Gl4. It is\na 21-dimensional algebra whose explicit presentation is given in Section 7.1\nof [GCT7]. We follow the notation as therein. Let Pi and Qi be as defined\nin the begining of that section. The monomials Pσ, where σ ranges over\nstrings in 1 and 2 of length k ≤10 with no consecutive 1’s or 2’s, form a\nbasis of B. This algebra has one trivial one-dimensinal representation, and\nfive nonisomorphic two-dimensional representations, so that\n21 = 1 + 22 + 22 + 22 + 22 + 22.\nLet\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(18)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x]\nbe the algebraic extension of Q(q) obtained by adjoining x. It is shown in\n[GCT7] that B admits a complete Wederburn-structure decomposition over\nK, but not Q(q). In what follows, we assume that B is defined over this\nbase field K.\nThe nonstandard canonical bases E(3) and Eopp(3) of B were computed\nin MATLAB/Maple using the algorithm in Section 3. They are as follows.\nLet Ui, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the\nmatrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni\n \n,\nwhere u1\n1 is as specified in Figure 6, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 7-9. Elements\nare specified in these figures by giving their nonzero coefficients in the {Qσ}\nbasis; the coefficient for Qσ is shown in front of σ. Let u2\ni , 1 ≤i ≤5, be the\nelement obtained from u1\ni by interchanging Q1 and Q2. Let u12\ni\n= u1\ni Q2,\nand u21\ni\n= Q2u1\ni , for 1 ≤i ≤5. Let u0 = 1 (this definition of u0 is different\nfrom that in [GCT7]). Then u0 and the entries of ui form the canonical\nbasis E(3) of BH\n3 (q). The left cells of E(e) are {u0} and the columns of ui.\nThe right cells are {u0} and the rows of ui. The representation supported\n31"},{"page":32,"text":"by {u0} is the trivial one-dimensional representation; let us denote it by\nΣ. The representations supported by the columns or rows of ui are two-\ndimensional representations of B, distinct for each i; let us denote them by\nχi.\nThe left cell {u0} is at the top of the ≤L partial order and the left\ncells corresponding to the columns Ui are at depth 1 from the top in this\npartial order (and mutually incomparable). The situation for the right cells\nis similar.\nLet\nvi =\n v1\ni\nv12\ni\nv21\ni\nv2\ni\n \n,\nwhere vα\ni is obtained from uα\ni by substituting Pσ for Qσ in the expression of\nuα\ni in the {Qσ} basis. Let v0 be the element whose coefficients in the {Pσ}\nbasis are as shown in Figures 10-11. Then v0 and the elements of vi form the\nopposite canonical basis Eopp(3) of BH\n3 (q). The left cells are {v0} and the\ncolumns of vi. The right cells are {v0} and the rows of vi. The left cell {v0}\nis at the bottom of the ≤L partial order and the left cells corresponding to\nthe columns of vi at height 1 from the bottom (and mutually incomparable).\nThe situation for the right cells is similar.\nLet Σ′ be the trivial one-dimensional representation of the subalgebra\nBH\n2 (q) ⊂B = BH\n3 (q) generated by P2. Let μ′ be the other one-dimensional\nrepresentation of BH\n2 (q) that specializes to the signed one-dimensional rep-\nresentation of the symmetric group S2 at q = 1. Then the nonstandard\nGelfand-Tsetlin tableau T(b)’s associated with the basis elements b’s of E(3)\nare as follows.\nIf b = u0, then T(b) = [Σ, Σ′]. If b = u1\ni , then T(b) = [χi, Σ′]. If b = u21\ni ,\nthen T(b) = [χi, μ′].\nIf b = u2\ni , then T(b) = [χi, Σ′].\nIf b = u12\ni , then\nT(b) = [χi, μ′].\nThe nonstandard Gelfand-Tsetlin tableau associated with the basis ele-\nments in Eopp(r) are similar.\n5.3.1\nViolation of standard balance\nLet R(3) be the AK-lattice generated by E(3), ̄R(3) = (R(3))−. Let Ropp(3)\nand ̄Ropp(3) be defined similarly. Then it turns out that the triple (BH\n3,Q, R(3), ̄R(3))\nassociated with the canonical basis E(3) is balanced, but the triple (BH\n3,Q, Ropp(3), ̄Ropp(3))\nassociated with the opposite canonical basis Eopp(3) is not balanced. Specif-\nically, the fibre ψ−1\nT (0) ̸= {0} when T = T(b) is the nonstandard tableau\nassociated with b = v2\ni , for any i (it is zero for all other b’s).\n32"},{"page":33,"text":"For example, with the help of computer it was found that the Q-module\nψ−1\nT (0), for b = v2\n5, T = T(b), is generated by the two elements w and x\nspecified in Figures 12-15, which give their nonzero cofficients in the {Pσ}\nbasis. Clearly w belongs to the KQ form BH\n3,Q since the coefficients belong\nto Q[q, q−1]. It is −-invariant, since the coefficients are −-invariant. It can\nbe verified that w ∈qRopp(3). Specifically, it can be shown that\nw = a0v0 + c1v1\n5 + c12v12\n5 + c2v2\n5 + c21v21\n5 ,\nwhere the coefficient vector [a0, c1, c12, c2, c21] is the following\n \n \n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n−\nq6\n(2 q8−2 q6+3 q4−2 q2+2)(q2+1)2\n(q4−q2+1)(q4+1)q\n(q2+1)(2 q8−2 q6+3 q4−2 q2+2)\n \n \nThis means ψT (w) = 0. It can be verified that w belongs to B≤T . Similarly,\nx is a −-invariant element of BH\n3,Q ∩B≤T that belongs to qRopp(3).\n5.4\nNonstandard globalization via minimization of degree\ncomplexity\nThus each element in ψ−1\nT ( ̄b), b = v2\n5, ̄b = ψ(b), T = T(b), is a linear combi-\nnation of b, w and x. It easy to see from the explicit formulae in Figures 12-15\nand Figure 9 that b = v2\n5 is the element of minimum degree complexity in\nψ−1\nT ( ̄b), where the degree complexity is defined internally as in Section 4.\n(Remember that v2\n5 is obtained from u1\n5 in Figure 9 by substituting Pi for Qi\nand then interchanging P1 and P2.) The same is also true for all b’s. This\nverifies the minimum degree conjecture (Conjecture 3.1); Conjecture 3.2 can\nalso be verified similarly.\nWe could not use the external definition of degree complexity as in Sec-\ntion 3.3 here, since the smallest product of Hecke algebras containing BH\n3 (q)\nis Z = H3(q)⊗9 with dimension 10077696 = 69. It is impossible to carry out\nsymbolic computations in an algebra of this size in MATLAB/Maple.\n33"},{"page":34,"text":"c(γ1\n1; γ1\n1; γ1\n1) = c(γ12\n1 ; γ1\n1; γ1\n1) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n2; γ1\n1; γ1\n21) = −(q2 + q + 1) ∗(q −1)2/q2;\nc(γ1\n21; γ1\n1; γ1\n21) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;\nc(γ1\n1; γ1\n1; Σ) = c(γ2\n1; γ1\n1; Σ) = c(γ1\n2; γ1\n1; Σ) = c(γ2\n2; γ1\n1; Σ) = 1/q ∗(1 + q)2;\nc(γ1\n12; γ1\n12; Σ) = c(γ1\n21; γ1\n12; Σ) = (1 + q)2 ∗(6 ∗q + 5 ∗q2 + 5)/q2;\nc(γ2\n21; γ2\n21; γ2\n21) = (1 + q2)2 ∗(q2 + q + 1) ∗(q −1)2/q4;\nc(γ2\n21; γ2\n21; Σ) = c(γ2\n12; γ2\n21; Σ) = (1 + q)2 ∗(2 ∗q2 + q + 1) ∗(q2 + q + 2)/q3.\nFigure 2: Multiplicative structural constants of the canonical basis of BH\n3 (q)\nin the Kronecker case, n = 2, r = 3\n34"},{"page":35,"text":"10\n4\n3\n44\n21\n7\n22\n17\n7\n1\n50\n30\n15\n2\n20\n12\n4\n14\n7\n3\n44\n31\n14\n5\n19\n12\n4\n1\n6\n5\n3\n1\n94\n64\n29\n4\n88\n65\n28\n7\n39\n24\n8\n1\n80\n45\n17\n2\n40\n32\n16\n4\n28\n21\n10\n3\n75\n45\n19\n5\n38\n31\n16\n5\n1\n11\n8\n4\n1\n122\n69\n23\n2\n62\n49\n23\n5\nFigure 3: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (ignoring a positive, unimodal factor)\n35"},{"page":36,"text":"126\n92\n47\n12\n2\n158\n93\n33\n4\n153\n93\n35\n7\n78\n63\n32\n9\n1\n160\n125\n62\n19\n2\n72\n48\n20\n4\n150\n120\n64\n24\n5\n69\n47\n21\n6\n1\n22\n19\n12\n5\n1\n212\n163\n74\n17\n244\n191\n92\n25\n2\n111\n72\n28\n5\n102\n71\n33\n9\n1\n104\n88\n52\n20\n4\n100\n85\n52\n22\n6\n1\n30\n23\n13\n5\n1\n128\n106\n59\n20\n3\n316\n251\n126\n37\n4\n306\n246\n128\n42\n7\n141\n95\n41\n10\n1\nFigure 4: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q), continued\n36"},{"page":37,"text":"144\n120\n68\n24\n4\n41\n31\n17\n6\n1\n344\n213\n79\n12\n375\n237\n91\n17\n162\n117\n59\n19\n3\n222\n183\n100\n33\n5\n204\n173\n104\n42\n10\n1\n192\n140\n72\n24\n4\n60\n53\n36\n18\n6\n1\n220\n191\n124\n58\n18\n3\n264\n188\n92\n28\n4\n82\n72\n48\n23\n7\n1\n280\n244\n160\n76\n24\n4\n83\n66\n41\n19\n6\n1\n750\n612\n328\n108\n17\n345\n237\n107\n28\n3\n324\n279\n176\n78\n22\n3\n113\n89\n54\n24\n7\n1\n106\n96\n71\n42\n19\n6\n1\n528\n452\n280\n120\n32\n4\nFigure 