structured/0709.0749.structured.json

{"paper_meta":{"paper_id":"arxiv:0709.0749","title":"0709.0749","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:0709.0749v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VII: Nonstandard\nquantum group for the plethysm problem\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\n(Technical Report TR-2007-14\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nNovember 3, 2018\nAbstract\nThis article describes a nonstandard quantum group that may be\nused to derive a positive formula for the plethysm problem, just as the\nstandard (Drinfeld-Jimbo) quantum group can be used to derive the\npositive Littlewood-Richardson rule for arbitrary complex semisimple\nLie groups. The sequel [GCT8] gives conjecturally correct algorithms\nto construct canonical bases of the coordinate rings of these nonstan-\ndard quantum groups and canonical bases of the dually paired non-\nstandard deformations of the symmetric group algebra.\nA positive\n#P-formula for the plethysm constant follows from the conjectural\nproperties of these canonical bases and the duality and reciprocity\nconjectures herein.\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1\n\n1\nIntroduction\nThe following is a fundamental problem in representation theory [GCT6,\nMc, St]:\nProblem 1.1 (Plethysm problem)\nFind an explicit positive (#P-) formula in the spirit of the Littlewood-\nRichardson rule for the plethysm constant aπ\nλ,μ. For given partitions λ, μ\nand π, this is the multiplicity of the irreducible representation Vπ(H) of\nH = GLn(C) in the irreducible representation Vλ(G) of G = GL(X), where\nX = Vμ = Vμ(H) is an irreducible representation of H.\nHere Vλ(G) is\nconsidered an H-module via the representation map ρ : H →G.\n(Generalized plethysm problem):\nThe same as above, letting H be any complex, semisimple (or, more\ngenerally, reductive) classical Lie group, λ a dominant weight of G, π and\nμ dominant weights of H.\nThis article describes a quantum group that may be used to derive such\na positive formula, just as the standard (Drinfeld-Jimbo) quantum group\n[Dri, Ji, RTF] can be used to derive the positive Littlewood-Richardson rule\nfor arbitrary complex semisimple Lie groups [Kas1, Li, Lu2]; the results here\nwere announced in [GCT4] (most of the results here also hold for nonclassical\nH, though we shall only worry about classical H here). For the significance\nof a positive formula in the context of geometric complexity theory, see\n[GCTflip1]. The approach that we wish to follow is:\n1. Find a quantization of the homomorphism\nH →G\n(1)\nof the form\nHq →GH\nq ,\n(2)\nwhere Hq is the standard Drinfeld-Jimbo quantization of H, and GH\nq\nis the new nonstandard quantization of G that we seek.\n2. Develop a theory of canonical (local/global crystal) bases for the rep-\nresentations of GH\nq in the spirit of the canonical bases [Kas1, Lu1] for\nthe representations of the standard quantum group.\n2\n\n3. Derive the required explicit positive formula for the plethysm constant\nfrom the properties of the canonical bases.\nThe following addresses the first step.\nTheorem 1.2 (cf.\nSection 2) There exists a possibly singular quantum\ngroup GH\nq\nsuch that the homomorphism (1) can be quantized in the form\n(2).\nFurthermore, all finite dimensional polynomial representations of GH\nq\nare completely reducible, and a quantum analogue of the Peter-Weyl theorem\nholds for the matrix coordinate ring of GH\nq .\nFor the precise meaning of the various terms here, see Section 2. Here and\nin what follows, we assume that the base field is C = C(q), q complex. But\na suitable algebraic extension of Q(q) will also suffice for our purposes; see\nSection 6 for a discussion on the base field.\nWhen H = G, GH\nq specializes to the standard quantum group Hq. When\nH = GL(V ) × GL(W), G = GL(X), X = V ⊗W with natural H-action, it\nreduces to the quantum group in [GCT4] for the Kronecker problem.\nWe call GH\nq\nthe nonstandard quantum group associated with the em-\nbedding (1). It can be singular in general. That is, its determinant may\nvanish, and hence, the antipode need not exist. Strictly speaking, it should\nhence be called a nonstandard quantum semi-group. We still use the term\ngroup, because this object has characteristic features of the standard quan-\ntum group, such as semisimplicity of polynomial representations, Peter-Weyl\ntheorem, and most importantly, conjectural existence of canonical bases for\nits representations and the matrix coordinate ring.\nWe also construct (Section 5) a nonstandard quantization BH\nr = BH\nr (q) of\nthe group algebra C[Sr] of the symmetric group Sr whose relationship with\nGH\nq\nis conjecturally akin to that of the Hecke algebra with the standard\nquantum group. Specifically, let Xq denote the irreducible representation\nVq,μ of Hq with highest weight μ; it is the usual quantization of X = Vμ.\nThen:\nConjecture 1.3 (Nonstandard duality)\n(1) The left action of GH\nq on X⊗r\nq\nand the right action BH\nr (q) on X⊗r\nq\nde-\ntermine each other.\n(2) There is a one-to-one correspondence between the irreducible polynomial\nrepresentations of GH\nq of degree r and the irreducible representations of BH\nr\n3\n\nso that, as a bimodule,\nX⊗r\nq\n=\nM\nα\nWq,α ⊗Tq,α,\n(3)\nwhere Wq,α runs over the irreducible polynomial representations of GH\nq\nof\ndegree r, and Tq,α denotes the irreducible representation of BH\nr (q) in corre-\nspondence with Wq,α.\nThe irreducible representations Wq,α here need not be q-deformations of\nthe irreducible representations of G, because GH\nq\nis, in general, a nonflat\ndeformation of G. This means the Poincare series of GH\nq need not coincide\nwith that of G. Our first goal is to associate with each Weyl module Vλ of\nG a possibly reducible representation V H\nq,λ of GH\nq , called the q-analogue of\nVλ, so that\nlimq→1V H\nq,λ ∼= Vλ\nas an H-module. In this context:\nConjecture 1.4 (Nonstandard reciprocity) Let λ be a partition of size\nr. Let\nV H\nq,λ =\nM\nα\nmα\nλWq,α,\nwhere mα\nλ denotes the multiplicity of the Specht module Sλ of the symmetric\ngroup Sr in Tq,α(1) = limq→1Tq,α, as defined in Section 6. Then V H\nq,λ is a\nq-analogue of Vλ in the sense defined above.\nThus the multiplicity of the GH\nq -module Wq,α in V H\nq,λ is equal to the mul-\ntiplicity of the Specht module Sλ in the specialization of Tq,α at q = 1.\nA more refined form of this conjecture is given in Section 6. Both duality\nand reciprocity are supported by experimental evidence; cf. Section 7.\nBy the conjectural reciprocity,\naπ\nλ,μ =\nX\nα\nmα\nλnα\nπ,\nwhere nα\nπ is the multiplicity of the irreducible Hq-module Vq,π in Wq,α. Hence\nProblem 1.1 can be decomposed into the following two subproblems:\n(P1): Find a positive (#P-) formula for the multiplicity nα\nπ.\n(P2): Find a positive (#P-) formula for the multiplicity mα\nλ.\n4\n\nThe article [GCT8] gives conjecturally correct algorithms to construct\na canonical basis of the matrix coordinate ring of GH\nq\nwhose conjectural\nproperties would imply a positive formula as needed in the first problem,\nand a canonical basis of BH\nr\nwhose conjectural properties would imply a\npositive formula as needed in the second problem.\nAt present, we cannot prove correctness of these algorithms nor the re-\nquired conjectural properties, because we are unable to deal with the high\ncomplexity of the nonstandard quantum group. Specifically, as we shall see\nin Section 4, the formulae for the minors of the nonstandard group turn out\nto be highly nonelementary in contrast to the elementary formulae for the\nminors of the standard quantum group. The coefficients of these formulae\ndepend on the multiplicative structural constants of canonical bases akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nconstructed by Kashiwara and Lusztig [Kas2, Lu2]. To get explicit formulae\nfor these structural constants, one needs interpretations for them akin to\nthe interpretations for the Kazhdan-Lusztig polynomials and multiplicative\nstructural constants of the canonical basis of the coordinate ring of the stan-\ndard quantum group in terms of perverse sheaves [KL2, Lu1, BBD]. Thus,\nthe linear algebra for the nonstandard quantum group–i.e. the theory of its\nminors–is already highly nonelementary in contrast to the linear algebra for\nthe standard quantum group. This is why its representation theory may turn\nout to be far more complex. In particular, we cannot explicitly construct\nnor classify its irreducible polynomial representations. Of course, all this\nand much more would follow if correctness of the algorithms in [GCT8] for\nconstructing canonical bases and their conjectural properties can be proved.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for helpful discussions,\nespecially for bringing the reference [Ro] to our attention, and for the help\nin explicit computations in Section 7.2 in MATLAB.\n2\nNonstandard quantum group\nWe describe in this section the construction of the nonstandard quantum\ngroup GH\nq\nin Theorem 1.2. The reader may refer to [GCT4] for the full\ndetails in a nontrivial special case of the plethysm problem, called the Kro-\nnecker problem. For the sake of simplicity, we assume here that H = GL(V )\n(type A). Let X = Vμ(H) be its irreducible polynomial representation. The\n5\n\ngoal is to quantize the homomorphism\nH = GL(V ) →G = GL(X).\nLet H and G be the Lie algebras of H and G. We follow the terminology in\n[Kli], which will be our standard reference on quantum groups.\nThe standard quantum group Hq = GLq(V ) associated with GL(V ) can\nbe defined by first constructing the coordinate algebra O(Mq(V )) of the stan-\ndard quantum matrix space Mq(V ) as a suitable FRT-algebra [RTF]. The\ncoordinate ring O(GLq(V )) of GLq(V ) is obtained by localizing O(Mq(V ))\nat the suitably defined quantum determinant. The Drinfeld-Jimbo univer-\nsal enveloping algebra Uq(G) [Dri, Ji] of GLq(V ) can then be defined dually.\nSpecifically, let J be the maximal ideal of the elements in O(Mq(V )) which\nvanish at the identity–i.e. on which ǫ, the counit, vanishes. Then Uq(G) can\nbe identified with the space of linear functions on O(Mq(V )) which vanish\non Jr for some integer r > 0 depending on the linear function.\nAnalogously, we first construct the nonstandard matrix coordinate ring\nO(MH\nq (X)) of the (virtual) nonstandard matrix space MH\nq (X), and then\ndefine the nonstandard quantized universal enveloping algebra U H\nq (G) by\ndualization. We define the nonstandard quantum group GH\nq as the virtual\nobject whose universal enveloping algebra is U H\nq (G). The construction would\nyield natural bialgebra homomorphisms from Uq(H) to U H\nq (G) and from\nO(MH\nq (X)) to O(Mq(V ), thereby giving the desired quantizations of the\nhomomorphisms U(H) →U(G) and O(M(X)) →O(M(V )). This is what\nis meant by the quantization (2) of the map (1). The determinant of GH\nq\nmay vanish, and hence, we cannot, in general, define its coordinate ring\nO(GH\nq ) by localizing O(MH\nq (X)). Fortunately, this will not matter since the\ncoordinate ring O(MH\nq (X)) and the nonstandard quantized algebra BH\nr (q)\n(Section 5) together contain conjecturally all the information that we need\n(cf. Conjecture 1.4), and have properties similar to that of the standard\nmatrix coordinate ring O(Mq(V )) and the Hecke algebra; cf. Theorem 2.1\nbelow.