5: The vectors associated with distinct nonzero coefficients of Σ ∈\nBH\n4 (q) (continued)\n37"},{"page":38,"text":"σ\nCoefficient\n1\n1/2 (q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(q2 + 1)4\n×(x + 3 q28 + 4 q24 −2 q22 + 10 q20 −2 q18 + 4 q16 + 3 q12)/q40\n121\n−1/2 (q2 + 1)2(2 q18 −295 q28 −516 q36 + x + 210 q26 + 3 q56 −3 q54 + 47 q46 + 9 q52 −q48\n+2 q50 −84 q24 −295 q40 + 604 q34 + 462 q30 −xq2 + 47 q22 −9 q20x + 19 q10x −q26x + q28x −3 q14\n+q24x + 4 q22x + 30 q14x + 462 q38 −516 q32 + 19 q18x + 210 q42 −q20 + 9 q16 −24 q16x −24 q12x\n−9 q8x + 4 q6x + q4x −84 q44 + 3 q12)/q36\n12121\n1/2 (q18 −2 q28 + 22 q36 + x + 45 q26 + 2 q46 + 3 q48 + 22 q24 + 24 q40 + 45 q34 + 92 q30 + 18 q22 + q20x\n+6 q10x + 2 q14 + q14x + 18 q38 −2 q32 + q42 + 24 q20 + 9 q16 + q16x + q6x + q4x + 9 q44 + 3 q12)/q30\n1212121\n−1/2 (22 q20 + 6 q16 + 6 q24 + 2 q26 + 2 q14 + 2 q30 + 2 q10 + 3 q28 −2 q22 −2 q18 + 3 q12 + x)/q20\n121212121\n1\nFigure 6: Coefficients of u1\n1\n38"},{"page":39,"text":"σ\nCoefficient\n1\n(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4/q32\n121\n−(1 −4 q10 + 14 q8 −30 q14 + 44 q28 + 73 q16 + 3 q2 + 14 q32 −30 q26 + 73 q24 + 3 q38\n+102 q20 −53 q18 + q40 + 44 q12 −53 q22 −4 q30 + 5 q4 + 5 q36)(q2 + 1)2(q4 −q2 + 1)2/q26\n12121\n(3 + 72 q18 + 14 q28 + 3 q36 + 20 q26 + 10 q24 + 2 q34 + 2 q30 + 36 q22 + 14 q8 + 7 q4\n+2 q2 + 7 q32 + 2 q6 −10 q20 −10 q16 + 10 q12 + 20 q10 + 36 q14)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + q4 + 3 q2 + 4 q6 + q20 + 4 q14 + 3 q18 + q16)/q10\n121212121\n1\nFigure 7: Coefficients of u1\n3\n39"},{"page":40,"text":"σ\nCoefficient\n1\n(q2 + 1)2(q4 −q2 + 1)2(q4 + 1)2(q8 −q6 + q4 −q2 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2/q30\n121\n−(1 + 75 q18 −49 q28 + 42 q36 + 206 q26 −q46 + q52 + 7 q48 −49 q24 + 40 q40 + 75 q34 + 158 q30 + 158 q22\n+22 q8 + 7 q4 + 17 q14 + 17 q38 + q32 −q6 + q20 + 42 q16 + 22 q44 + 40 q12)q26\n12121\n(3 + 80 q18 + 10 q28 + 3 q36 + 26 q26 −3 q24 + q34 + 5 q30 + 52 q22 + 10 q8 + 5 q4 + 52 q14 + q2 + 5 q32\n+5 q6 −19 q20 −19 q16 −3 q12 + 26 q10)/q18\n1212121\n−(1 −2 q12 + 14 q10 −2 q8 + 3 q2 + 5 q6 + q20 + 5 q14 + 3 q18)/q10\n121212121\n1\nFigure 8: Coefficients of u1\n4\n40"},{"page":41,"text":"σ\nCoefficient\n1\n(q2 + 1)2(q4 + 1)2(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q4 −q2 + 1)4/q26\n121\n−(q36 + 3 q34 + 10 q32 + 19 q30 + 33 q28 + 53 q26 + 64 q24 + 91 q22 + 84 q20 + 116 q18 + 84 q16 + 91 q14\n+64 q12 + 53 q10 + 33 q8 + 19 q6 + 10 q4 + 3 q2 + 1)(q4 −q2 + 1)2/q22\n12121\n(80 q16 + 3 q26 + 26 q24 + 4 q22 + q32 + 7 q28 + 50 q20 + 3 q6 + 3 q2 + 50 q12 + 1 + 3 q30 + 7 q4 −14 q18\n−14 q14 + 4 q10 + 26 q8)/q16\n1212121\n−(3 + 5 q12 −2 q10 + 14 q8 + 5 q4 + q2 −2 q6 + q14 + 3 q16)/q8\n121212121\n1\nFigure 9: Coefficients of u1\n5\n41"},{"page":42,"text":"σ\nCoefficient\n∅\n(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n×(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2\n×(q4 + 1)2(q −1)4(q + 1)4(q2 + 1)4(q4 −q2 + 1)5/q46\n2\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n1\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42\n+12 q20 −12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30\n+11 q52 + q22 + 13 q48 −12 q46 + 13 q44 −4 q58 + 7 q56 −9 q54 + 2 q60)/q41\n12\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n21\n(2 + 13 q12 −12 q10 + 11 q8 + 7 q4 −4 q2 −9 q6 + q36 −10 q42 + 12 q20 −12 q50 −11 q32 −12 q14 −10 q18\n+13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52 + q22 + 13 q48 −12 q46 + 13 q44\n−4 q58 + 7 q56 −9 q54 + 2 q60)(q2 + 1)2(q4 −q2 + 1)2/q36\n212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30\n+2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q31\n121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20\n−4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52\n−77 q22 + 2 q48 −2 q46 −2 q44)/q31\n1212\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18\n+q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\n2121\n(2 −3 q12 −9 q10 −2 q8 + 2 q4 −4 q2 −2 q6 + q36 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16\n−110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44)/q26\nFigure 10: First nine coefficients of v0 in {Pσ} basis\n42"},{"page":43,"text":"σ\nCoefficient\n21212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n12121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 −q6 + q4 + 1)(q8 + q4 −q2 + 1)/q21\n121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n212121\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n2121212\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n1212121\n(q2 + 1)(q4 −q2 + 1)(q4 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n12121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n21212121\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n212121212\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n121212121\n−(q2 + 1)(q4 −q2 + 1)(q4 + 1)/q5\n1212121212\n1\n2121212121\n1\nFigure 11: Last twelve coefficients of v0 in {Pσ} basis\n43"},{"page":44,"text":"σ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(q4 + 1)2(q2 −q + 1)2(1 + q2 + q)2(q −1)4(q + 1)4(q2 + 1)2\n(q4 −q2 + 1)5)/q40\n2\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40\n−14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n1\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)3(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10\n+q40 −14 q26 −8 q34 −q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q35\n12\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34\n−q32 −3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n21\n((q4 −q2 + 1)2(1 −q20 −3 q38 + q44 −3 q14 + q4 −q2 + q12 + q8 −q6 −2 q10 + q40 −14 q26 −8 q34 −q32\n−3 q24 −10 q22 −3 q28 + q48 −10 q30 + q52 −8 q18 −q46 −q50 −2 q42))/q30\n212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q25\n1212\n((q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1))/q20\nFigure 12: A −-invariant element w in qRopp(3): the first eight coefficients\n44"},{"page":45,"text":"σ\nCoefficient\n2121\n(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16\n−13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)/q20\n21212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n12121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1))/q15\n121212\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n212121\n(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)/q10\n2121212\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n1212121\n−((q2 + 1)(q4 + 1)(q4 −q2 + 1))/q5\n12121212\n1\n21212121\n1\nFigure 13: A −-invariant element w in qRopp(3): the last nine coefficients\n45"},{"page":46,"text":"σ\nCoefficient\n∅\n((q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q2 + 1)3(q4 −q2 + 1)5)/q45\n2\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56\n−4 q58 + 12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n1\n−((q2 + 1)2(q4 −q2 + 1)3(q4 + 1)(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −12 q46 + 13 q12 −12 q10 + q24\n−10 q18 + 13 q16 −12 q14 + 16 q26 + 13 q48 + 13 q44 + 28 q30 −10 q42 + 11 q52 + 2 q60 −9 q54 + 7 q56 −4 q58\n+12 q20 −12 q50 −11 q28 + 16 q34 + q22 + q36 −11 q32 + q38 + 12 q40))/q40\n12\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8\n−q46 + q12 −2 q10 −3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50\n−3 q28 −8 q34 −10 q22 −q32 −3 q38 + q40))/q35\n21\n((q2 + 1)(q4 −q2 + 1)2(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(1 + q4 −q2 −q6 + q8 −q46 + q12 −2 q10\n−3 q24 −8 q18 −3 q14 −14 q26 + q48 + q44 −10 q30 −2 q42 + q52 −q20 −q50 −3 q28 −8 q34 −10 q22\n−q32 −3 q38 + q40))/q35\n212\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n121\n−((q4 −q2 + 1)(q4 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −2 q46 −3 q12 −9 q10 + 53 q24 −48 q18 + q16\n−13 q14 −110 q26 + 2 q48 −2 q44 −77 q30 −9 q42 + 2 q52 + 27 q20 −4 q50 + 53 q28 −48 q34 −77 q22\n+q36 + 27 q32 −13 q38 −3 q40))/q30\n1212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26\n+25 q24 −22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\nFigure 14: A −-invariant element x in qRopp(3): the first eight coefficients\n46"},{"page":47,"text":"σ\nCoefficient\n2121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q40 −3 q38 + 4 q36 −4 q34 + 7 q32 −10 q30 + 19 q28 −13 q26 + 25 q24\n−22 q22 + 40 q20 −22 q18 + 25 q16 −13 q14 + 19 q12 −10 q10 + 7 q8 −4 q6 + 4 q4 −3 q2 + 1)(q2 + 1))/q25\n21212\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n12121\n−((q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1))/q20\n121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))/q15\n212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q20 −3 q18 + q16 −3 q14 + 3 q12 −10 q10 + 3 q8 −3 q6 + q4 −3 q2 + 1)\n(q2 + 1))//q15\n2121212\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n1212121\n((q4 + 1)(q4 −q2 + 1)(−3 q6 −3 q10 + 12 q8 −q2 + 3 + 3 q4 + 3 q16 −q14 + 3 q12))/q12\n12121212\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n21212121\n((2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + 1))/q5\n212121212\n−((q4 + 1)(q4 −q2 + 1))/q4\n121212121\n−((q4 + 1)(q4 −q2 + 1))/q4\nFigure 15: A −-invariant element x in qRopp(3): the last eleven coefficients\n47"},{"page":48,"text":"5.4.1\nNonstandard positivity\nNow we shall describe evidence for Conjecture 3.6 in the case under consid-\neration.\nSo far we are assuming that Pi’s are as defined in the begining of Section\n7.1 in [GCT7]. But as explained towards the end of that section, the actual\nPi’s differ from these (chosen for convenience and simplicity) by a positive,\nunimodal factor ˆfp(q) ∈Q[q, q−1] given there. As it turns this does not\nmatter in the calculations so far, but for rescaling of the picture, but it does\nin the study of the positivity properties below. Rescaling of Pi by ˆfp(q)\nimplies that each structural coefficient or constant c(q) computed so far\nhas to be multiplied by an appropriate power of ˆfp(q)m(c), where m(c) is a\nnonnegative integer depending on c. In what follows, it is implicitly assumed\nthat each c computed so far has been rescaled by a suitable power ˆfp(q)n(c),\nwhere n(c) ≤m(c) is the smallest nonnegative integer chosen so that the\n(nonstandard) positivity property of the coefficients of c becomes apparent\nThe picture remains the same even if we were to multiply by ˆfp(q)m(c), but\nwe choose n(c) as small as possible to keep the the degrees of the polynomials\nfrom blowing up.\nFigures 16-20 show the vectors associated with the nonzero coefficients\nof v0 in the {Pσ} basis. The vector (as defined in the begining of Section 5)\nfor the coefficient corresponding to each σ is obtained by concatenating the\nrows in front of that σ. Figure 21 shows the vectors associated with the\ntraces of the nonzero coefficients of v1\n1 and Figures 22-23 show their norms.\nHere the trace and norm of an element in the algerbraic extension K of\nQ(q) is defined in the usual fashion as the sum and product of its images\nunder the Frobenius automorphisms of K over Q(q); they are coefficients\nof the minimal polynomial of the element. It can be seen that all vectors\nin Figures 16-23 are positive and nonincreasing, except for the vectors as-\nsociated with v0 for σ = 212, 121, 1212, 2121, which are almost positive and\nnonincreasing. It was verified with the help of computer that the vectors\nassociated with the coefficients of other canonical basis elements are simi-\nlarly either positive and nonincreasing or almost positive and nonincreasing.\nThis is in accordance with Conjectures 3.5-3.6.\nFigures 24-35 show vectors associated with a few multiplicative con-\nstants, taking norms and traces whenever necessary; the coefficient of bb′\nwith respect to the basis element b′′ is denoted by c(b, b′; b′′). Again it can\nbe seen that these vectors are positive and nonincreasing, except a few,\nwhich are almost positive and nonincreasing. It was verified with the help\n48"},{"page":49,"text":"of computer that the picture is the same for other multiplicative constants\nas well. This too is in accordance with Conjectures 3.5-3.6.\n49"},{"page":50,"text":"σ\nvector\n∅\n34116640\n34028832\n33766665\n33333910\n32736719\n31983492\n31084702\n30052720\n28901524\n27646408\n26303647\n24890162\n23423208\n21920062\n20397697\n18872456\n17359791\n15874002\n14428069\n13033484\n11700135\n10436190\n9248100\n8140602\n7116788\n6178174\n5324820\n4555450\n3867635\n3257976\n2722277\n2255718\n1853025\n1508640\n1216881\n972102\n768790\n601662\n465735\n356396\n269443\n201126\n148131\n107564\n76935\n54142\n37439\n25404\n16892\n10988\n6976\n4308\n2576\n1484\n821\n434\n217\n100\n40\n12\n2\n2\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\nFigure 16: The vectors associated with the coefficients of v0\n50"},{"page":51,"text":"σ\nvector\n1\n13180\n13086\n12992\n12744\n12496\n12124\n11752\n11225\n10698\n10112\n9526\n8890\n8254\n7584\n6914\n6294\n5674\n5083\n4492\n3979\n3466\n3036\n2606\n2256\n1906\n1638\n1370\n1178\n986\n840\n694\n603\n512\n450\n388\n335\n282\n259\n236\n206\n176\n153\n130\n116\n102\n85\n68\n54\n40\n34\n28\n21\n14\n9\n4\n4\n4\n2\n12\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\nFigure 17: The vectors associated with the coefficients of v0 (continued)\n51"},{"page":52,"text":"σ\nvector\n21\n3432\n3432\n3379\n3326\n3242\n3158\n3033\n2908\n2744\n2580\n2417\n2254\n2069\n1884\n1709\n1534\n1371\n1208\n1062\n916\n797\n678\n581\n484\n411\n338\n287\n236\n202\n168\n143\n118\n108\n98\n84\n70\n65\n60\n56\n52\n43\n34\n30\n26\n23\n20\n15\n10\n7\n4\n4\n4\n2\n212\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n121\n13352\n13285\n13218\n12957\n12696\n12341\n11986\n11462\n10938\n10358\n9778\n9131\n8484\n7812\n7140\n6498\n5856\n5240\n4624\n4088\n3552\n3079\n2606\n2236\n1866\n1552\n1238\n1026\n814\n646\n478\n373\n268\n198\n128\n93\n58\n35\n12\n4\n−4\n−4\n−4\n−4\n−4\n−4\n−4\n−2\n1212\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\n2121\n3472\n3472\n3427\n3382\n3293\n3204\n3093\n2982\n2810\n2638\n2483\n2328\n2132\n1936\n1772\n1608\n1423\n1238\n1098\n958\n820\n682\n587\n492\n398\n304\n249\n194\n153\n112\n85\n58\n37\n16\n10\n4\n2\n0\n−2\n−4\n−4\n−4\n−2\nFigure 18: The vectors associated with the coefficients of v0 (continued)\n52"},{"page":53,"text":"σ\nvector\n21212\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n12121\n5496\n5453\n5410\n5286\n5162\n5008\n4854\n4600\n4346\n4068\n3790\n3497\n3204\n2894\n2584\n2295\n2006\n1749\n1492\n1280\n1068\n889\n710\n586\n462\n366\n270\n215\n160\n117\n74\n56\n38\n27\n16\n11\n6\n3\n121212\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n212121\n1434\n1434\n1406\n1378\n1346\n1314\n1267\n1220\n1128\n1036\n961\n886\n799\n712\n633\n554\n470\n386\n334\n282\n231\n180\n146\n112\n82\n52\n42\n32\n24\n16\n11\n6\n3\n2121212\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\nFigure 19: The vectors associated with the coefficients of v0 (continued)\n53"},{"page":54,"text":"σ\nvector\n1212121\n1004\n992\n980\n957\n934\n908\n882\n829\n776\n716\n656\n597\n538\n472\n406\n346\n286\n240\n194\n158\n122\n94\n66\n51\n36\n26\n16\n11\n6\n3\n12121212\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n21212121\n258\n258\n252\n246\n245\n244\n237\n230\n208\n186\n168\n150\n132\n114\n95\n76\n60\n44\n37\n30\n23\n16\n11\n6\n3\n212121212\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\n121212121\n68\n67\n66\n65\n64\n63\n62\n58\n54\n49\n44\n39\n34\n28\n22\n17\n12\n9\n6\n4\n2\n1\nFigure 20: The vectors associated with the coefficients of v0 (continued)\n54"},{"page":55,"text":"σ\nvector\n1\n15864\n15864\n15680\n15496\n15160\n14824\n14334\n13844\n13228\n12612\n11932\n11252\n10505\n9758\n9008\n8258\n7534\n6810\n6126\n5442\n4834\n4226\n3702\n3178\n2738\n2298\n1948\n1598\n1333\n1068\n868\n668\n534\n400\n310\n220\n165\n110\n80\n50\n34\n18\n12\n6\n3\n121\n14690\n14690\n14484\n14278\n13930\n13582\n13065\n12548\n11880\n11212\n10509\n9806\n9032\n8258\n7498\n6738\n6031\n5324\n4684\n4044\n3498\n2952\n2515\n2078\n1729\n1380\n1124\n868\n698\n528\n405\n282\n212\n142\n101\n60\n42\n24\n15\n6\n3\n12121\n4974\n4974\n4886\n4798\n4683\n4568\n4380\n4192\n3914\n3636\n3367\n3098\n2803\n2508\n2217\n1926\n1669\n1412\n1205\n998\n825\n652\n529\n406\n316\n226\n172\n118\n89\n60\n41\n22\n14\n6\n3\n1212121\n708\n708\n694\n680\n673\n666\n642\n618\n565\n512\n464\n416\n365\n314\n262\n210\n170\n130\n106\n82\n62\n42\n30\n18\n11\n4\n2\nFigure 