\nThe nonstandard matrix coordinate ring O(MH\nq (X)) is constructed as\nfollows.\nLet ˆRH\nX,X be the ˆR matrix of Xq = Vq,μ considered as an Hq-\nmodule [Kli]. Here and in what follows, we sometimes denote Xq by X; the\nmeaning should be clear from the context. It is well known that ˆRH\nX,X is\ndiagonalizable and that its each eigenvalue is of the form + or −qa/2 for\nsome integer a [Kli]. Let\nI = P +,H\nX,X + P −,H\nX,X ,\n(4)\n6\n\nbe the associated spectral decomposition of the identity, where P +,H\nX,X and\nP −,H\nX,X denote the projections of Xq ⊗Xq on the eigenspaces of ˆRH\nX,X for\neigenvalues with + and −sign, respectively. Let u be a variable matrix\nspecifying a generic transformation from X to X. Let ui\nj denote its variable\nentries. Then O(MH\nq (X)) is defined to be the FRT bialgebra [RTF] associ-\nated with the transformation P +,H\nX,X , or equivalently, P −,H\nX,X . That is, it is the\nquotient of C⟨ui\nj⟩modulo the relations\nP +,H\nX,X (u ⊗u) = (u ⊗u)P +,H\nX,X ,\n(5)\nor equivalently,\nP −,H\nX,X (u ⊗u) = (u ⊗u)P −,H\nX,X .\n(6)\nAn alternative definition of O(MH\nq (X)) is as follows. Let SH\nq (X ⊗X),\nthe symmetric subspace of X ⊗X, be the image of P +,H\nX,X , and AH\nq (X ⊗X),\nthe antisymmetric subspace of X ⊗X, the image of P −,H\nX,X [Kli]. In other\nwords, SH\nq (X ⊗X) is defined by the equation\nP −,H\nX,X x1x2 = 0,\n(7)\nwhere x1 = x ⊗I and x2 = I ⊗x, and AH\nq (X⊗X) is defined by the equation\nP +,H\nX,X x1x2 = 0.\n(8)\nThe braided symmetric algebra [BZ, Ro] CH\nq [X] of X is defined to be\nthe algebra over the entries xi’s of x subject to the relation (7). It will be\ncalled the coordinate ring of the virtual quantum space XH\nsym. Similarly, the\nbraided exterior algebra ∧H\nq [X] of X is defined to be the algebra over the\nentries xi’s of x subject to the relation (8). It will called the coordinate ring\nof the virtual quantum space XH\n∧. Let CH,r\nq\n[X] and ∧H,r\nq\n[X] be the degree\nr components of CH\nq [X] and ∧H\nq [X], respectively.\nIt is known [BZ] that\nthe dimensions of CH,r\nq\n[X] and ∧H,R\nq\nare bounded by the dimensions of the\nclassical Cr[X] and ∧r[X], respectively. But unlike in the standard setting,\nthe dimensions can be strictly less [BZ, Ro]. That is, CH\nq [X] and ∧H\nq [X]\nare, in general, nonflat deformations of the classical symmetric and exterior\nalgebras C[X] and ∧[X]. For example, ∧H,3\nq\n[X] = 0 when H = sl2(C) and\nX is the four dimensional irreducible representation of sl2(C) [BZ].\nThe equation (5) or (6) after reformulation just says that the defining\nrelation (7) of XH\nsym–or equivalently, the defining relation (8) of XH\n∧–is pre-\nserved under the left and right actions of u on x given by x →ux and\nxt →xtu.\n7\n\nThis means CH\nq [X] and ∧H\nq [X] have left and right coactions of O(MH\nq (X)).\nWe define the left and right nonstandard minors of GH\nq to be the matrix co-\nefficients (in a suitable basis specified later) of the left and right coactions\non ∧H\nq [X]. If ∧H,dim(X)\nq\n[X] ̸= 0, then we define the determinant of GH\nq to\nbe the matrix coefficient of the action of O(MH\nq (X)) on ∧H,dim(X)\nq\n[X]. But\nit can vanish, as it does for H = sl2(C), dim(X) = 4. The nonstandard\nminors will be discussed in more detail in Section 4.\nLet J be the ideal of elements in O(MH\nq (X)) on which the counit ǫ van-\nishes. Then the nonstandard universal enveloping algebra U H\nq (G) is defined\nto be the space of linear functions of O(MH\nq (X)) which vanish on Jr for\nsome r > 0 depending on the linear function.\nThe following is a precise form of Theorem 1.2.\nTheorem 2.1 (1) There is a natural bialgebra homomorphism from O(MH\nq (X))\nto O(Mq(V )).\nThis gives the desired quantization of the homomorphism\nO(M(X)) →O(M(V )).\n(2) The matrix coordinate ring O(MH\nq (X)) of GH\nq is cosemisimple. Hence,\nits every finite dimensional corepresentation is completely reducible as a di-\nrect sum of irreducible corepresentations.\n(3) The q-analogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)).\n(4) The nonstandard enveloping algebra U H\nq (G) is a bialgebra with a compact\nreal form (a ∗-structure) such that X⊗r\nq\nis its unitary representation with\nrespect to the Hermitian form on X⊗r\nq\ninduced by the standard Hermitian\nform on Xq. There is a bialgebra homomorphism form Uq(H) to U H\nq (G).\nThis gives a desired quantization of the homomorphism U(H) →U(G).\nHere the standard Hermitian form on Xq is the one that is Uq-invariant,\nwhere Uq ⊆Hq is the compact real form (the unitary subgroup) of Hq in the\nsense of Woronowicz [W]. The special case of this theorem in the context of\nthe Kronecker problem was proved in [GCT4] on the basis of Woronowicz’s\nwork [W]. The latter is no longer applicable in the general context here,\nsince the determinant of GH\nq may vanish, and hence, we cannot, in general,\nconvert O(MH\nq (X)) into a Hopf algebra by localization at the determinant.\n8\n\nFortunately, this does not matter since U H\nq (G) still has a compact real form,\nwhose existence can be proved using the spectral properties of ˆRH\nX,X.\nWe also call Wq,α here a polynomial representation of GH\nq . By a poly-\nnomial representation of U H\nq (G) we mean a representation that is induced\nby a (finite dimensional) corepresentation of O(MH\nq (X)). It is completely\nreducible by cosemsimplicity of O(MH\nq (X)).\nIt may be conjectured that\nevery finite dimensional representation of U H\nq (G) is completely reducible (as\nin the standard case), though we shall not need this more general fact.\nThe standard Drinfeld-Jimbo enveloping algebra has an explicit presen-\ntation in the form of explicit generators (ei, fi, Ki) and explicit relations\namong them. It will be interesting to find an analogous explicit presenta-\ntion for U H\nq (G); cf. Section 4 for the problems that arise in this context.\n3\nNonstandard q-Schur algebra\nIn the standard setting, the q-Schur algebra Ar = Ar(q) is defined to be the\ndual O(Mq(V ))∗r of the degree r component O(Mq(V ))r of the standard\nmatrix coordinate algebra O(Mq(V )). Thus Ar(q) acts on V ⊗r from the\nleft. It is known [Kli] that it is the centralizer in End(V ⊗r) of the right\naction of the Hecke algebra Hr(q) on V ⊗r.\nAnalogously, we define the nonstandard q-Schur algebra AH\nr\n= AH\nr (q)\nto be the dual O(MH\nq (X))∗r of the degree r component O(MH\nq (X))r of\nthe nonstandard matrix coordinate algebra O(MH\nq (X)). Thus AH\nr (q) acts\non X⊗r from the left.\nAs per the nonstadard duality conjecture (Con-\njecture 1.3), it is the centralizer in End(X⊗r) of the right action of the\nnonstandard quantized algebra BH\nr (q) (cf. Section 5) on X⊗r.\nEvery irreducible corepresentation Wq,α of O(MH\nq (X)) of degree r can\nalso be considered as a representation of AH\nr (q), and conversely, every irre-\nducible representation of AH\nr (q) arises in this way. Theorem 2.1 now imme-\ndiately implies:\nTheorem 3.1 (1) The nonstandard q-Schur algebra AH\nr (q) is semisimple.\nHence, its every finite dimensional representation is completely reducible as\na direct sum of irreducible representations.\n(2) The q-analogue of the Peter-Weyl theorem in this case is the Wederburn\n9\n\nstructure theorem for AH\nr (q):\nAH\nr (q) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible representations of AH\nr (q).\n(3) The nonstandard q-Schur algebra AH\nr (q) has a compact real form (a ∗-\nstructure) such that X⊗r\nq\nis its unitary representation with respect to the\nHermitian form on X⊗r\nq\ninduced by the standard Hermitian form on Xq.\n4\nNonstandard minors\nIn this section, we give a conjectural formula for the Laplace expansion\nof the minors of GH\nq .\nThe Laplace expansion for the standard quantum\ngroup GLq(V ) is based on the simple relation defining the standard exterior\nalgebra ∧q[V ], namely\nv2\ni = 0\nand\nvivj = −q−1vjvi,\nfor\ni < j.\nThis explains why the Laplace expansion in the standard setting is obtained\nfrom the classical Laplace expansion by simply substituting −q for −1. We\nneed a similar explicit formula for multiplication in CH\nq [X] to get an explicit\nformula for Laplace expansion in the nonstandard setting.\n4.1\nKronecker problem\nWe begin with a special case that arises in the context of the Kronecker\nproblem [GCT4] when H = GL(V ) × GL(W) and X = V ⊗W, with the\nnatural H-action. The article [GCT4] gives a formula for the column or row\nexpansion of the minor of GLH\nq (X) in this special case in terms of fundamen-\ntal Clebsch-Gordon coefficients for the standard quantum groups GLq(V )\nand GLq(W). But this formula cannot be extended for the general Laplace\nexpansion since Clebsch-Gordon coefficients are not well defined when the\nunderlying tensor products do not have multiplicity-free decompositions as\nin the fundamental case. Here we give a formula for general Laplace expan-\nsion of the minors of GLH\nq (X) in this case.\nWe begin by recalling that when V = W ∗the braided symmetric al-\ngebra CH\nq [X] = CH[W ∗⊗W] is isomorphic to the matrix coordinate ring\nO(Mq(W)) of the standard matrix space Mq(W) [GCT4]. For this, we have:\n10\n\nTheorem 4.1 (Kashiwara and Lusztig [Kas2, Lu2]) The coordinate ring\nO(Mq(W)) has an (upper) canonical basis.\nThis can be naturally and easily extended to:\nTheorem 4.2 The braided symmetric coordinate algebra CH\nq [X] = CH\nq [V ⊗\nW], H = GL(V ) × GL(W), has an (upper) canonical basis.\nThe exterior form of this result is:\nTheorem 4.3 The exterior coordinate algebra ∧H\nq [V ⊗W], H = GL(V ) ×\nGL(W), also has an (upper) canonical basis.\nLusztig [Lu2] has conjectured that the multiplicative and comultiplica-\ntive structural constants of the canonical basis of O(Mq(W)) are polynomials\nin q and q−1 with nonnegative integer coefficients; i.e., belong to N[q, q−1].\nAnalogous conjecture can be made for ∧H\nq [V ⊗W]. Specifically, it can be\nconjectured that for any canonical basis elements b and b′ in ∧H\nq [V ⊗W]:\nbb′ =\nX\nb′′\nǫ(b, b′, b′′)cb′′\nb,b′b′′,\n(9)\nwhere the sign ǫ(b, b′, b′′) is 1 or −1 and the coefficient cb′′\nb,b′ ∈N[q, q−1]. And\nconversely, any b′′ ∈∧H,r′′\nq\n[V ⊗W] can be expressed as:\nb′′ =\nX\nb,b′\nǫ′(b, b′, b′′)db,b′\nb′′ bb′,\n(10)\nwhere b and b′ run over elements of ∧H,r\nq\n[V × W] and ∧H,r′\nq\n[V × W] respec-\ntively with r′′ = r + r′, the sign ǫ′(b, b′, b′′) is 1 or −1, and db,b′\nb′′ ∈N[q, q−1].\nTo prove nonnegativity of the coefficients of cb′′\nb,b′ and db,b′\nb′′ , one needs in-\nterpretations for them in terms of perverse sheaves [BBD] in the spirit of\nKazhdan-Lusztig [KL2] and Lusztig [Lu1].\nWe now define the (left or right) minors of GH\nq\nwith respect to the\ncanonical basis ∧H,r\nq\n[V ⊗W] to be the matrix coefficients of the (left or right)\ncoaction of O(MH\nq (V ⊗W)). We shall call them (left or right) canonical\nminors. Then:\n11\n\nTheorem 4.4 A canonical minor of degree r′′ of GLH\nq (X), H = GL(V ) ×\nGL(W), admits a Laplace expansion in terms of canonical minors of degree\nr and r′ with r′′ = r + r′. The coefficients of this Laplace expansion are\nquadratic forms in the structural constants cb′′\nb,b′ and db,b′\nb′′\nabove.\nAn explicit formula for Laplace expansion here (omitted) is similar to the\none in Proposition 6.1 of [GCT4] with these structural constants in place of\nthe Clebsch-Gordon coefficients there (which are not well defined for general\nLaplace expansion).\n4.2\nGeneral nonstandard setting\nNow let us turn to the general case. The conjecturally correct algorithm\nin [GCT8] for constructing a canonical basis of O(MH\nq (X)) also yields, as\na byproduct, conjectural canonical bases of ∧H\nq [X] and CH\nq [X] as implicitly\nsought in [BZ]. We define the (left or right) minors of GH\nq in general to be\nthe matrix coefficients of the (left or right) coaction of O(MH\nq (X)) in this\ncanonical basis of ∧H\nq [X]. We call these nonstandard canonical minors, or\nsimply nonstandard minors.\nOne can define structural constants cb′′\nb,b′ and db,b′\nb′′ analogous to the ones\nin (9) and (10) in this case. With this:\nTheorem 4.5 Analogue of Theorem 4.4 holds in general.\nLaplace expansion in the standard setting is used as a straightening\nrelation to construct standard monomial bases of the coordinate ring and\nirreducible representations of GLq(X). In this sense, Laplace expansion is\na mother relation that governs the representation theory of the standard\nquantum group. Similarly, the nonstandard Laplace expansions in Theo-\nrems 4.4 and 4.5 are expected to be mother relations governing the rep-\nresentation theory of the nonstandard quantum group GH\nq . In particular,\nan explicit interpretation for the structural coefficients cb′′\nb,b′ and db,b′\nb′′\nakin\nto the ones based on perverse sheaves for the Kazhdan-Lusztig polynomials\n[KL2] and the multiplicative structural constants of the canonical basis for\nthe standard quantum group [Lu2] is necessary to get fully explicit formulae\nfor the nonstandard minors, and hence, for constructing explicit bases for\nthe irreducible polynomial representations and the matrix coordinate ring\nof GH\nq . In particular, this seems necessary for proving correctness of the\n12\n\nalgorithms in [GCT8] for constructing nonstandard canonical bases for the\npolynomial representations and the matrix coordinate ring of GH\nq . This also\nseems necessary for finding an explicit presentation of the nonstandard uni-\nversal enveloping algebra U H\nq (G) in the spirit of the explicit presentation of\nthe Drinfeld-Jimbo enveloping algebra. Specifically, we expect the coeffi-\ncients occuring in such an explicit presentation to depend on the structural\nconstants such as cb′′\nb,b′ and db,b′\nb′′ above.\n5\nNonstandard quantized algebra\nWe now construct a nonstandard quantization BH\nr (q) of the symmetric\ngroup ring C[Sr] which conjecturally has the same relationship with GH\nq\nthat the Hecke algebra Hr(q), the standard deformation of C[Sr], has with\nthe standard quantum group. For the sake of simplicity, we assume that\nH = GL(V ).\nChoose a standard embedding of X = Vμ(H) in V ⊗d, where d is the size\nof the partition μ. That is, choose a Young symmetrizer cμ ∈C[Sr] such\nthat V ⊗d ·cμ, the image of V ⊗d under the right action of cμ, is isomorphic to\nX = Vμ(H). Let zμ ∈Hd(q) be the quantization of cμ such that V ⊗d\nq\n· zμ ∼=\nXq = Vq,μ. Here Vq denotes the quantization of V and Vq,μ the irreducible\nHq module with highest weight μ.\nAn explicit expression of zμ may be\nfound in [DJ]. Let Zq = V ⊗d\nq\n. Let ˆRH\nZ,Z denote the ˆR-matrix of Zq as an\nHq-module. Let rZ ∈H2d(q), 1 ≤i < r, be the element whose right action\non Zq ⊗Zq = V ⊗2d\nq\ncoincides with the action of ˆRH\nZ,Z. One can easily write\ndown an explicit expression for rZ in terms of the generators of H2d(q).\nNow consider the right action of Hs(q), s = dr, on Z⊗r\nq\n= V ⊗s\nq\n, which\ncommutes with the left action of Hq = GLq(V ). Let rZ,i ∈Hs(q), 1 ≤i < r,\nbe the element whose right action on Z⊗r\nq\ncoincides with the action of ˆRH\nZ,Z\non the product of the i-th and (i + 1)-st factors of Z⊗r\nq . Thus rZ,i is the\nimage of rZ under the obvious embedding of H2d(q) in Hs(q) depending on\ni. One can thus write down an explicit expression for rZ,i in terms of the\ngenerators of Hs(q). Let\nrH\nX,i = zλ,i · zλ,i+1 · rZ,i,\nwhere zλ,i ∈Hs(q) denotes an explicit element whose action on the i-th\nfactor of Z⊗r\nq\ncoincides with the action of zλ on that factor–it is the image of\nzλ under the obvious embedding of Hd(q) in Hs(q) depending on i. Then the\nright action of rH\nX,i on Z⊗r\nq\ncorresponds to the action of ˆRH\nX,X on the product\n13\n\nof the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq . Let p+,H\nX,i , p−,H\nX,i ∈Hs(q) be\nthe polynomials in rH\nX,i whose actions on Z⊗r\nq\ncorrespond to the actions of\nthe positive and negative projection operators P +,H\nX,X and P −,H\nX,X in eq. (4) on\nthe tensor product of the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq ; one can\nwrite down these polynomials explicitly, using the known explicit spectral\nform of ˆrH\nX,i.\nWe define the nonstandard quantized algebra BH\nr (q) to be the subalgebra\nof Hs(q) generated by the explicit elements p+,H\nX,i , or equivalently, p−,H\nX,i . In\ngeneral, it is a nonflat deformation of C[Sr]. That is, its dimension can be\nlarger than that of C[Sr]. It can be shown to be semisimple. Its right action\non X⊗r\nq\ncommutes with the left action GH\nq by the defining equation (5) of\nGH\nq . Conjecture 1.3 says that its relationship with GH\nq\nis akin to that of\nHr(q) with the standard quantum group Gq = GLq(X).\nThe Hecke algebra has an explicit presentation in terms of explicit re-\nlations among its generators.\nIt will be interesting to find an analogous\nexplicit presentation for BH\nr (q). Its complexity would be much higher than\nthat of the Hecke algebra as indicated by the concrete computations in\n[GCT4].\nSpecifically, we expect an explicit presentation for BH\nr (q) with\ndefining relations whose coefficients are akin to the structural constants cb′′\nb,b′\nand db,b′\nb′′ in Section 4 and have a topological interpretation akin to the one\nfor Kazhdan-Lusztig polynomials. Such an explicit presentation is needed\nto prove correctness of the algorithm in [GCT8] to construct a canonical\nbasis of BH\nr .\nRemark: We can also define a (possibly singular) quantum group ˆGH\nq , in-\nstead of GH\nq , by substituting ˆRH\nX,X in place of P +,H\nX,X in the defining equation\n(5). One can then define a deformation ˆBH\nq (r) of C[Sr] that is conjecturally\npaired with ˆGH\nq , as GH\nq is with BH\nq (r). The main results (semisimplicity, and\nq-analogue of the Peter-Weyl theorem) also hold for these objects. Further-\nmore, variants of the algorithms in [GCT8] can be conjectured to provide\ncanonical bases for these as well.\nHowever, the Poincare series of ˆGH\nq\nis\nmuch smaller than that of GH\nq , and for this and other reasons, it does not\nseem possible to use these objects in the context of the plethysm problem.\nHowever, these may be interesting intermediate quantum objects to study\nnevertheless.\n14\n\n6\nRefined reciprocity\nWe now describe a refinement of the reciprocity conjecture (Conjecture 1.4)\nthat specifies precisely how the decomposion (3) of X⊗r\nq ,\nX⊗r\nq\n=\nM\nWq,α ⊗Tq,α,\n(11)\nas a GH\nq × BH\nr (q)-bimodule, tends to the decomposition\nX⊗r =\nM\nλ\nVλ ⊗Sλ\n(12)\nof X⊗r as a G × Sr-bimodule, as q →1, and gives an explicit realization\nwithin X⊗r\nq\nof the q-analogue V H\nq,λ of Vλ as in Conjecture 1.4. Here, as usual,\nVλ denotes the Weyl module of G, and Sλ the Specht module of Sr.\nFirst, we have to define the multiplicity mα\nλ of a Specht module Sλ in the\nspecialization Tq,α(1) of Tq,α at q = 1. In this context, it may be remarked\nthat though B = BH\nr (q) is semisimple, its specialization B(1) at q = 1 need\nnot be semisimple; see Section 7.1 for an example. Clearly, every represen-\ntation of Sr is also a representation of B(1), though not always conversely.\nBut it may be conjectured that every irreducible B(1)-representation is also\nan irreducible Sr-representation, i.e., a Specht module. Fix any (maximal)\ncomposition series of Tq,α(1) as a B(1)-module. We define the multiplic-\nity mα\nλ to be the number of factors in this (or any such) series that are\nisomorphic to the specht module Sλ.\nSince B is semisimple (cf. Section 5), it admits a Wederburn structure\ndecomposition of the form\nB =\nM\nU α,\nU α = Tq,α,L ⊗Tq,α,R,\n(13)\nwhere α is as in (11), and Tq,α,L and Tq,α,R denote the left and right irre-\nducible B-modules indexed by α. We call this a complete Wederburn struc-\nture decomposition. Here we are assuming that the base field is C = C(q),\nq complex. This complete decomposition would also hold if the base field\nis instead an appropriate algebraic extension K of Q(q). In the standard\nsetting of Hecke algebras, K = Q(q) suffices. This need not be so in the\nnonstandard setting. That is, an algebraic extension of Q(q) may be ac-\ntually necessary for a complete decomposition of the above form to hold;\nsee Section 7.1 for an example. If the base field is Q(q), each U α in the\nWederburn structure decomposition need not be, in general, of the form\n15\n\nTq,α,L ⊗Tq,α,R as above, but rather it would be isomorphic to the endomor-\nphism ring of Tq,α over the division algebra EndB(Tq,α). One has to take\nsimilar variations of the nonstandard q-analogue of the Peter-Weyl theorem\n(Theorem 2.1 (3)) and the duality conjecture (Conjecture 1.3) if the base\nfield is Q(q). However, for the reciprocity conjecture, it is necessary to take\nthe base field as C(q), q complex, or an algebraic extension K of Q(q) as\ndescribed above. We assume this in the rest of this section. See Section 7.1\nfor an example wherein reciprocity fails over Q(q).\nFix any right cell, i.e., an irreducible right B-subrepresentation within\nU α. Let us denote it by Tq,α,R again. Fix a maximal composition series as\na B(1)-module of the specialization Tq,α,R(1) of Tq,α,R at q = 1:\nˆTα,0 ⊂ˆTα,1 ⊂· · · ⊂ˆTα,l(α) = Tq,α,R(1).\nLet {xi} denote the upper canonical basis of Xq as an Hq-module.\nConjecture 6.1 (Nonstandard refined reciprocity)\nThere exists a basis Zα of Tq,α,R for each α with a filtration\nZα,0 ⊂Zα,1 ⊂· · · ⊂Zα,l(α) = Zα,\nsuch that:\n1. The specialization Zα,i(1) of Zα,i at q = 1 is a basis of ˆTα,i.\n2. Let zj\nα,i denote the basis elements in Zα,i \\ Zα,i−1.\nLet λα,i be the\npartition such that ˆTα,i/ ˆTα,i−1 ∼= Sλα,i as a B(1)-module (or equiv-\nalently as an Sr-module).\nFor any α, i, define the left GH\nq -module\nWq,α,i = ∪jX⊗r\nq\n· zj\nα,i. By the duality conjecture (Conjecture 1.3),\nWq,α,i ⊆Wq,α ⊗Tq,α ⊆X⊗r\nq .\n(14)\nWe define its specialization W1,α,i at q = 1, also denoted by Wq,α,i(1),\nas follows. Let a(α, i) be the largest nonnegative integer such that the\nlimit vector\nlimq→1xi1 ⊗· · · ⊗xir.zj\nα,i/(q −1)a(α,i),\nis well defined for any i1, . . . , ir and j. We define W1,α,i to be the span\nof such limits at q = 1. Then, W1,α,i is a left G-module contained\nwithin the component Vλα,i ⊗Sλα,i ⊆X⊗r in (12).\n16\n\n3. For any fixed partition λ,\nM\nα\nM\ni\nW1,α,i = Vλ ⊗Sλ ⊆X⊗r,\n(15)\nwhere, for a given α, i ranges over all indices such that λα,i = λ.\nFurthermore, it may be conjectured that the canonical basis of Tq,α,R\nin terms of the P-monomials as defined in [GCT8] has this property–this\nwould make everything in the conjecture above explicit.\nThe refined reciprocity conjecture basically says that there is no infor-\nmation loss in the nonstandard setting despite the lack of flatness. In fact,\nit can be thought of as a variant of flatness.\n7\nEvidence for duality and reciprocity\nHere we describe some concrete computations carried out in MATLAB/Maple\nthat support duality and reciprocity conjectures.