21: The vectors associated with the traces of the coefficients of v1\n1\n55"},{"page":56,"text":"σ\nvector\n1\n942062408\n940617136\n936306916\n929157344\n919242745\n906637444\n891460752\n873831980\n853909410\n831851324\n807845688\n782080468\n754764061\n726104864\n696321516\n665632656\n634256120\n602409744\n570301452\n538139168\n506112609\n474411492\n443199968\n412642188\n382872865\n354026712\n326205988\n299512952\n274016717\n249786396\n226859750\n205274540\n185039618\n166163836\n148631230\n132425836\n117511844\n103853444\n91399832\n80100204\n69893842\n60720028\n52513000\n45206996\n38734986\n33029940\n28027078\n23661620\n19873490\n16602612\n13795174\n11397364\n9362691\n7644664\n6204292\n5002584\n4007841\n3188364\n2519160\n1975236\n1537471\n1186744\n908860\n689624\n518854\n386368\n285084\n207920\n150131\n106972\n75388\n52324\n35881\n24160\n16048\n10432\n6681\n4164\n2550\n1508\n874\n484\n262\n132\n64\n28\n12\n4\n1\n121\n835471628\n833946584\n829404381\n821877948\n811459617\n798241720\n782371271\n763995284\n743308085\n720504000\n695809491\n669451020\n641674579\n612726160\n582858876\n552325840\n521370802\n490237512\n459149620\n428330776\n397976239\n368281268\n339402663\n311497224\n284682092\n259074408\n234750212\n211785544\n190216267\n170078244\n151372598\n134100452\n118233841\n103744800\n90582172\n78694800\n68016456\n58480912\n50013272\n42538640\n35978861\n30255780\n25293821\n21017408\n17356415\n14240716\n11608322\n9397244\n7554889\n6028664\n4775532\n3752456\n2925473\n2260620\n1732150\n1314316\n988221\n734968\n541214\n393616\n283165\n200852\n140775\n97032\n65989\n44012\n28926\n18556\n11701\n7160\n4299\n2484\n1405\n752\n391\n188\n88\n36\n14\n4\n1\nFigure 22: The vectors associated with the norms of the coefficients of v1\n1\n56"},{"page":57,"text":"12121\n100225178\n100001236\n99336179\n98236776\n96719406\n94800448\n92504960\n89858000\n86893481\n83645316\n80151751\n76451032\n72582611\n68585940\n64501436\n60369516\n56227388\n52112260\n48055958\n44090308\n40241933\n36537456\n32995876\n29636192\n26469934\n23508632\n20757113\n18220204\n15896382\n13784124\n11876674\n10167276\n8645972\n7302804\n6125315\n5101048\n4216788\n3459320\n2815838\n2273536\n1820593\n1445188\n1137127\n886216\n684162\n522672\n395267\n295468\n218449\n159384\n114887\n81572\n57144\n39308\n26622\n17644\n11492\n7284\n4521\n2704\n1573\n868\n461\n224\n102\n40\n15\n4\n1\n1212121\n2212002\n2205476\n2186350\n2155076\n2112378\n2058980\n1995675\n1923256\n1842975\n1756084\n1663849\n1567536\n1468102\n1366504\n1263855\n1161268\n1059834\n960644\n864459\n772040\n684050\n601152\n523831\n452572\n387494\n328716\n276117\n229576\n188839\n153652\n123592\n98236\n77108\n59732\n45640\n34364\n25488\n18596\n13346\n9396\n6493\n4384\n2894\n1848\n1142\n672\n377\n196\n95\n40\n15\n4\n1\nFigure 23: The vectors associated with the norms of the coefficients of v1\n1\n(continued)\n57"},{"page":58,"text":"9846\n9820\n9750\n9644\n9501\n9320\n9093\n8812\n8493\n8152\n7779\n7364\n6917\n6448\n5966\n5480\n4989\n4492\n4001\n3528\n3080\n2664\n2274\n1904\n1569\n1284\n1038\n820\n631\n472\n346\n256\n186\n120\n69\n44\n31\n16\n4\nFigure 24: The vector for the multiplicative constant c(v0, v1\n5; v0)\n13026\n12964\n12777\n12464\n12048\n11552\n10964\n10272\n9523\n8764\n7990\n7196\n6398\n5612\n4867\n4192\n3569\n2980\n2444\n1980\n1583\n1248\n967\n732\n539\n384\n267\n188\n131\n80\n42\n24\n16\n8\n2\nFigure 25: The vector for the multiplicative constant c(v0, v1\n4; v0)\n14180\n14088\n13832\n13432\n12894\n12224\n11448\n10592\n9682\n8744\n7800\n6872\n5976\n5128\n4342\n3632\n2998\n2440\n1956\n1544\n1202\n928\n706\n520\n374\n272\n198\n136\n88\n56\n36\n24\n16\n8\n2\nFigure 26: The vector for the multiplicative constant c(v0, v1\n3; v0)\n58"},{"page":59,"text":"1356922\n1356922\n1341857\n1326792\n1297628\n1268464\n1227083\n1185702\n1133960\n1082218\n1023821\n965424\n902616\n839808\n776306\n712804\n650849\n588894\n531320\n473746\n421861\n369976\n325182\n280388\n243022\n205656\n175677\n145698\n122582\n99466\n82260\n65054\n52971\n40888\n32575\n24262\n19002\n13742\n10480\n7218\n5402\n3586\n2573\n1560\n1107\n654\n431\n208\n135\n62\n35\n8\n4\nFigure 27: The vector for the trace of the multiplicative constant c(v0, v1\n1; v0)\n59"},{"page":60,"text":"128607887512140\n128408739182416\n127813071267725\n126826189536408\n125456853005552\n123717149426056\n121622324577366\n119190568251368\n116442761279740\n113402187959064\n110094219510476\n106545974210632\n102785960441146\n98843708903656\n94749400393129\n90533495522016\n86226372383558\n81857978142544\n77457499613617\n73053057887224\n68671430858636\n64337807515464\n60075576278933\n55906149694176\n51848826289840\n47920690427296\n44136549385454\n40508906927184\n37047971392480\n33761696363504\n30655850888930\n27734116255008\n24998205742188\n22448003806144\n20081720818165\n17896059499880\n15886389332178\n14046925218184\n12370907081114\n10850777077832\n9478351722036\n8244986211000\n7141729026165\n6159464877872\n5289044837698\n4521402501856\n3847655805468\n3259194107520\n2747750779948\n2305461534304\n1924909254250\nFigure 28:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0)\n60"},{"page":61,"text":"1599156102128\n1321763961652\n1086804278768\n888858554957\n723010747256\n584832826172\n470364742664\n376089969502\n298907782312\n236103258694\n185315973800\n144508153155\n111933043504\n86104032184\n65765045520\n49862540548\n37519404368\n28010898646\n20742786784\n15231640340\n11087321280\n7997549136\n5714462144\n4043023409\n2831123144\n1961213430\n1343311944\n909211769\n607734400\n400883734\n260758832\n167107927\n105406152\n65367982\n39805384\n23766532\n13889944\n7930388\n4412904\n2386644\n1250232\n631716\n306184\n141372\n61592\n25006\n9272\n3049\n848\n190\n32\n4\nFigure 29:\nThe vector for the norm of the multiplicative constant\nc(v0, v1\n1; v0) (continued)\n61"},{"page":62,"text":"612764\n609940\n601539\n587774\n568990\n545654\n518332\n487666\n454355\n419136\n382753\n345926\n309336\n273610\n239300\n206862\n176658\n148958\n123938\n101678\n82176\n65362\n51106\n39226\n29503\n21696\n15554\n10828\n7279\n4686\n2851\n1604\n800\n316\n52\n−68\n−101\n−90\n−64\n−38\n−18\n−6\n−1\nFigure 30: The vector for the multiplicative constant c(v1\n5, v1\n5; v1\n5)\n9738\n9694\n9563\n9348\n9052\n8676\n8228\n7732\n7209\n6658\n6077\n5484\n4900\n4332\n3784\n3268\n2794\n2360\n1962\n1604\n1292\n1028\n809\n626\n471\n344\n246\n172\n116\n76\n50\n32\n17\n6\n1\nFigure 31: The vector for the multiplicative constant c(v1\n4, v1\n4; v1\n4)\n5.5\nExperimental evidence for crystalization\nLet (Lρ\nα, Bρ\nα) be an upper crystal basis of W ρ\nq,α as in Conjecture 2.2. Since\nit is also a local crystal basis of W ρ\nq,α as an Hq-module, there is a crystal\ngraph over Bρ\nα whose connected components correspond to irreducible Hq-\nsubmodules of W ρ\nq,α. The elements of Bρ\nα that correspond to the highest\nweight nodes of these connected components are called the highest weight\ncrystal elements of the upper crystal base (Lρ\nα, Bρ\nα) with respect to Hq.\n5.6\nKronecker problem: n = 2, r = 3\nConsider again the special case of the Kronecker problem for n = 2 and\nr = 3 as in Section 5.1. Thus H = Gl2 × Gl2, G = Gl4 with H embedded\ndiagonally, X = Xq = Vq ⊗Wq is the standard four dimensional represen-\ntation of Hq, where Vq ∼= Wq is the standard two representation of GLq(2).