\nNotation: We denote the q-Weyl module of Gq for a partition λ by Vq,λ(Gq).\nWe denote Vq,λ(GLq(Cn)) by Vq,λ(n).\n7.1\nExample 1\nLet H = GL(C2), H = gl(C2), X = V(3)(H) is its four dimensional irre-\nducible representation, and G = GL(X) = GL(C4). Then Hq = GLq(C2),\nGq = GLq(C4), and Hq = glq(C2). We shall verify duality and reciprocity\nin this case for r = 3. This example is interesting because, as shown in [BZ],\nthe degree three component ∧H,3\nq\n[X] of the braided exterior algebra vanishes\nin this case. We expect that the results in this section can be extended to\nany irreducible representation X of H. But we shall confine ourselves to the\ncase dim(X) = 4, since this seems to be the gist.\nLet ˆR = ˆRH\nX,X be the ˆR-matrix associated with Xq. Let P = P H\nX,X and\nQ = QH\nX,X be the projections on the eigenspaces in Xq ⊗Xq for the positive\nand negative eigenvalues of ˆRH\nX,X, respectively. Let xi = f ix0, where f is\nthe usual operator in Hq, and x0 is the highest weight vector in Xq. Matrices\nof ˆR, P and Q in the basis xi ⊗xj of Xq ⊗Xq can be calculated from the\nknown explicit formulae; cf. [Kass, Kli]. The eigenvalues of ˆR turn out to be\nq9/2, −q−3/2, q−11/2 and −q−15/2. Explicit matrix of P in the basis xi ⊗xj\nof Xq ⊗Xq is given by\n17\n\nP = 1\nf P,\n(16)\nwhere\nf = (q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n(17)\nand the matrix of P is as specified in Figure 1 with the following sparse\nrepresentation: the entry (j, v) in the i-row in Figure 1 means P(i, j) = v.\nThus the entry (5, (q4 + 1)/q2) in the second row there means P(2, 5) =\n(q4 +1)/q2. The entries of P-matrix not shown in Figure 1 are all zero. The\nscaling factor f here is chosen so that the entries of P-matrix are polynomials\nin q and q−1. Explicit matrix of\nQ = fQ\n(18)\nis similar.\n7.1.1\nExplicit presentation of B\nLet P1 and P2 denote the P operators on the first two and the last two\nfactors X⊗3, respectively; Q1 and Q2 are defined similarly. We have the\ntrivial relations:\nQ2\ni = fQi,\nand P2\ni = fPi.\nThe first nontrivial basic relation among Qi’s, as determined with the help\nof a computer, is:\nX\nσ\naσQσ = 0,\n(19)\nwhere σ ranges over the various strings of 1’s and 2’s as shown in Figure 2,\naσ ∈Q[q, q−1] are as specified there, and, for a string σ = i1i2 · · · , Qσ\ndenotes the monomial Qi1Qi2 · · · .\nThe second relation is obtained from\nthis by simply interchaning Q1 and Q2. Simialrly, the first nontrivial basic\nrelation among Pi’s is\nX\nσ\nbσPσ = 0,\n(20)\nwhere σ ranges over strings of 1’s and 2’s as in Figures 3-4, bσ’s are as shown\nthere, and Pσ is defined similarly. The second relation is obtained from this\nby simply interchanging P1 and P2. All coefficients in Figures 2-4 as well\nas other figures in this section are shown in factored forms, i.e., as products\nof irreducible polynomials. One may ask if these coefficients have a nice\ninterpretation; we shall turn to this question in Section 7.1.7.\n18\n\nLet B = BH\n3 (q) be the nonstandard algebra in this case, as defined in\nSection 5. It is isomorphic to the algebra generated by Pi’s subject to the\ntwo basic nontrivial relations among Pi’s described above and the trivial\nrelations P2\ni = fPi, or equivalently, to the algebra generated by Qi’s subject\nto the two basic nontrivial relations among Qi’s described above, and the\ntrivial relations Q2\ni = fQi.\nIt is clear from these basic defining relations that {Pσ} or {Qσ}, where σ\nranges over all strings of 1’s and 2’s of length at most 10 without consecutive\n1’s or 2’s, is a basis of B. Its dimension is 21.\n7.1.2\nWederburn structure decomposition\nUnlike for the Hecke algebras, for the complete Wederburn structure decom-\nposition as in (13) to hold for B, the base field has to contain the algebraic\nextension K of Q(q) defined as follows. Let\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(21)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x] be\nthe algebraic extension of Q(q) obtained by adjoining x. We assume that\nB is defined over this base field. It was found by computer that B has one\none-dimensional irreducible representation T0, and five two-dimensional irre-\nducible representations Ti, 1 ≤i ≤5, with a complete Wederburn structure\ndecomposition\nB =\nM\ni\nUi,\nUi = Ti,L ⊗Ti,R,\n(22)\nwhere the basis elements of the various B ⊗B-bimodules Ui and the explicit\nrepresentation matrices of the irreducible B-representations Ti are as follows.\nLet U0 be the K-span of u0 ∈B, where u0 is as specified in Figures 5-6.\nThe coefficients in these and the following figures are in the basis {Qσ}. Let\nUi, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the matrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni\n \n,\nwhere u1\n1 is as specified in Figure 7, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 8-10. Let u2\ni ,\n19\n\n1 ≤i ≤5, be the element obtained from u1\ni by interchanging Q1 and Q2.\nLet u12\ni\n= u1\ni Q2, and u21\ni\n= Q2u1\ni , for 1 ≤i ≤5.\nThen it can be shown that each Ui has a left and right action of B, and\nas a B ⊗B-bimodule\nB =\nM\ni\nUi.\n(23)\nThe columns of ui correspond to the left cells and the rows to right cells;\ni.e., the span of each column (row) is a left (resp. right) B-module, which\nwe shall denote by Ti,L (resp. Ti,R). Thus,\nB =\nM\ni\nTi,L ⊗Ti,R.\n(24)\nHere T0, the span of u0, is the trivial one dimensional representation of\nB, since it can be verified that:\nQju0 = 0,\nfor j = 1, 2.\nThe representation matrices M1\ni and M2\ni of Q1 and Q2 in the basis {u1\ni , u21\ni }\nof Ti,L, 1 ≤i ≤5, are as follows:\nM1\ni =\n 0\n1\n0\nf\n \n,\nwhere f is the scaling factor in (16),\nM2\ni =\n f\ngi\n0\n0\n \n,\nwhere gi are as shown in Figure 11; g2 is obtained from g1 by substituting\n−x for x.\nLet Ti(1) denote the specialization of Ti at q = 1. It is a representation\nof B(1), the specialization of B at q = 1. Then T0(1) corresponds to the\ntrivial one-dimensional representation of S3. There is no one dimensional\nrepresentation of B that specializes to the alternating (signed) one dimen-\nsional representation of S3. This implies that the degree three component\n∧H,3\nq\n[X] of the braided exterior algebra ∧H\nq [X] in this case is zero–as was\nalready observed and proved by other means in [BZ].\nAt q = 1, the values of f = f(q) and gi = gi(q) are as follows:\nf(1) = g1(1) = g3(1) = g4(1) = g5(1) = 4, and g2(1) = 16.\n20\n\nHence the B(1)-modules T1(1), T3(1), T4(2) and T5(2) are all isomorphic, and\nit can be verified that they are isomorphic to the Specht module S(2,1) of the\nsymmetric group S3 for the partition (2, 1). The module T2(1) is reducible.\nBecause it can be verified that it contains an irreducible B(1)-module T 1\n2 (1)\nisomorphic to the trivial one dimensional Specht module S(3) of the sym-\nmetric group S3, and the quotient T 2\n2 (1) = T2(1)/T 1\n2 (1) is isomorphic to\nthe one dimensional signed representation S(1,1,1) of S3. But T2(1) is not\ncompletely reducible as a B(1) module. That is, T2(1) ̸∼= T 1\n2 (1) ⊕T 2\n2 (1),\nsince it does not contain a submodule isomorphic to S(1,1,1). Thus, though\nB is semisimple for generic q, its specialization B(1) is not semisimple.\n7.1.3\nDuality\nPick an element ui from each Ui, 1 ≤i ≤5; say, ui = u1\ni , and u0 is as before.\nFor 0 ≤i ≤5, let Wi = X⊗3\nq\n· ui, which has a left action of the nonstandard\nquantum group GH\nq . These are nonisomorphic irreducible representations\nof GH\nq . Their explicit decompositions as Hq-modules, Hq = GLq(C2), were\ndetermined with the help of computer. They are as follows.\nThe module W0 is isomorphic to the sixteen dimensional degree three\ncomponent CH,3\nq\n[X] of the braided symmetric algebra [BZ] with the following\ndecomposition as an Hq-module:\nW0 = Vq,(9)(2) ⊕Vq,(7,2)(2);\nrecall that Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to\nthe partition λ. This decomposition of CH,3\nq\n[X] in this case agrees with the\none obtained in [BZ] by other means.\nThe modules Wi, i > 0, are distinct irreducible representations of GH\nq\nwith the following decompositions as Hq-modules:\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(25)\nTheir dimensions are 4, 4, 8, 2 and 6, respectively. Though W1 and W2 are\nisomorphic as Hq-modules, they are nonisomorphic as GH\nq -modules; the ma-\ntrix coefficients of W2 are obtained from those for W1 by substituting −x\nfor x.\n21\n\nIt can be verified that, as a GH\nq × B-bimodule,\nX⊗3\nq\n∼= ⊕iWi ⊗Ti,\n(26)\nas per the duality conjecture (Conjecture 1.3).\n7.1.4\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of the symmetric\ngroup S3 in the B(1)-module Ti. Then, we see that\nm0\n(3) = 1,\nm1\n(2,1) = m3\n(2,1) = m4\n(2,1) = m5\n(2,1) = 1,\nm2\n(3) = m2\n(1,1,1) = 1.\nFurthermore, it can be verified that the various Gq-modules, Gq =\nGLq(C4), decompose as follows when considered as Hq-modules:\nVq,(3)(4)\n∼=\nm0\n(3)W0 ⊕m2\n(3)W2,\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2) ⊕Vq,(6,3)(2),\nand\nVq,(2,1)(4)\n∼=\nm1\n(2,1)W1 ⊕m3\n(2,1)W3 ⊕m4\n(2,1)W4 ⊕m5\n(2,1)W5\n∼=\nVq,(6,3)(2) ⊕Vq,(8,1)(2) ⊕Vq,(5,4)(2) ⊕Vq,(7,2)(2).\nThis verifies the nonstandard reciprocity conjecture (Conjecture 1.4) in this\ncase.\n7.1.5\nRefined reciprocity\nFix a right cell within U2 isomorphic to the representation T2,R; say, the one\nspanned by u1\n2 and u12\n2 . We shall denote it by T2,R again. Let z0 ∈T2,R be\nthe element such that z0Q2 = 0. Its coefficients are shown in Figure 12 in\nthe basis {Qσ}. Let z1 = u1\n2. Then the basis Z = {z0, z1} of T2,R admits a\nfiltration\nZ0 = {z0} ⊆Z1 = {z0, z1},\nthat yields at q = 1 a composition series of T2,R(1) as a B(1)-module:\nˆT2,0 ⊂ˆT2,1 = T2,R(1),\n22\n\nwhere ˆT2,0, spanned by the specialization z0(1) of z0, is the one dimensional\ntrivial representation of S3, and ˆT2,1/ ˆT2,0 is the one-dimensional signed rep-\nresentation of S3.\nLet W2,1 = X⊗3\nq\n· z1 and W2,0 = X⊗3\nq\n· z0 be the GH\nq -submodules of X⊗3\nq ,\nand W2,1(1), W2,0(1) their specializations at q = 1. It can be verified that\nat q = 1 we get:\nW2,1(1) = ∧3(X) ⊆X⊗3,\nand\nW0(1) ⊕W2,0(1) = Sym3(X) ⊆X⊗3,\n(27)\nwhere ∧3(X) and Sym3(X) are the Weyl modules of G = GL(X) for the\npartitions (1, 1, 1) and (3), respectively, and W0(1) the specialization of W0\nat q = 1.\nFor example, Figures 13-16 show the nonzero coefficients of the elements\na = (x1 ⊗x2 ⊗x0) · z1 and b = (x1 ⊗x2 ⊗x0) · z0 in the monomial basis\n{xi ⊗xj ⊗xk} of X⊗3\nq . It can be verified that the specialization a(1) at\nq = 1 of a indeed belongs to the subspace ∧3(X) ⊆X⊗3. The specialization\nb(1) of b, as it is, just vanishes, since its coefficients are divisible by (q −1)2.\nBut instead we consider the basis element b′ = b/(q −1)2 of W2,0. Then\nits specialization b′(1) at q = 1 indeed belongs to the subspace Sym3(X) of\nX⊗3. The equation (27) can be verified similarly.\nSimilarly it can be verified that\nlim\nq→1\nM\ni=1,3,4,5\n(X⊗3\nq\n· u1\ni ∪X⊗3\nq\n· u12\ni ) = V(2,1) ⊗S(2,1) ⊆X⊗3.\nThis verifies the refined reciprocity conjecture in this case. In particular,\nit explains what happens to the exterior and symmetric algebra components\nhere. Specifically, though the braided exterior algebra component ∧H,3\nq\n[X] =\n0,\nW2,1(1) = ∧H,3[X].\nThus the q-deformation of ∧H\n3 [X] has simply relocated itself as W2,1 in the\ndecomposition\nX⊗3\nq\n= ⊕Wi ⊗Ti.\nSimilarly, the symmetric algebra component CH,3[X] splits in two parts, and\nthe q-deformations of these parts, namely W0 and W2,0, get distributed in\nthis decomposition. The situation for V2,1 is similar. Thus, overall, there\nis no information loss; the information has only been redistributed. As per\nthe refined reciprocity conjecture, this is a general phenomenon.\n23\n\n7.1.6\nBase field Q(q)\nLet us now see what happens if the base field is Q(q) instead.\nThe B-\nrepresentations T0, T3, T4, T5 are already defined over Q(q).\nBut T1 and\nT2 merge into a four dimensional B-representation T12 defined over Q(q).\nExplicitly, it can be realized within B as the linear span of the elements\nv1\n=\n(u1\n1 + u1\n2)/2,\nv2\n=\n(u1\n1 −u1\n2)/(2x),\nv3\n=\n(u21\n1 + u21\n2 )/2,\nv3\n=\n(u21\n1 −u21\n2 )/(2x).\nRepresentation matrices of left multiplication by Q1 and Q2 in the basis\n{vi} are, respectively,\nM1 =\n \n \nq10+q6+q4+1\nq5\n0\na\n1/2 q−20\n0\nq10+q6+q4+1\nq5\nb\na\n0\n0\n0\n0\n0\n0\n0\n0\n \n \n,\nwith\na\n=\n1/2 3 q16+4 q12−2 q10+10 q8−2 q6+4 q4+3\nq8\n,\nb\n=\n1/2 q4(5 q32 + 8 q28 −4 q26 + 28 q24 −4 q22 + 24 q20 −8 q18 + 46 q16\n−8 q14 + 24 q12 −4 q10 + 28 q8 −4 q6 + 8 q4 + 5),\nand\nM2 =\n \n \n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\nq10+q6+q4+1\nq5\n0\n0\n1\n0\nq10+q6+q4+1\nq5\n \n \n.\nSimilarly, the GH\nq -modules W0, W3, W4, W5 are already defined over Q(q).\nThe modules W1 and W2 merge into an eight-dimensional GH\nq -module W12 ∼=\nX⊗3\nq\n· vi, for any i–this is defined over Q(q). As an Hq-module,\nW1,2 ∼= 2 · Vq,(6,3)(2).\n24\n\nA variant of the duality also holds. Specifically, the components W1⊗T1 and\nW2⊗T2 in the decomposition (26) of X⊗3\nq\nmerge into one sixteen dimensional\nGH\nq × B-bimodule defined over Q(q). As a GH\nq module, it is a direct sum of\ntwo copies of W12, and as a B-module a direct sum of four copies of T12. But\nfor the reciprocity to hold, the base field has to be K = Q(q)[x] as before or\nlarger. Indeed, it can be seen here that the reciprocity conjecture fails over\nthe base field Q(q). This illustrates the need for base extension in general.\nIt may be illuminating to compare the r = 3 case here with the one for\nthe Kronecker problem treated in [GCT4]. The one here is basically a more\ncomplex version of the one in [GCT4], because the basic defining relations\nhere (Figures 2-4) are more complex versions of the ones in [GCT4].\n7.1.7\nOn r > 3 and positivity\nSimilar symbolic computations for r = 4 seem beyond the reach of desktop\nMATLAB/Maple. Fortunately, this case for the Kronecker problem is within\nthe reach, and will be treated in the next section. The r = 4 case, H =\nGL2(C), X four dimensional, is expected to be its more complex version\njust as for r = 3.\nBut it does not seem possible to progress much beyond r = 3 using the\nbrute force computer-based approach that we are following here. What is\nneeeded is an explicit presentation for BH\nr akin to the explicit presentation\nfor the Hecke algebra, or the one for r = 3 in Section 7.1.1. That is, we need\nan explicit set of generating relations among Qi or Pi’s, each of the form\nX\naσQσ = 0,\n(28)\nor\nX\nbσPσ = 0,\n(29)\nwhere Qσ and Pσ, for a string σ = i1i2 · · · of symbols in {1, · · · , r −1},\ndenote the monomials Qi1Qi2 · · · and Pi1Pi2 · · · , respectively, and each aσ\nand bσ has an explicit interpretation (formula).\nThe coefficients aσ and bσ in Figures 2-4 for the r = 3 case do not\nseem to have any obvious elementary interpretation. Hence, in general, one\ncan only expect nonelementary interpretations for the coefficients aσ and\nbσ in (28)-(29). The following numerical analysis of these coefficients for\nthe r = 3 case suggests that BH\nr , in general, may plausibly have an explicit\npresentation, the coefficients aσ and bσ of whose generating relations have\nnonelementary interpretations in the spirit of the one for Kazhdan-Lusztig\n25\n\npolynomials. By this we mean that each aσ has an explicit formula of the\nform of an alternating sum\naσ = (−1)d(σ)(q1/2 −q−1/2)d′(σ)(\ns(σ)\nX\nj=0\n(−1)jaj\nσ),\n(30)\nfor some nonnegative integers d(σ), d′(σ), s(σ), where\n1. s(σ) is small, say bounded by a polynomial of a fixed degree in r and\ndim(X) in the present case when H = GL2(C), and in r, the rank of\nH and the size of μ in the general plethysm problem (Problem 1.1),\n2. each aj\nσ is a −-invariant (note that aσ is −-invariant), positive and\nunimodal polynomial in q and q−1; positive means each coefficient of\naj\nσ is nonnegative, and unimodal means, if aj\nσ(−k), . . . , aj\nσ(k) are the\ncoefficients of aj\nσ, then\naj\nσ(−k) ≤aj\nσ(−k + 1) ≤· · · ≤aj\nσ(−1) ≤aj\nσ(0) ≤aj\nσ(1) ≤· · · aj\nσ(k),\n3. each aj\nσ(s) has a topological interpretation akin to that for Kazhdan-\nLusztig polynomials, i.e., as the rank of an appropriate cohomology\ngroup. Then the duality aj\nσ(−s) = aj\nσ(s) as per the −-invariance of aj\nσ\nshould come out as a consequence of some form of Poincare duality\nand the unimodality as a consequence of some form of Hard Lefschetz,\nand each bσ has a similar explicit formula of the form\nbσ = (−1)\n ̄d(σ)(q1/2 −q−1/2)\n ̄d′(σ)(\n ̄s(σ)\nX\nj=0\n(−1)jbj\nσ).\n(31)\nWe shall call such an interpretation for aσ or bσ, if it exists, a positive,\nunimodal, and topological interpretation.\nIdeally speaking, one would like each s(σ) and ̄s(σ) above to be zero, but\nthis may not always be possible for the reasons given below. It is plausible\nthat there exists some notion of cohomological depth that measures the\nextent of nonflatness, and which provides an upper bound on s(σ) and ̄s(σ) in\nsuch a topological interepretation, if it exists. For example, in the Kronecker\nproblem, the braided symmetric and exterior algebras CH\nq [X] and ∧H\nq [X] are\nflat deformations of the classical algebras C[X] and ∧[X]. In this case, one\ncan expect an explicit presentation for BH\nq whose coefficients aσ and bσ have\n26\n\npositive topological interpretation with s(σ), ̄s(σ) = 0 in (30) and (31). This\nis because aσ and bσ here are akin to the structural constants cb′′\nb,b′, db,b′\nb′′\nin\nTheorem 4.4, which occur in the defining Laplace relations for GH\nq , and\nwhich, in the Kroncker problem, are conjecturally polynomials in q and q−1\nwith nonnegative coefficients for the reasons indicated there. But in general\nwhen CH\nq [X] and ∧H\nq [X] are nonflat deformations, such cohomological depth\nwould not vanish, and hence s(σ) and ̄s(σ) may be nonzero, but still small\nas indicated above.\nWe now turn to the analysis of the coefficients in the r = 3 case men-\ntioned above which suggests that such an interpretation may plausibly exist.\nFirst let us oberve that the scaling factor f in (17) used in the analysis so far\nis formally not the correct scaling factor. To get the latter, we have to look\nat the formal expressions for P and Q in terms of ˆR. Since the eigenvalues\nof ˆR in the present case are\nq1 = q9/2,\nq2 = −q−3/2,\nq3 = q−11/2,\nand\nq4 = −q−15/2,\nwe have\nP = ( ˆR −q2)( ˆR −q3)( ˆR −q4)\n(q1 −q2)(q1 −q3)(q1 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q4)\n(q3 −q1)(q3 −q2)(q3 −q4),\n(32)\nand\nQ = ( ˆR −q1)( ˆR −q3)( ˆR −q4)\n(q2 −q1)(q2 −q3)(q2 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q3)\n(q4 −q1)(q4 −q2)(q4 −q3).\n(33)\nHence, formally we should have defined the rescaled versions P and Q of P\nand Q by the equations\nP = fpP,\nfp = (q1 −q2)(q1 −q3)(q1 −q4)(q3 −q1)(q3 −q2)(q3 −q4), (34)\nand\nQ = fqQ,\nfq = (q2 −q1)(q2 −q3)(q2 −q4)(q4 −q1)(q4 −q3)(q4 −q2), (35)\ninstead of the equations (16) and (18). The scaling factor f in (17) was\nthe smallest factor chosen so that the matrix coefficients of P and Q after\nrescaling become polynomials in q, q−1. But this choice was dependendent on\nthe accidental cancellations in the numerators and denominators in (32) and\n27\n\n(33). The choice of scaling makes no essential difference in Sections 7.1.1-\n7.1.6. But it does matter in the study of positivity below.\nHence, let us redefine P and Q as per (34) and (35). Let us denote the\ncoefficients of the old defining relations (19) and (20) among Qi’s and Pi’s\nby a′\nσ and b′\nσ, and the coefficients of the defining relations among the new\nQi’s and Pi’s by a′′\nσ and b′′\nσ. Then we have\na′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄aσ,\nwith\n ̄aσ = ( ˆfq)11−l(σ)a′\nσ,\nand\nb′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄bσ,\nwith\n ̄bσ = ( ˆfp)11−l(σ)b′\nσ,\nwhere l(σ) denotes the length of σ,\nˆfp\n=\n−q\n(q−1)2\nfp\nf\n=\n10 + 8 q + 2 q4 + 12 q−1 + 18 q−6 + 6 q2 + 4 q3 + q5\n+14 q−2 + 18 q−4 + 16 q−12 + 16 q−8 + q−27 + 16 q−11\n+16 q−10 + 17 q−7 + 6 q−24 + 18 q−5 + 2 q−26 + 16 q−3\n+16 q−9 + 10 q−22 + 14 q−20 + 4 q−25 + 8 q−23 + 12 q−21 + 18 q−18\n+16 q−19 + 18 q−16 + 18 q−17 + 17 q−15 + 16 q−14 + 16 q−13,\nand\nˆfq\n=\n−q\n(q−1)2\nfq\nf\n=\n10 q−12 + 2 q−8 + q−31 + 8 q−27 + 8 q−11 + 6 q−10 + q−7\n+2 q−30 + 10 q−24 + 6 q−28 + 10 q−26 + 4 q−9 + 6 q−22\n+2 q−20 + 10 q−25 + 8 q−23 + 4 q−29 + 4 q−21 + 2 q−18\n+2 q−19 + 6 q−16 + 4 q−17 + 8 q−15 + 10 q−14 + 10 q−13.\nBoth fp and fq are positive polynomials.\nLet us define\nˆaσ\n=\nˆf 2\nq a′\nσ,\nfor\nσ = 121212121,\n=\nˆfqa′\nσ,\notherwise,\n(36)\nand\nˆbσ\n=\nb′\nσ,\nfor\nσ = ∅, and 2,\n=\nˆfpb′\nσ,\notherwise.\n(37)\n28\n\nSince fp and fq are positive, the positivity properties of ˆaσ and ̄aσ (also\nˆbσ and ̄bσ) are similar; it turns out that the unimodularity properties are\nalso similar. Hence we shall focus on ˆaσ and ˆbσ in what follows. Since ˆaσ\nis −-invariant, it is of the form ˆaσ(0) + P\nt>0 ˆaσ(t)(qt + q−t). Let ˆAσ be\nthe vector [aσ(0), aσ(1), . . .]; the vector ˆBσ is defined similarly. Figure 17\nshows ˆAσ for the various σ in Figure 2; the vector for each σ is obtained by\nconcatenating the rows in front of that σ. Figures 18-20 similarly show ˆBσ\nfor the various σ in Figures 3-4; only the distinct ˆBσ’s are shown. It may\nbe seen the ˆAσ’s are positive and nonincreasing. Thus all aσ are positive\nand unimodal, and hence, of the form (30) with s(σ) = 0. All ˆBσ’s are\npositive and nonincreasing, except for σ = 121, 1212 and 21212, for which\neach ˆBσ is positive and unimodal except at the tail. Thus all bσ, for σ ̸=\n121, 1212, 21212, are positive and unimodal, and hence of the form (31) with\n ̄s(σ) = 0. For σ = 121, 1212, 21212, bσ seems to be of the form (31) with\n ̄s(σ) = 1, both b0\nσ and b1\nσ being positive and unimodal, b0\nσ being the dominant\npolynomial that accounts for bσ’s mostly positive and unimodal behaviour,\nand b1\nσ the error polynomial that accounts for the deviation at the tail.\nThe (co)multiplicative structural constants cb′′\nb,b′ and db,b′\nb′′ for the canoni-\ncal basis of the braided exterior algebra ∧H,r\nq\n[X], which occur in the Laplace\nrelations for the general nonstandard quantum group GH\nq (cf. Theorem 4.5),\nare akin to the structure constants aσ and bσ in (28) and (29). Hence, we can\nexpect a similar positive topological interpretation for cb′′\nb,b′ and db,b′\nb′′ (but not\nnecessarily unimodality since cb′′\nb,b′ and db,b′\nb′′\nneed not be −-invariant). The\nexperimental evidence in [GCT8] suggests that the structure constants as-\nsociated with the canonical bases of the matrix coordinate ring of GH\nq and\nthe ring BH\nq defined there may also have similar positive topological inter-\npretations (additionally unimodal for BH\nq ).