\n62"},{"page":63,"text":"81298\n80886\n79660\n77650\n74908\n71510\n67546\n63110\n58306\n53254\n48074\n42870\n37740\n32786\n28097\n23732\n19734\n16144\n12987\n10258\n7938\n6010\n4449\n3212\n2251\n1526\n999\n628\n374\n208\n107\n50\n20\n6\n1\nFigure 32: The vector for the multiplicative constant c(v1\n3, v1\n3; v1\n3)\n976672\n974152\n971632\n954184\n936736\n915476\n894216\n860888\n827560\n791463\n755366\n713035\n670704\n627743\n584782\n540877\n496972\n454258\n411544\n373144\n334744\n298301\n261858\n232239\n202620\n175584\n148548\n128938\n109328\n91580\n73832\n62540\n51248\n41348\n31448\n25972\n20496\n15650\n10804\n8712\n6620\n4729\n2838\n2215\n1592\n986\n380\n312\n244\n113\n−18\n−9\n0\n−15\n−30\n−15\nFigure 33: The vector for the trace of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n63"},{"page":64,"text":"80255448992640\n80138274925167\n79787740098774\n79206792680861\n78400301248652\n77374989989221\n76139349486502\n74703523453335\n73079174468928\n71279332267508\n69318226392392\n67211105097154\n64974044241996\n62623750255631\n60177360010954\n57652240221271\n55065789702876\n52435247936361\n49777512793214\n47108969815061\n44445335040340\n41801513256271\n39191473712706\n36628144773627\n34123327701708\n31687629536553\n29330415837966\n27059783595895\n24882552858520\n22804275396566\n20829260005316\n18960613784310\n17200296928344\n15549188413159\n14007161465134\n12573167695619\n11245327257264\n10021022340390\n8896992772108\n7869432671924\n6934086054972\n6086339325247\n5321309766266\n4633929430149\n4019023248184\n3471380269320\n2985817677720\n2557237529092\n2180675978000\n1851344837246\n1564665615708\nFigure 34: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n64"},{"page":65,"text":"1316296386448\n1102151942468\n918417720699\n761557915658\n628318302423\n515724544156\n421076738262\n341940657408\n276136144552\n221723424848\n176988052223\n140424814566\n110720869883\n86738676456\n67499233418\n52165764220\n40027907728\n30486730148\n23040841355\n17273591498\n12841266667\n9462395940\n6908272529\n4994579014\n3573971135\n2529616304\n1769689525\n1222700770\n833511103\n559989972\n370279871\n240567010\n153252303\n95475892\n57962133\n34120478\n19338377\n10435196\n5256321\n2374094\n864025\n140844\n−155240\n−236220\n−220838\n−171968\n−120043\n−77166\n−46045\n−25460\n−12974\n−6040\n−2518\n−900\n−255\n−50\n−5\nFigure 35: The vector for the norm of the multiplicative constant c(v1\n1, v1\n1; v1\n1)\n(continued)\n65"},{"page":66,"text":"Let\nx1 = v1 ⊗w1, x2 = v1 ⊗w2, x3 = v2 ⊗w1, x4 = v2 ⊗w2,\nbe the standard basis of Xq. Let b1, . . . , b4 be the corresponding standard\ncrystal basis of L(Xq). The irreducible representations of GH\nq that occur in\nX⊗3\nq\nare:\n1. CH,3\nq\n(X), the 16-dimensional the degree-three component of the braided\nsymmetric algebra of GH\nq ,\n2. ∧H,3\nq\n(X), the four dimensional degree three component of the braided\nexterior algebra of GH\nq ,\n3. two copies of the 16-dimensional GH\nq -representation Wq,(2,1),1(X) de-\nfined in [GCT4] (it is denoted by Vq,(2,1),1(X) there), and\n4. two copies of the 4-dimensional representation Wq,(2,1),2(X) of GH\nq as\nalso defined there (it is called Vq,(2,1),2(X) there).\nEmbeddings of the braided symmetric and exterior algebra components in\nX⊗3\nq\nare uniquely defined. We denote their embedded images by CH,3\nq\n(X)\nand ∧H,3\nq\n(X) again.\nWe choose appropriate embeddings of Wq,(2,1),2(X)\nand Wq,(2,1),2(X) in X⊗3\nq\nand denote them by the same symbols again. As\nHq = GLq(V ) × GLq(W)-modules,\nCH,3\nq\n(X)\n=\nVq,(3)(V ) ⊗Vq,(3)(W) ⊕Wq,(2,1)(V ) ⊗Vq,(2,1)(W),\n∧H,3\nq\n(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W),\nWq,(2,1),1(X)\n=\nVq,(2,1)(V ) ⊗Vq,(3)(W) ⊕Vq,(3)(V ) ⊗Vq,(2,1)(W)\nWq,(2,1),2(X)\n=\nVq,(2,1)(V ) ⊗Vq,(2,1)(W).\nIt was verified by computer that they have upper crystal bases as per\nConjecture 2.2.\nThe highest weight crystal elements with respect to Hq\nfor these upper crystal bases, as shown separately for each module, are as\nfollows; we denote the monomial basis element bi1 ⊗bi2 ⊗bi3 of B(X⊗3\nq ) by\nbi1i2i3.\n66"},{"page":67,"text":"CH,3\nq\n(X) :\n{b114 + b141, b111}.\n∧H,3\nq\n(X) : {b123 + b132}.\nWq,(2,1),1(X) :\n{b121, b131}.\nWq,(2,1),2(X) :\n{b114 −b141}.\nThe highest weight crystal elements whose monomial support have size\ntwo correspond to the four dimensional Hq-module Vq,(2,1)(V ) ⊗Vq,(2,1)(W).\nThe element b111 corresponds to the Hq-module Vq,(3)(V ) ⊗Vq,(3)(W), the\nelement b121 to the Hq-module Vq,(3)(V ) ⊗Vq,(2,1)(W), and b131 to the Hq-\nmodule Vq,(2,1)(V ) ⊗Vq,(3)(W). Notice that not all highest weight crystal\nelements have monomial supports of size one as in the standard setting.\n5.6.1\nH = sl2, G = sl4\nNow we consider the case when Hq = Glq(2), Xq its four dimensional irre-\nducible representation, and GH\nq\nas in Section 5.3. Let W0, . . . , W5 be the\nirreducible representations of GH\nq occuring in X⊗3\nq\nas defined in Section 6.1.2\nof [GCT7], with W0 = CH,3\nq\n[X].\nAs Hq-modules,\nW0\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2),\nW1\n∼=\nVq,(6,3)(2),\nW2\n∼=\nVq,(6,3)(2),\nW3\n∼=\nVq,(8,1)(2),\nW4\n∼=\nVq,(5,4)(2),\nW5\n∼=\nVq,(7,2)(2),\n(19)\nwhere Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to the\npartition λ. Their dimensions are 16, 4, 4, 8, 2 and 6, respectively. Though\nW1 and W2 are isomorphic as Hq-modules, they are nonisomorphic as GH\nq -\nmodules.\nIt was verified by computer that they–or rather their embeddings in\nX⊗3\nq –have upper crystal bases as per Conjecture 2.2. The highest weight\ncrystal elements of of the embedding of W1, . . . , W5 have monomial supports\nof size one. Let bi1i2i3 = bi1 ⊗bi2 ⊗bi3 denote the monomial basis elements of\nB(X⊗3\nq ). Then the highest weight crystal elements of the uniquely defined\nembedding of W0 are b111 and b = b113 + b131. The latter element b here\nhas monomial support of size two, a phenomenon not seen in the standard\nsetting.\nNonzero coefficients of the element x in the lattice L(C3,H\nq\n(X))\n67"},{"page":68,"text":"whose crystalization is b is shown in Figure 36, wherein x1, . . . , x4 are the\nstandard basis vectors of Xq.\n6\nComplexity theoretic properties of the canonical\nbasis\nIn the standard setting, elements of the canonical basis of Vq,λ are indexed\n(labelled) by semistandard tableau, for H = GL(V ), and by LS-paths [Li],\nfor general semisimple H. And combinatorial analogues of Kashiwara’s crys-\ntal operators [Li] on these labels can be computed efficiently [GCT6]. This\nis enough to imply a #P-formula for the generalized Littlewood-Richardson\ncoefficient though the canonical basis of Vq,λ is hard to compute.\nIn the same spirit, it may be conjectured that the canonical basis of\nO(MH\nq (X) (or rather the set of its labels) has additional complexity the-\noretic properties (to be described in the full version), based on its cellular\nand refined sub-cellular decomposition (Conjecture 2.6), that imply a pos-\nitive #P-formula for the multiplicity nα\nπ of the irreducible Hq-module Vq,π\nin Wq,α. This would solve the problem P1 in [GCT7].\nSimilarly, let mα\nλ denote the multiplicity of the Specht module Sλ of\nthe symmetric group Sr corresponding to the partition λ in Limq→1Tq,α,\nconsidered as an Sr-module. It may be conjectured that the canonical basis\nBH\nr (q) (or rather the set of its labels) has similar additional complexity\ntheoretic properties (to be described in the full version) based on its cellular\nand quasi-subcellular decompositions (Conjecture 3.3). This would imply a\npositive #P-formula for the multiplicity mπ\nλ, as needed in the problem P2\nin [GCT7].\nReferences\n[BBD]\nA.\nBeilinson,\nJ.\nBernstein,\nP.\nDeligne,\nFaisceaux\npervers,\nAst ́erisque 100, (1982), Soc. Math. France.\n[BZ]\nA. Berenstein, S. Zwicknagl, Braided symmetric and exterior alge-\nbras, arXiv:math/0504155v3, April, 2007.