\n7.2\nExample 2\nNow we verify the duality and reciprocity conjectures for the special case of\nthe Kronecker problem (Section 4.1), when H = GL(V )×GL(W), V = W =\nC2 and G = GL(X), X = V ⊗W ∼= C4, and r = 4. Thus Gq = GLq(C4), and\nHq = GLq(C2) × GLq(C2). Let B = BH\nr be the nonstandard algebra in this\ncase and Pi = p+,H\nX,i , Qi = p−,X\nX,i , i < r, the positive and negative projection\noperators as in Section 5. Let Pi and Qi be the rescaled versions of Pi and\nQi as defined in [GCT4]. Then B is generated by Pi, or equivalently, Qi.\nThe explicit generating relations among Pi’s and Qi’s turn out to be very\n29\n\n1, (q4+1)(q4−q2+1)(q2+1)\nq5\n2, q\n q4 + 1\n \n5, q4+1\nq2\n3, (q2+1)(q8−q6+q4−q2+1)\nq5\n6,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, q2+1\nq\n4, q8−q6+q4+1\nq3\n7,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n10,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n13, q4+1\nq2\n2, q4+1\nq2\n5, q4+1\nq5\n3, (q2+1)\n2(q−1)(q+1)\nq2\n6, 2 q2+1\nq\n9, −(q2+1)\n2(q−1)(q+1)\nq2\n4,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n7, 2 q8+q6−2 q2+1\nq5\n10, −q8−q6−2 q4−q2+1\nq4\n13, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n8, (q2+1)(q8−q6+q4−q2+1)\nq5\n11, (q2+1)\n2(q−1)(q+1)\nq4\n14, q2+1\nq\n3, q2+1\nq\n6, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, (q2+1)(q8−q6+q4−q2+1)\nq5\n4,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n7, −q8−q6−2 q4−q2+1\nq4\n10, q8−2 q6+q2+2\nq3\n13,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n8,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n11, 2 q2+1\nq\n14, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n12, q\n q4 + 1\n \n15, q4+1\nq2\n4, q4+1\nq2\n7, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n10,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n13, q8+q4−q2+1\nq5\n8, q2+1\nq\n11, −(q2+1)\n2(q−1)(q+1)\nq4\n14, (q2+1)(q8−q6+q4−q2+1)\nq5\n12, q4+1\nq2\n15, q4+1\nq5\n16, (q4+1)(q4−q2+1)(q2+1)\nq5\n \n \nFigure 1: P-matrix\n30\n\nσ\naσ\n1\n−(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2/q36\n121\n(1 + 7 q4 + q2 + 5 q6 + 18 q8 + 21 q42 −107 q20 + q50 −107 q32 + 73 q14 + 187 q18 −14 q16 + 402 q26\n−197 q28 + 20 q40 + 187 q34 + 73 q38 −197 q24 + 328 q30 + q52 + 328 q22 + 7 q48 + 5 q46 + 18 q44\n+20 q12 + 21 q10 −14 q36)(q2 + 1)2(q4 −q2 + 1)2/q32\n12121\n−(1 + 8 q4 + 3 q2 + 4 q6 + 33 q8 + 12 q42 + 80 q20 + 3 q50 + 80 q32 + 27 q14 + 113 q18 + 115 q16 + 360 q26\n−9 q28 + 76 q40 + 113 q34 + 27 q38 −9 q24 + 253 q30 + q52 + 253 q22 + 8 q48 + 4 q46 + 33 q44 + 76 q12\n+12 q10 + 115 q36)/q26\n1212121\n(3 q36 + 2 q34 + 8 q32 + 5 q30 + 17 q28 + 30 q26 + 11 q24\n+61 q22 −15 q20 + 108 q18 −15 q16 + 61 q14 + 11 q12 + 30 q10 + 17 q8 + 5 q6 + 8 q4 + 2 q2 + 3)/q18\n121212121\n−(q20 + 3 q18 + q16 + 5 q14 −2 q12 + 16 q10 −2 q8 + 5 q6 + q4 + 3 q2 + 1)/q10\n12121212121\n1\nFigure 2: Coefficients of the basic generating relation among Qi’s\n31\n\nσ\nbσ\n∅\n−(q2 + 1)5(q4 + 1)3(q4 −q2 + 1)6(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)\n×(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)\n×(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q51\n2\n(q2 + 1)4(q4 −q2 + 1)5(q4 + 1)2(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n×(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n×(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\n1\n(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q46\n12\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n21\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n212\n(q2 + 1)2(q4 −q2 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q36\n121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(2 + 2 q4 −4 q2 −2 q6 −2 q8 −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 −3 q12 −9 q10 + q36)/q36\n1212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\nFigure 3: The first eight terms of the basic generating relation among Pi’s\n32\n\nσ\nbσ\n2121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\n21212\n(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26\n+53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44 −3 q12\n−9 q10 + q36)/q26\n12121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q26\n121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n2121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n1212121\n−(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q18\n12121212\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n21212121\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n212121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n121212121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2/q10\n1212121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n2121212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n21212121212\n1\nFigure 4: The last fourteen terms of the basic generating relation among Pi’s\n33\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 5: First nine coefficients of u0\n34\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 6: Last twelve coefficients of u0\n35\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 7: Coefficients of u1\n1\n36\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 8: Coefficients of u1\n3\n37\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 9: Coefficients of u1\n4\n38\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 10: Coefficients of u1\n5\n39\n\ng1\n−1/2 −3 q28−4 q24+2 q22−10 q20+2 q18−4 q16−3 q12+x\nq20\ng3\n(q4+1)\n2\nq4\ng4\n(q2+1)\n2(q4−q2+1)\n2\nq6\ng5\n(q2+1)\n2(q8−q6+q4−q2+1)\n2\nq10\nFigure 11: The elements gi\ncomplicated. For example, Figures 21-23 reproduced from [GCT4] shows\na typical generating relation among Qi’s with 74 terms. There are several\ndozen such relations. Because of the nature of these generating relations,\nthere is no good “standard monomial basis” for B as for the Hecke algebra or\nfor the r = 3 case in Section 7.1.1. Fortunately, this makes no difference as\nfar as duality and reciprocity is concerned, as we shall see here, and also as\nfar as existence of a canonical basis is concerned, as we shall see in [GCT8].\nIt was verified by computer that B is of dimension 114 [GCT4]. Since it\nis semisimple, it admits a Wederburn structure decomposition. It turns out\nthat a complete Wederburn structure decomposition of the form (13) works\nover Q(q) itself; i.e., no algebraic extension of Q(q) is necessary here, just\nas in the case of Hecke algebras. This may be conjectured to be the case\nfor the Kronecker problem in general, though it is not so for the plethysm\nproblem in general as we already saw in Section 7.1.\nSo let\nB = ⊗iTi,L ⊗Ti,R,\n(38)\nbe the complete Wederburn structure decomposition of B, where Ti = Ti,L\nranges over all irreducible left B-modules.\n7.2.1\nIrreducible representations\nWe describe these Ti next. There are two distinct irreducible representations\nof B of dimension 1, 2, 3 and 5 each, and one of dimension 6. Since\n114 = 12 + 12 + 22 + 22 + 32 + 32 + 52 + 52 + 62,\nthis is consistent with the Wederburn structure decomposition in (38).\n40\n\nσ\nCoefficient\n1\n1/2 (q2 + 1)5(q4 −q2 + 1)3(q8 −q6 + q4 −q2 + 1)2(q4 + 1)3(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q45\n12\n−1/2 (q2 + 1)4(q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q40\n121\n−1/2 (q2 + 1)3(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)(q4 + 1)(q4 −q2 + 1)/q41\n1212\n1/2 (q2 + 1)2(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)/q36\n12121\n1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)\n×(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q35\n121212\n−1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)/q30\n1212121\n−1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)(q2 + 1)\n×(q4 + 1)(q4 −q2 + 1)/q25\n12121212\n1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)/q20\n121212121\n−(q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n1212121212\n1\nFigure 12: Nonzero coefficients of z0\n41\n\nMonomial\nCoefficient\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24 + 11 q48\n−45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8 + 8 xq16\n−7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q51\nx1 ⊗x2 ⊗x0\n1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8\n+8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q46\nx0 ⊗x3 ⊗x0\n−1/2 (q + 1)(q −1)(q4 + 1)(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(3 x + 727 q32 −500 q38\n−330 q22 + 191 q20 + 460 q24 + 59 q48 −359 q42 + 192 q44 −110 q46 −603 q26 −138 q18 + 76 q16 −35 q14\n−20 xq22 + 5 xq24 + 70 xq12 −58 xq10 −77 xq14 + 45 xq8 + 48 xq16 −46 xq6 + 23 xq20 + 30 xq4 −15 xq2\n−32 xq18 + 587 q28 + 402 q40 −780 q34 + 584 q36 −44 q50 + 11 q52 −685 q30 + 7 q12)/q53\nx2 ⊗x0 ⊗x1\n1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20\n+117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34\n+117 q36 −310 q30 + 11 q12)/q48\nx1 ⊗x1 ⊗x1\n−1/2 (q2 + 1)2(q4 + 1)(q −1)(q + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22\n+91 q20 + 117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q47\nx0 ⊗x2 ⊗x1\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(−3 x −1316 q32 + 1364 q38 + 419 q22 −200 q20 −577 q24\n−464 q48 + 957 q42 −617 q44 + 613 q46 + 748 q26 + 5 q30x + 134 q18 −76 q16 + 35 q14 −25 xq28 + 35 xq26\n+67 xq22 −28 xq24 −117 xq12 + 97 xq10 + 110 xq14 −48 xq8 −85 xq16 + 44 xq6 −112 xq20 −30 xq4 + 15 xq2\n+123 xq18 −816 q28 −1224 q40 + 1325 q34 −1132 q36 + 257 q50 + 85 q54 −55 q56 + 11 q58 −108 q52\n+1220 q30 −7 q12)/q50\nFigure 13: First five nonzero coefficients of a ∈X⊗3\nq\n42\n\nMonomial\nCoefficient\nx0 ⊗x0 ⊗x2\n−1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14\n+19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q46\nx0 ⊗x1 ⊗x2\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(3 x + 951 q32 −1060 q38 −363 q22 + 176 q20 + 449 q24 + 248 q48\n−592 q42 + 395 q44 −451 q46 −532 q26 −97 q18 + 65 q16 −35 q14 + 8 xq28 −27 xq26 −31 xq22 + 16 xq24\n+81 xq12 −85 xq10 −62 xq14 + 40 xq8 + 59 xq16 −27 xq6 + 64 xq20 + 25 xq4 −15 xq2 −97 xq18 + 654 q28\n+797 q40 −898 q34 + 828 q36 −129 q50 −61 q54 + 18 q56 + 52 q52 −998 q30 + 7 q12)/q49\nx0 ⊗x0 ⊗x3\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q50\nFigure 14: Last four nonzero coefficients of a\n43\n\nMonomial\nCoefficient\nx3 ⊗x0 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(5 x + 1214 q32 −847 q38 −525 q22\n+289 q20 + 714 q24 + 107 q48 −525 q42 + 289 q44 −178 q46 −847 q26 −178 q18 + 107 q16 −55 q14 −25 xq22\n+5 xq24 + 130 xq12 −113 xq10 −113 xq14 + 75 xq8 + 75 xq16 −60 xq6 + 45 xq20 + 45 xq4 −25 xq2 −60 xq18\n+920 q28 + 714 q40 −1139 q34 + 920 q36 −55 q50 + 11 q52 −1139 q30 + 11 q12)/q57\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−5 x −1170 q32 + 977 q38 + 425 q22\n−207 q20 −565 q24 −371 q48 + 636 q42 −451 q44 + 397 q46 + 523 q26 + 5 q30x + 106 q18 −89 q16 + 55 q14\n−20 xq28 + 20 xq26 + 43 xq22 −20 xq24 −119 xq12 + 97 xq10 + 71 xq14 −45 xq8 −59 xq16 + 28 xq6 −95 xq20\n−37 xq4 + 25 xq2 + 87 xq18 −676 q28 −1021 q40 + 891 q34 −849 q36 + 173 q50 + 52 q54 −44 q56 + 11 q58\n−82 q52 + 1002 q30 −11 q12)/q56\nx1 ⊗x2 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−8 x −1447 q32 + 1598 q38 + 590 q22\n−304 q20 −648 q24 −496 q48 + 927 q42 −742 q44 + 696 q46 + 778 q26 + 5 q30x + 153 q18 −89 q16 + 72 q14\n−25 xq28 + 40 xq26 + 58 xq22 −40 xq24 −124 xq12 + 138 xq10 + 96 xq14 −76 xq8 −114 xq16 + 39 xq6 −116 xq20\n−33 xq4 + 32 xq2 + 152 xq18 −1064 q28 −1329 q40 + 1319 q34 −1386 q36 + 244 q50 + 96 q54 −55 q56\n+11 q58 −134 q52 + 1516 q30 −18 q12)/q55\nx0 ⊗x3 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(−3 x −1693 q32 + 1146 q38 + 641 q22 −404 q20\n−1020 q24 −178 q48 + 814 q42 −567 q44 + 296 q46 + 1268 q26 + 266 q18 −155 q16 + 53 q14 + 5 xq26 + 45 xq22\n−25 xq24 −164 xq12 + 123 xq10 + 170 xq14 −108 xq8 −131 xq16 + 96 xq6 −60 xq20 −65 xq4 + 23 xq2\n+78 xq18 −1376 q28 −1006 q40 + 1709 q34 −1491 q36 + 107 q50 + 11 q54 −55 q52 + 1449 q30 −7 q12)/q58\nx2 ⊗x0 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(5 x + 964 q32 −627 q38\n−431 q22 + 224 q20 + 582 q24 + 100 q48 −431 q42 + 224 q44 −143 q46 −627 q26 −143 q18 + 100 q16 −55 q14 −25 xq22\n+5 xq24 + 108 xq12 −87 xq10 −87 xq14 + 50 xq8 + 50 xq16 −45 xq6 + 42 xq20 + 42 xq4 −25 xq2 −45 xq18 + 645 q28\n+582 q40 −860 q34 + 645 q36 −55 q50 + 11 q52 −860 q30 + 11 q12)/q53\nFigure 15: First five nonzero coefficients of b\n44\n\nMonomial\nCoefficient\nx1 ⊗x1 ⊗x1\n1/2 (q2 + 1)2(q2 −q + 1)(q2 + q + 1)(q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(3 x + 277 q32 −621 q38 −165 q22\n+97 q20 + 83 q24 + 125 q48 −291 q42 + 291 q44 −299 q46 −255 q26 −47 q18 −17 q14 + 5 xq28 −20 xq26 −15 xq22\n+20 xq24 + 5 xq12 −41 xq10 −25 xq14 + 31 xq8 + 55 xq16 −11 xq6 + 21 xq20 −4 xq4 −7 xq2 −65 xq18 + 388 q28\n+308 q40 −428 q34 + 537 q36 −71 q50 −44 q54 + 11 q56 + 52 q52 −514 q30 + 7 q12)/q52\nx0 ⊗x2 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(−3 x −1651 q32 + 1803 q38\n+486 q22 −287 q20 −679 q24 −567 q48 + 1181 q42 −1013 q44 + 814 q46 + 920 q26 + 5 q30x + 147 q18 −72 q16\n+35 q14 −25 xq28 + 45 xq26 + 78 xq22 −60 xq24 −133 xq12 + 122 xq10 + 138 xq14 −87 xq8 −167 xq16\n+49 xq6 −131 xq20 −28 xq4 + 15 xq2 + 170 xq18 −1270 q28 −1556 q40 + 1669 q34 −1825 q36 + 296 q50\n+107 q54 −55 q56 + 11 q58 −178 q52 + 1547 q30 −7 q12)/q55\nx1 ⊗x0 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10\n−15 xq14 + 22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q51\nx0 ⊗x1 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14 + 22 xq8\n+3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q54\nx0 ⊗x0 ⊗x3\n1/2 (q2 + 1)(q4 + 1)2(q4 −q2 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32\n−94 q38 −220 q22 + 132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q55\nFigure 16: Last five nonzero coefficients of b\n45\n\nLet Sq,λ denote the q-Specht module of the Hecke algebra Hr(q) for\nthe partition λ, and KLλ its Kazhdan-Lusztig basis ordered appropriately.\nSince, in this case, B = BH\nr (q) ⊆Hr(q)⊗Hr(q), the tensor product Sq,λ⊗Sq,μ\nis a representation of B. In particular,\nTq,λ = Sq,λ ⊗Sq,(r) ∼= Sq,(r) ⊗Sq,λ,\nwhere Sq,(r) is the trivial one dimensional q-Specht module, is an irreducible\nB-module, which specializes at q = 1 to the Specht module Sλ of the sym-\nmetric group Sr.\nLet\nT0\n=\nTq,(4),\nT1\n=\nTq,(1,1,1,1),\nT2\n=\nTq,(2,2),\nT3\n=\nTq,(2,1,1),\nT4\n=\nTq,(3,1),\nT5\n=\nSq,(3,1) ⊗Sq,(2,2) ∼= Sq,(2,1,1) ⊗Sq,(2,2).\n(39)\nThese are irreducible B-modules. Their dimensions are 1, 1, 2, 3, 3 and 6\nrespectively.\nTo get the other two dimensional irreducible B-module, we analyze how\nthe tensor product Sq,(2,2) ⊗Sq,(2,2) decomposes as a B-module. It decom-\nposes as:\nSq,(2,2) ⊗Sq,(2,2) ∼= Tq,(4) ⊕Tq,(1,1,1,1) ⊕T6,\nwhere T6 is the other two dimensional irreducible B-module that we were\nlooking for. Explicitly, a basis of T6 in terms of the Kazhdan-Lusztig basis\nKL(2,2) ⊗KL(2,2) of Sq,(2,2) ⊗Sq,(2,2) is given by the rows of the matrix\n\"\n1\n1+q\n2q1/2\n1+q\n2q1/2\n0\n0\n1+q\n2q1/2\n1+q\n2q1/2\n1\n#\n.\nMatrix representations of the right action of the generators Qi’s of B on\nthis basis are:\nQ1 = Q3 =\n (1 + q)2/q\n0\n(1 + q2)/q\n0\n \nQ2 =\n 0\n(1 + q2)/q\n0\n(1 + q)2/q\n \n46\n\nThe specialization of T6 at q = 1 is isomorphic to the Specht module\nS(2,2) of S4. But T6 is nonisomorphic to T2, whose specialization at q = 1 is\nthe same.\nTo get the five dimensional irreducible B-modules, we analyze how the\ntensor products Sq,(2,1,1) ⊗Sq,(2,1,1) and Sq,(3,1) ⊗Sq,(2,1,1) decompose as B-\nmodules. We have\nSq,(2,1,1) ⊗Sq,(2,1,1) ∼= Tq,(2,1,1) ⊕Tq,(4) ⊕T7,\nwhere T7 is the first five dimensional irreducible B-representation that we\nwere looking for. Explicitly, its basis in terms of the Kazhdan-Lusztig basis\nKL(2,1,1) ⊗KL(2,1,1) is given by the rows of the matrix:\nw1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n0\n0\n0]\nw2 = [\n−1\n−(1 + q)/(2q1/2)\n0\n0\n(1 + q)/(2q1/2)\n1]\nw3 = [\n0\n0\n−(1 + q)2/(2q)\n0\n−(1 + q)/(2q1/2)\n0]\nv1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n−1\n−(1 + q)/(2q1/2)\n0]\nv2 = [\n1\n(1 + q)/(2q1/2)\n1\n0\n(1 + q)/(2q1/2)\n1]\nMatrix representations of the right action of Qi’s in this basis are:\nQ1 =\n \n \n(1 + q)2/q\n0\n0\n0\n0\n(1 + q2)/q\n0\n0\n−(1 + q2)/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n(q −1)2/q\n0\n0\n−(1 + q2)/q\n0\n \n \nQ2 =\n \n \n0\n(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n(1+q)2\nq\n0\n0\n0\n0\n−(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n0\n0\n0\n−1+q2\nq\n0\n0\n0\n0\n(1+q)2\nq\n \n \nQ3 =\n \n \n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n−(1 + q2)/q\n(1 + q2)/q\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n(q −1)2/q\n−(1 + q2)/q\n0\n \n \n47\n\nLet V denote the span of the vectors v1 and v2, and V (1) its specialization\nat q = 1. It can be checked that V (1) is isomorphic to the Specht module\nS(2,2) of S4, and the quotient T7(1)/V , where T7(1) denotes the specialization\nof T7 at q = 1, is isomorphic to the Specht module S(3,1) of S4.\nFinally,\nSq,(3,1) ⊗Sq,(2,1,1) ∼= Tq,(3,1) ⊕Tq,(1,1,1,1) ⊕T8,\nwhere T8 ̸∼= T7 is the second five dimnsional irreducible B-representation\nthat we were looking for. Its basis and representation matrices are similar.\nThis specifies all irreducible representations of B.\n7.2.2\nDuality\nUsing the explicit representations Ti above, the Wederburn structure decom-\nposition (38) of B was explicitly determined with the help of a computer.\nThe explicit bases of the structure components Ui = Ti,L ⊗Ti,R in (38) are\nfar too complex to be given here.\nFix any ui ∈Ui, 0 ≤i ≤8, and let Wi = X⊗r\nq\n· ui be the corre-\nsponding left representation of the nonstandard quantum group GH\nq . Com-\nputer experiments indicate that these are nonisomorphic irreducible repre-\nsentations of GH\nq with the following decompositions as Hq-modules, Hq =\nGLq(C2)×GLq(C2). (Recall that Vq,λ(n) is the q-Weyl module of GLq(Cn)).\nW0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nW1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW2\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2),\nW3\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2),\nW4\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2),\nW5\n∼=\nVq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nW6\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW7\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2).\nTheir dimensions are 35, 1, 10, 9, 30, 6, 1 and 9, respectively. The module\nW8 turns out to be zero when dim(V ) = dim(W) = 2, as here; however, it\nwould be nonzero for general dim(V ) and dim(W). Furthermore,\nX⊗4\nq\n=\nM\ni\nWi ⊗Ti,\n48\n\nin accordance with the duality conjecture.\nRemark: These computations are not final. The main problem is that the\nsymbolic computations needed here are too heavy for MATLAB/Maple to\nhandle. Hence, in some of the computations q was set to a fixed real value\n(such as .5). This introduces floating point errors in various calculations.\nAs far as we can see, this does not affect the decomposition above. But this\nhas to be double checked by other means.\n7.2.3\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of S4 in Ti. Then\nit can be verified that\nm0\n(4) = 1,\nm1\n(1,1,1,1) = 1,\nm2\n(2,2) = 1,\nm3\n(2,1,1) = 1,\nm4\n(3,1) = 1,\nm5\n(3,1) = m5\n(2,1,1) = 1,\nm6\n(2,2) = 1,\nm7\n(3,1) = m7\n(2,2) = 1,\nm8\n(2,1,1) = m8\n(2,2) = 1.\nAll other mi\nμ’s are zero. It can now be seen that, as Hq-modules, Hq =\n49\n\nGLq(2) × GLq(2), we have\nVq,(4)(4)\n∼=\nm0\n(4)W0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(3,1)(4)\n∼=\nm4\n(3,1)W4 ⊕m5\n(3,1)W5 ⊕m7\n(3,1)W7\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2),\nVq,(2,2)(4)\n∼=\nm2\n(2,2)W2 ⊕m7\n(2,2)W7 ⊕m6\n(2,2)W6\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(2,1,1)(4)\n∼=\nm3\n(2,1,1)W3 ⊕m5\n(2,1,1)W5\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nVq,(1,1,1,1)(4)\n∼=\nm1\n(1,1,1,1)W1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nin accordance with the reciprocity conjecture.\nWe are unable to verify the refined reciprocity conjecture on computer\nsince the necessary symbolic computations turn out to be beyond the reach\nof the desktop MATLAB/Maple.\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[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n50\n\nσ\nˆAσ\n1\n17738\n17738\n17550\n17362\n16994\n16626\n16114\n15602\n14933\n14264\n13550\n12836\n12008\n11180\n10392\n9604\n8790\n7976\n7226\n6476\n5806\n5136\n4518\n3900\n3418\n2936\n2504\n2072\n1762\n1452\n1202\n952\n776\n600\n482\n364\n280\n196\n152\n108\n77\n46\n34\n22\n14\n6\n4\n2\n1\n121\n20322\n20322\n20083\n19844\n19354\n18864\n18211\n17558\n16668\n15778\n14890\n14002\n12934\n11866\n10916\n9966\n8962\n7958\n7092\n6226\n5470\n4714\n4047\n3380\n2895\n2410\n1982\n1554\n1287\n1020\n804\n588\n463\n338\n253\n168\n122\n76\n53\n30\n19\n8\n5\n2\n1\n12121\n9078\n9078\n8973\n8868\n8623\n8378\n8051\n7724\n7245\n6766\n6335\n5904\n5363\n4822\n4382\n3942\n3443\n2944\n2562\n2180\n1851\n1522\n1267\n1012\n831\n650\n506\n362\n284\n206\n151\n96\n70\n44\n28\n12\n7\n2\n1\n1212121\n1918\n1918\n1913\n1908\n1866\n1824\n1742\n1660\n1523\n1386\n1277\n1168\n1042\n916\n821\n726\n603\n480\n395\n310\n246\n182\n145\n108\n83\n58\n40\n22\n14\n6\n3\n121212121\n25032\n24784\n24124\n23136\n21978\n20808\n19710\n18768\n17934\n17160\n16358\n15440\n14384\n13168\n11849\n10484\n9139\n7880\n6725\n5692\n4765\n3928\n3170\n2480\n1868\n1344\n914\n584\n345\n188\n93\n40\n15\n4\n1\nFigure 17: Positivity and unimodality of ˆAσ’s\n51\n\nσ\nˆBσ\n∅\n390464\n389128\n385120\n378581\n369652\n358471\n345176\n330055\n313396\n295506\n276692\n257207\n237304\n217316\n197576\n178283\n159636\n141795\n124920\n109152\n94632\n81349\n69292\n58469\n48888\n40497\n33244\n27011\n21680\n17198\n13512\n10505\n8060\n6095\n4528\n3314\n2408\n1731\n1204\n812\n540\n357\n232\n147\n84\n43\n24\n16\n8\n2\n2\n102390\n101996\n100847\n98976\n96425\n93236\n89466\n85172\n80462\n75444\n70190\n64772\n59280\n53804\n48429\n43240\n38271\n33556\n29145\n25088\n21393\n18068\n15099\n12472\n10185\n8236\n6586\n5196\n4040\n3092\n2333\n1744\n1286\n920\n640\n440\n300\n200\n129\n76\n41\n24\n16\n8\n2\n1\n50420\n50420\n49799\n49178\n48066\n46954\n45325\n43696\n41665\n39634\n37420\n35206\n32782\n30358\n27969\n25580\n23303\n21026\n18902\n16778\n14947\n13116\n11553\n9990\n8713\n7436\n6455\n5474\n4724\n3974\n3416\n2858\n2490\n2122\n1831\n1540\n1350\n1160\n1031\n902\n779\n656\n582\n508\n441\n374\n313\n252\n213\n174\n144\n114\n86\n58\n47\n36\n27\n18\n11\n4\n4\n4\n2\n12\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 18: The vectors ˆBσ\n52\n\nσ\nˆBσ\n212\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\n121\n51252\n51252\n50661\n50070\n48941\n47812\n46219\n44626\n42589\n40552\n38328\n36104\n33645\n31186\n28756\n26326\n23948\n21570\n19376\n17182\n15217\n13252\n11581\n9910\n8522\n7134\n6030\n4926\n4106\n3286\n2664\n2042\n1632\n1222\n941\n660\n497\n334\n233\n132\n89\n46\n25\n4\n−4\n−12\n−10\n−8\n−6\n−4\n−4\n−4\n−2\n1212\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\n21212\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\n12121\n20922\n20922\n20625\n20328\n19815\n19302\n18558\n17814\n16848\n15882\n14868\n13854\n12740\n11626\n10537\n9448\n8430\n7412\n6509\n5606\n4830\n4054\n3439\n2824\n2349\n1874\n1526\n1178\n945\n712\n550\n388\n301\n214\n155\n96\n70\n44\n30\n16\n11\n6\n3\nFigure 19: The vectors ˆBσ (cont.)