\n[DJM]\nM. Date, M. Jimbo, T. Miwa, Representations of Uq(ˆgl(n, C)) at\nq = 0 and the Robinson-Schensted correspondence, in Physics and\n68"},{"page":69,"text":"Monomial\nCoefficient\nx1 ⊗x1 ⊗x3\n−(q4 + 1)2(q2 + 1)4(q4 −q2 + 1)5(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q20 −2 q18 + q16 + q10 −q8 + q6 + q4 −q2 + 1)\n(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\nx1 ⊗x2 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q37\nx1 ⊗x3 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n(q6 −q5 + q4 −q3 + q2 −q + 1)(q16 + q14 −q12 + q10 + q4 −q2 + 1)(q4 + 1)2(q2 −q + 1)2(q2 + q + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q46\nx2 ⊗x1 ⊗x2\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)\n(q4 −q3 + q2 −q + 1)(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5\n(q4 −q2 + 1)5/q40\nx2 ⊗x2 ⊗x1\n−(q2 + 1)5(2 q6 + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q4 −q3 + q2 −q + 1)\n(q4 + q3 + q2 + q + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2(q −1)5(q + 1)5(q4 −q2 + 1)5/q43\nx3 ⊗x1 ⊗x1\n−(q2 + 1)4(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5\n+q4 + q3 + q2 + q + 1)(q12 −q10 + q8 + q6 + q4 −q2 + 1)(q2 + q + 1)2(q2 −q + 1)2(q4 + 1)2\n(q −1)4(q + 1)4(q4 −q2 + 1)5/q42\nFigure 36: Nonzero coefficients of x in the lattice L(C3,H\nq\n(X))\n69"},{"page":70,"text":"Mathematics of Strings, World Scientific, Singapore, 1990, pp. 185-\n211.\n[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n[Dri]\nV. Drinfeld, Quantum groups, Proc. Int. Congr. Math. Berkeley,\n1986, vol. 1, Amer. Math. Soc. 1988, 798-820.\n[GCTflip1] K. Mulmuley, On P. vs. NP, geometric complexity theory, and\nthe flip I: a high-level view, Technical Report TR-2007-13, Com-\nputer Science Department, The University of Chicago, September\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu\n[GCT4] K.\nMulmuley,\nM.\nSohoni,\nGeometric\ncomplexity\nthe-\nory\nIV:\nquantum\ngroup\nfor\nthe\nKronecker\nproblem,\ncs.\nArXiv preprint cs. CC/0703110,\nMarch,\n2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu\n[GCT6] K. Mulmuley, Geometric complexity theory VI: the flip via sat-\nurated and positive integer programming in representation the-\nory and algebraic geometry, Technical report TR 2007-04, Comp.\nSci. Dept., The University of Chicago, May, 2007. Available at:\nhttp://ramakrishnadas.cs.uchicago.edu.\n[GCT7] K. Mulmuley, Geometric complexity theory VII: Nonstandard\nquantum group for the plethysm problem, technical report TR-\n2007-14, computer science dept., The university of Chicago, Sept.\n2007. Available at: http://ramakrishnadas.cs.uchicago.edu.\n[Ji]\nM. Jimbo, A q-difference analogue of U(G) and the Yang-Baxter\nequation, Lett. Math. Phys. 10 (1985), 63-69.\n[Kas1]\nM. Kashiwara, Crystalizing the q-analogue of universal enveloping\nalgebras, Comm. Math. Phys. 133 (1990), 249-260.\n[Kas1]\nM. Kashiwara, On crystal bases of the q-analogue of universal en-\nveloping algebras, Duke Math. J. 63 (1991), 465-516.\n[Kas2]\nM. Kashiwara, Global crystal bases of quantum groups, Duke\nMathematical Journal, vol. 69, no.2, 455-485.\n[KL1]\nD. Kazhdan, G. Lusztig, Representations of Coxeter groups and\nHecke algebras, Invent. Math. 53 (1979), 165-184.\n70"},{"page":71,"text":"[KL2]\nD. Kazhdan, G. Lusztig, Schubert varieties and Poincare duality,\nProc. Symp. Pure Math., AMS, 36 (1980), 185-203.\n[Kli]\nA. Klimyk, K. Schm ̈udgen, Quantum groups and their representa-\ntions, Springer, 1997.\n[Li]\nP. Littelmann, Paths and root operators in representation theory,\nAnn. of Math. 142 (1995), 499-525.\n[Lu1]\nG. Lusztig, Canonical bases arising from quantized enveloping al-\ngebras, J. Amer. Math. Soc. 3, (1990), 447-498.\n[Lu2]\nG. Lusztig, Introduction to quantum groups, Birkh ̈auser, 1993.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[So]\nW. Soergel, Kazhdan-Lusztig polynomials and a combinatoric for\ntilting modules, Representation theory 1, (1997)\n[St]\nR. Stanley, Positivity problems and conjectures in algebraic com-\nbinatorics, In Mathematics: frontiers and perspectives, 295-319,\nAmer. Math. Soc. Providence, RI (2000).\n[W]\nS. Woronowicz: Compact matrix pseudogroups, Commun. Math.\nPhys. 111 (1987), 613-665.\n71"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Let H be a complex connected classical reductive group, X = Vμ(H) its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"irreducible polynomial representation with highest weight μ, G = GL(X),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"case, i.e., when H = G, this specializes to the standard quantum group, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"in the Kronecker case, i.e., when H = GL(V ) × GL(W), X = V ⊗W with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"expansion at q = 1–the Laplace expansion of a minor of a nonstandard","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"For the sake of simplicity, let us assume that H = GL(V ). Let Hq =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"algebra of Hq = GLq(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"To quantize the homomorphism ρ : H →G = GL(X) as in (1), the arti-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"q (X)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"the ring of rational functions in q regular at q = 0 and q = ∞, respectively.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"V ∼= Q(q) ⊗Q[q,q−1] VQ ∼= Q(q) ⊗A L0 ∼= Q(q) ⊗ ̄","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"(a) E = VQ ∩L0 ∩L∞→L0/qL0 is an isomorphism.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"Let R = O(Mq(V )).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"L = L(R) ⊂R, an involution −of R, a Q[q, q−1]-submodule RQ ⊂R, and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"R = ⊕πV ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"as a bi-GLq(V )-module, where Vq,π = Vq,π(V ) is the irreducible polynomial","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"(L, B) = ⊕π(L∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"q,π = ρ(Vq,π) be-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"q,π is a GLq(V )-module. Thus V ⊗r = V ρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"π = L(V )⊗r ∩V ρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"= L(V )⊗r ∩V ρ,⊥","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"It follows from Kashiwara’s work [Kas1] that L(V )⊗r = Lρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"Let B(V ) = {bi = ψ(vi)} denote the basis of L(V )/qL(V ), where ψ :","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"L(V ) →L(V )/qL(V ) is the natural projection. Let B(V )⊗r = {bi1 ⊗· · · ⊗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"b =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"(Lπ, Bπ) = (Lρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"depends on the existence of a compact real form Uq(V ) ⊆GLq(V ) = Hq.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"goal is to construct an analogous triple for the matrix coordinate ring S =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"functions in K that are regular at q = 0 and q = ∞, respectively. Let KQ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"be the integral closure of Q[q, q−1] in K. Clearly, KQ ∩AK ∩ ̄AK = ˆQ, where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"lattice at q = 0 we mean an AK-lattice, by a lattice at q = ∞a ̄AK-lattice,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"q . Let Xq = Vq,μ denote the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"quantization of X = Vμ; i.e, the Hq-module with highest weight μ. Since","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"module. We assume that limq→0(zi −xi) = 0; this can always be arranged.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"If r = 1, Wq,α = X, so this is trivial. Otherwise, choose any β of degree","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"Wq,β ⊗X = ⊕βjWq,βj,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"any j such that Wq,βj ∼= Wq,α. This fixes an embedding of Wq,α in Wq,β ⊗X.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"q,α = ρ(Wq,α) be its image.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"q -module. Thus X⊗r = W ρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"α = L(X)⊗r ∩W ρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"= L(X)⊗r ∩W ρ,⊥","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"Proposition 2.1 L(X)⊗r = Lρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"real number as q →0. This means ˆy is regular at q = 0 and hence belongs","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"Let B(X) = {bi = φ(xi)} denote the basis of the ̃Q-module L(X)/qL(X),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"section are different from the bi’s in Section 2.1). Let B(X)⊗r = {bi1 ⊗· · ·⊗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"b =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"Assuming this, we let (Lα, Bα) = (Lρ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"S = O(MH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"(L(S), B(S)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"at q = 0 (with appropriate normalization). Let SQ be the KQ-forms (KQ-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"(λr, λl) and (μr, μl), let uv = q(λr,μr)−(λl,μl) ̄v ̄u, where ( , ) denotes the usual","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"j = ui","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"j, and ̄q = q−1. This","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"at q = ∞. In analogy with the standard setting, we can now ask:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"the map ψ : E = SQ ∪L(S) ∪ ̄L(S) →L(S)/qL(S) of ˆQ-modules an isomor-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"Fix r. Let Sr denote the degree r component of S. Let A = AH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"r =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"A = ⊕αW ∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"L(A) = {a ∈A | ⟨a, L(Sr)⟩⊆AK}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"⟨ ̄a, s⟩= ⟨a, ̄s⟩−.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"AQ = {a ∈A | ⟨a, Sr,Q⟩⊆KQ},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"where Sr,Q = Sr ∩SQ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"(L(A), B(A)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"the triple T(b) = (α, μ1, μ2). We call it the type of b. The types T(b)’s can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"type T = (α, μ1, μ2). Put a partial order, which we shall again denote by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"Next fix a b ∈B(A). Let T = T(b) be its type. We shall associate a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"L(A≤T) = L(A) ∩A≤T .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"is a represention of A = AH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"η : A ֒→Z = End(X⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"q ) = (X⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"Let L(Z) = L(X⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"zi1,...,ir;j1,...,jr = x∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"y ∈A, let w = η(y). Express w in the canonical basis of Z:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"w = η(y) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"defined degree d(y, z) at q = 0 (the same as the order of its pole at q = ∞);","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"if a(y, z) = 0, we define d(y, z) = −∞. We define the degree complexity d(y)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"Let U = Uq(H) be the Drinfeld-Jimbo enveloping algebra of Hq, U −the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"subalgebra generated by its generators Fi’s. For any string ν = ν1, ν2, . . . of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"of Xq, we define its length to be |x| = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"letting w = η(y), w′ = η(y′), we say that d(y) ≤d(y′) if for every z: either","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"the canonical basis CB(A) of A. The canonical basis CB(Sr) = {xb} of Sr","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"a natural projection from Z = End(X⊗d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"that injects Sr,Q = Sr ∩SQ into ZQ. So the analogue of Proposition 2.4 does","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"∆(b) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"Conjecture 2.7 (Positivity) In the Kronecker case–i.e. when H = GL(V )×","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"GL(W), X = V ⊗W with the natural H-action–each coefficient g(q) of any","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"bb′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"[X]) = dim(Cr[X]). But when CH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"b(q) = (−1)e(q −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"behaviour of b(q) at q = 1 (cf. Section 6), so this relaxed saturation form","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"Lusztig polynomials at q = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"r = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"In the standard setting–i.e., when H = G–this basis would conjecturally","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"r (q) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"GT(O(Mq(V ))) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"We proceed by induction on r, the case r = 1 being easy. The following","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"in this setting. It is defined to be the tuple (αr, αr−1, . . .), with αr = α, of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"C(r) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"ψ : R(r) →R(r)/q1/2R(r). Let B(r) = {bT } be the basis of R(r)/q1/2R(r).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"Let R≤T = R(r) ∩B≤T , ̄R≤T = ̄R(r) ∩B≤T, and B≤T","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"= BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"algebras, say Z = Hk1(q) ⊗· · · ⊗Hkl(q), where Hj(q) denotes the Hecke","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"r,Q ⊆ZQ = Hk1,Q × · · · × Hkl,Q,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"y =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"q = ∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"If a(y, z) = 0, we define d(y, z) = −∞.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"We call eT the canonical basis element associated with T, and E(r) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"any rational a, let ca,T = qac′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"This fixes aT . Furthermore, ψT = ψaT ,T , bT = baT ,T , and eT in Conjec-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"c = c0 +","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"s(q) = (−1)e(q1/2 −q−1/2)e′g(q)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"ee′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"with nonzero coefficient in the expansion of ee′ specialized at q = 1. Let ≺L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"q = 1. That is, when the BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"is specialized at q = 1, so as to become a representation Yq=1 of C[Sr], the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"partial order on its left quasi-sub-cells induces a composition series of Yq=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"in correspondence with C in this composition series of Yq=1, where λ(C)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"specialization at q = 1 and projection, yield a basis of Sλ(C). It may be con-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"cialization at q = 1 of the Kazhdan-Lusztig basis of the quantized Specht","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"a =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"if, for each i, the monomials m of length i with am ̸= 0 in this expres-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"q = 0) of am for any m of length i, and l is the maximum length of m","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"with am ̸= 0 in the expression (16); by definition, ˆdi(a) = −∞if there is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"no m in (16) of length i with am ̸= 0. We order these degree complexities","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"ial special cases of the nonstandard algebra B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"First, some notation. Given a string σ = i1 · · · ik of positive integers,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"Kronecker problem: n = 2, r = 3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"Consider B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"3 (q) in the special case of the Kronecker problem for n = 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"and r = 3. Thus H = Gl2 ×Gl2, and G = Gl4 with H embedded diagonally.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"Let Pi, i = 1, 2, be as in Section 3 and [GCT4]. The nonstandard canonical","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"basis of B was computed in [GCT4] by an ad hoc method for r = 3, but it","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"c1 = q6 + 2q5 + 3q4 + 4q3 + 3q2 + 2q + 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"c2 = q4 + q3 + 4q2 + q + 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"b1 = −(q2 + 1)2/q2, and b2 = (q + 1)2/q.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"Σ = c1P1 −c2P121 + P12121,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"1 = b1P1 + P121,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"i = 1, 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"12 = b1P12 + P1212,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"i = 1, 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"2 = b1P2 + P212,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"i = 1, 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"21 = b1P21 + P2121,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"i = 1, 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"Uμ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"ize at q = 1 to the two-dimensional Specht module of the symmetric group","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"Kronecker case, H = SL2, r = 4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"For the Kronecker case, H = Gl2 × GL2, G = GL4, and r = 4, we could","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"H = sl2, G = sl4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"Now we study the nonstandard algebra B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"3 (q), when H = Gl2, X is its","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"four dimensional irreducible representation, and G = GL(X) = Gl4. It is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"disc =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"x = disc1/2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"Since disc is not a square, x does not belong to Q(q). Let K = Q(q)[x]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"ui =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"= u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"= Q2u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"i , for 1 ≤i ≤5. Let u0 = 1 (this definition of u0 is different","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"vi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"2 (q) ⊂B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"resentation of the symmetric group S2 at q = 1. Then the nonstandard","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"If b = u0, then T(b) = [Σ, Σ′]. If b = u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"i , then T(b) = [χi, Σ′]. If b = u21","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"then T(b) = [χi, μ′].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"If b = u2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"i , then T(b) = [χi, Σ′].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"If b = u12","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"T(b) = [χi, μ′].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"Let R(3) be the AK-lattice generated by E(3), ̄R(3) = (R(3))−. Let Ropp(3)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"T (0) ̸= {0} when T = T(b) is the nonstandard tableau","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"associated with b = v2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"T (0), for b = v2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"5, T = T(b), is generated by the two elements w and x","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"w = a0v0 + c1v1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"This means ψT (w) = 0. It can be verified that w belongs to B≤T . Similarly,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"T ( ̄b), b = v2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"5, ̄b = ψ(b), T = T(b), is a linear combi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"and Figure 9 that b = v2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"is Z = H3(q)⊗9 with dimension 10077696 = 69. It is impossible to carry out","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"1) = c(γ12","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"1) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"21) = −(q2 + q + 1) ∗(q −1)2/q2;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"21) = −(1 + q)2 ∗(q2 + q + 1) ∗(q −1)2/q3;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"1; Σ) = c(γ2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"1; Σ) = c(γ1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"1; Σ) = c(γ2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"1; Σ) = 1/q ∗(1 + q)2;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"12; Σ) = c(γ1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"12; Σ) = (1 + q)2 ∗(6 ∗q + 5 ∗q2 + 5)/q2;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"21) = (1 + q2)2 ∗(q2 + q + 1) ∗(q −1)2/q4;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"21; Σ) = c(γ2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"21; Σ) = (1 + q)2 ∗(2 ∗q2 + q + 1) ∗(q2 + q + 2)/q3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"in the Kronecker case, n = 2, r = 3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"sociated with v0 for σ = 212, 121, 1212, 2121, which are almost positive and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"Kronecker problem: n = 2, r = 3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"Consider again the special case of the Kronecker problem for n = 2 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"r = 3 as in Section 5.1. Thus H = Gl2 × Gl2, G = Gl4 with H embedded","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"diagonally, X = Xq = Vq ⊗Wq is the standard four dimensional represen-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"tation of Hq, where Vq ∼= Wq is the standard two representation of GLq(2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"x1 = v1 ⊗w1, x2 = v1 ⊗w2, x3 = v2 ⊗w1, x4 = v2 ⊗w2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"Hq = GLq(V ) × GLq(W)-modules,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"H = sl2, G = sl4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"Now we consider the case when Hq = Glq(2), Xq its four dimensional irre-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"of [GCT7], with W0 = CH,3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"of size one. Let bi1i2i3 = bi1 ⊗bi2 ⊗bi3 denote the monomial basis elements of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"embedding of W0 are b111 and b = b113 + b131. The latter element b here","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"(labelled) by semistandard tableau, for H = GL(V ), and by LS-paths [Li],","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"q = 0 and the Robinson-Schensted correspondence, in Physics and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":106982,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}