\n53\n\nσ\nˆBσ\n121212\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\n2121212\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\n1212121\n3800\n3800\n3735\n3670\n3573\n3476\n3326\n3176\n2974\n2772\n2567\n2362\n2138\n1914\n1692\n1470\n1277\n1084\n921\n758\n624\n490\n396\n302\n238\n174\n131\n88\n65\n42\n29\n16\n11\n6\n3\n12121212\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\n212121212\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\n1212121212\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 ˆBσ (cont)\n54\n\n[DJ]\nR. Dipper and G. James, Representations of Hecke algebras of\ngeneral linear groups, Proc. London Math. Soc. (3). 52 (1986), 20-\n52.\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\nat: http://ramakrishnadas.cs.uchicago.edu. Revised version to be\navailable here.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups, revised version under\npreparation.\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, 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[Kass]\nC. Kassel, Quantum groups, Springer-Verlag, 1995.\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.\n55\n\n[Li]\nP. Littelmann: A Littlewood-Richardson rule for symmetrizable\nKac-Moody Lie algebras, Invent. math. 116 (1994), 329-346.\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[Mc]\nI. Macdonald, Symmetric functions and Hall polynomials, Oxford\nScience Publications, 1995.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[Ro]\nO. Rossi-Doria, A Uq(sl(2))-representation with no quantum sym-\nmetric algebra, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.\nRend. Lincei (9), Mat. Appl. 10 10 (1999), no. 1, 5-9.\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.\n56\n\nNumber\nCoefficient\nσ\n1\n20.q0+104.q1+256.q2−113.q3−49.q4−113.q5+256.q6+104.q7+20.q8\n2.q3+12.q4+2.q5\n1\n2\n−16.q0−64.q1−128.q2−192.q3−224.q4−192.q5−128.q6−64.q7−16.q8\n2.q3+12.q4+2.q5\n2\n3\n−4.q0−16.q1−28.q2−32.q3−28.q4−16.q5−4.q6\n2.q3\n3\n4\n1.q0−4.q2+6.q4−4.q6+1.q8\n2.q3+2.q5\n12\n5\n−1.q0−18.q1−65.q2−128.q3−190.q4−220.q5−190.q6−128.q7−65.q8−18.q9−1.q10\n2.q4+12.q5+2.q6\n13\n6\n1.q0+5.q1+17.q2+36.q3+46.q4+46.q5+46.q6+36.q7+17.q8+5.q9+1.q10\n2.q4+2.q6\n21\n7\n7.q0+26.q1+75.q2+152.q3+174.q4+156.q5+174.q6+152.q7+75.q8+26.q9+7.q10\n2.q3+12.q4+4.q5+12.q6+2.q7\n23\n8\n−1.q0−8.q1−20.q2−24.q3−22.q4−24.q5−20.q6−8.q7−1.q8\n2.q3+2.q5\n32\n9\n−22.q0−92.q1−170.q2−200.q3−170.q4−92.q5−22.q6\n2.q2+12.q3+2.q4\n121\n10\n2.q0+2.q1+12.q2+14.q3+4.q4+14.q5+12.q6+2.q7+2.q8\n2.q3+2.q5\n132\n11\n−2.q0−12.q1−40.q2−52.q3−44.q4−52.q5−40.q6−12.q7−2.q8\n2.q3+12.q4+2.q5\n212\n12\n−1.q0−2.q1−12.q2−14.q3−6.q4−14.q5−12.q6−2.q7−1.q8\n2.q3+2.q5\n213\n13\n1.q0+22.q1+88.q2+170.q3+206.q4+170.q5+88.q6+22.q7+1.q8\n2.q3+12.q4+2.q5\n232\n14\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q2\n323\n15\n3.q0+6.q1+5.q2+4.q3+5.q4+6.q5+3.q6\n2.q2+2.q4\n1212\n16\n12.q0+32.q1+40.q2+32.q3+12.q4\n2.q1+12.q2+2.q3\n1213\n17\n−3.q0−2.q1−5.q2−12.q3−5.q4−2.q5−3.q6\n2.q2+2.q4\n1232\n18\n1.q0+4.q1+11.q2+16.q3+11.q4+4.q5+1.q6\n2.q3\n1321\n19\n8.q0+12.q1+24.q2+40.q3+24.q4+12.q5+8.q6\n2.q2+12.q3+2.q4\n1323\n20\n−6.q0−8.q1−4.q2−8.q3−6.q4\n2.q1+2.q3\n2121\n21\n−5.q0−4.q1−44.q2−60.q3−30.q4−60.q5−44.q6−4.q7−5.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n2123\n22\n−1.q0−5.q1−11.q2−14.q3−11.q4−5.q5−1.q6\n2.q3\n2321\n23\n−3.q0−6.q1−5.q2−4.q3−5.q4−6.q5−3.q6\n2.q2+2.q4\n2323\n24\n2.q0+4.q1+4.q2+4.q3+2.q4\n2.q2\n3212\n25\n−1.q0−4.q1−6.q2−4.q3−1.q4\n2.q2\n3213\n26\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q1+2.q3\n3232\n27\n16.q0+32.q1+16.q2\n2.q0+12.q1+2.q2\n12121\n28\n4.q0+8.q1+40.q2+8.q3+4.q4\n2.q1+12.q2+2.q3\n12123\n29\n−3.q0−8.q1−4.q2−8.q3+46.q4−8.q5−4.q6−8.q7−3.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n12132\n30\n−8.q0\n2.q0\n12321\nFigure 21: A relation in BH\n4 from GCT4\n57\n\nNumber\nCoefficient\nσ\n31\n−4.q0−8.q1−40.q2−8.q3−4.q4\n2.q1+12.q2+2.q3\n12323\n32\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n13212\n33\n−9.q0−6.q1−55.q2+12.q3−55.q4−6.q5−9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n13232\n34\n9.q0+6.q1+55.q2−12.q3+55.q4+6.q5+9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n21213\n35\n1.q0−1.q1+3.q2−6.q3+3.q4−1.q5+1.q6\n2.q2+2.q4\n21232\n36\n−1.q0+2.q2−1.q4\n2.q2\n21321\n37\n2.q0+3.q1+6.q2−3.q3−16.q4−3.q5+6.q6+3.q7+2.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n21323\n38\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n23213\n39\n−16.q0−32.q1−16.q2\n2.q0+12.q1+2.q2\n23232\n40\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n32121\n41\n8.q0\n2.q0\n32123\n42\n1.q0−2.q2+1.q4\n2.q2\n32132\n43\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n32321\n44\n−8.q0−16.q1−8.q2\n2.q0+12.q1+2.q2\n121213\n45\n−1.q0−14.q1−15.q2−4.q3−15.q4−14.q5−1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n121232\n46\n−2.q0−4.q1−2.q2\n2.q1\n121321\n47\n−2.q0+4.q2−2.q4\n2.q1+12.q2+2.q3\n123213\n48\n8.q0+16.q1+8.q2\n2.q0+12.q1+2.q2\n123232\n49\n−1.q0−2.q1−1.q2\n2.q1\n132121\n50\n2.q0−4.q2+2.q4\n2.q1+12.q2+2.q3\n132123\n51\n2.q0+8.q1+12.q2+8.q3+2.q4\n2.q1+12.q2+2.q3\n212132\n52\n2.q0+4.q1+2.q2\n2.q1\n212321\n53\n1.q0+14.q1+15.q2+4.q3+15.q4+14.q5+1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n212323\n54\n3.q0+8.q1+10.q2+8.q3+3.q4\n2.q1+12.q2+2.q3\n213212\n55\n−2.q0−8.q1−12.q2−8.q3−2.q4\n2.q1+12.q2+2.q3\n213232\n56\n−1.q0−2.q1−1.q2\n2.q1\n232121\n57\n−3.q0−8.q1−10.q2−8.q3−3.q4\n2.q1+12.q2+2.q3\n232132\n58\n1.q0+2.q1+1.q2\n2.q1\n232321\n59\n−2.q0−4.q1−2.q2\n2.q1\n321232\n60\n2.q0+4.q1+2.q2\n2.q1\n321323\nFigure 22: A relation in BH\n4 from GCT4 continued.\n58\n\nNumber\nCoefficient\nσ\n61\n1.q0+2.q1+1.q2\n2.q1\n323212\n62\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1212132\n63\n2.q0\n2.q0\n1213213\n64\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1213232\n65\n2.q0\n2.q0\n1232121\n66\n2.q0−4.q1+2.q2\n2.q0+12.q1+2.q2\n1232132\n67\n16.q1\n2.q0+12.q1+2.q2\n1321232\n68\n−2.q0\n2.q0\n1321323\n69\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2121323\n70\n−4.q0−8.q1−4.q2\n2.q0+12.q1+2.q2\n2123212\n71\n−16.q1\n2.q0+12.q1+2.q2\n2123213\n72\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2123232\n73\n−2.q0+4.q1−2.q2\n2.q0+12.q1+2.q2\n2132123\n74\n4.q0+8.q1+4.q2\n2.q0+12.q1+2.q2\n2321232\nFigure 23: A relation in BH\n4 from GCT4 continued.\n59","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.0749v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VII: Nonstandard\nquantum group for the plethysm problem\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\n(Technical Report TR-2007-14\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nNovember 3, 2018\nAbstract\nThis article describes a nonstandard quantum group that may be\nused to derive a positive formula for the plethysm problem, just as the\nstandard (Drinfeld-Jimbo) quantum group can be used to derive the\npositive Littlewood-Richardson rule for arbitrary complex semisimple\nLie groups. The sequel [GCT8] gives conjecturally correct algorithms\nto construct canonical bases of the coordinate rings of these nonstan-\ndard quantum groups and canonical bases of the dually paired non-\nstandard deformations of the symmetric group algebra.\nA positive\n#P-formula for the plethysm constant follows from the conjectural\nproperties of these canonical bases and the duality and reciprocity\nconjectures herein.\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1"},{"paragraph_id":"p2","order":2,"text":"1\nIntroduction\nThe following is a fundamental problem in representation theory [GCT6,\nMc, St]:\nProblem 1.1 (Plethysm problem)\nFind an explicit positive (#P-) formula in the spirit of the Littlewood-\nRichardson rule for the plethysm constant aπ\nλ,μ. For given partitions λ, μ\nand π, this is the multiplicity of the irreducible representation Vπ(H) of\nH = GLn(C) in the irreducible representation Vλ(G) of G = GL(X), where\nX = Vμ = Vμ(H) is an irreducible representation of H.\nHere Vλ(G) is\nconsidered an H-module via the representation map ρ : H →G.\n(Generalized plethysm problem):\nThe same as above, letting H be any complex, semisimple (or, more\ngenerally, reductive) classical Lie group, λ a dominant weight of G, π and\nμ dominant weights of H.\nThis article describes a quantum group that may be used to derive such\na positive formula, just as the standard (Drinfeld-Jimbo) quantum group\n[Dri, Ji, RTF] can be used to derive the positive Littlewood-Richardson rule\nfor arbitrary complex semisimple Lie groups [Kas1, Li, Lu2]; the results here\nwere announced in [GCT4] (most of the results here also hold for nonclassical\nH, though we shall only worry about classical H here). For the significance\nof a positive formula in the context of geometric complexity theory, see\n[GCTflip1]. The approach that we wish to follow is:\n1. Find a quantization of the homomorphism\nH →G\n(1)\nof the form\nHq →GH\nq ,\n(2)\nwhere Hq is the standard Drinfeld-Jimbo quantization of H, and GH\nq\nis the new nonstandard quantization of G that we seek.\n2. Develop a theory of canonical (local/global crystal) bases for the rep-\nresentations of GH\nq in the spirit of the canonical bases [Kas1, Lu1] for\nthe representations of the standard quantum group.\n2"},{"paragraph_id":"p3","order":3,"text":"3. Derive the required explicit positive formula for the plethysm constant\nfrom the properties of the canonical bases.\nThe following addresses the first step.\nTheorem 1.2 (cf.\nSection 2) There exists a possibly singular quantum\ngroup GH\nq\nsuch that the homomorphism (1) can be quantized in the form\n(2).\nFurthermore, all finite dimensional polynomial representations of GH\nq\nare completely reducible, and a quantum analogue of the Peter-Weyl theorem\nholds for the matrix coordinate ring of GH\nq .\nFor the precise meaning of the various terms here, see Section 2. Here and\nin what follows, we assume that the base field is C = C(q), q complex. But\na suitable algebraic extension of Q(q) will also suffice for our purposes; see\nSection 6 for a discussion on the base field.\nWhen H = G, GH\nq specializes to the standard quantum group Hq. When\nH = GL(V ) × GL(W), G = GL(X), X = V ⊗W with natural H-action, it\nreduces to the quantum group in [GCT4] for the Kronecker problem.\nWe call GH\nq\nthe nonstandard quantum group associated with the em-\nbedding (1). It can be singular in general. That is, its determinant may\nvanish, and hence, the antipode need not exist. Strictly speaking, it should\nhence be called a nonstandard quantum semi-group. We still use the term\ngroup, because this object has characteristic features of the standard quan-\ntum group, such as semisimplicity of polynomial representations, Peter-Weyl\ntheorem, and most importantly, conjectural existence of canonical bases for\nits representations and the matrix coordinate ring.\nWe also construct (Section 5) a nonstandard quantization BH\nr = BH\nr (q) of\nthe group algebra C[Sr] of the symmetric group Sr whose relationship with\nGH\nq\nis conjecturally akin to that of the Hecke algebra with the standard\nquantum group. Specifically, let Xq denote the irreducible representation\nVq,μ of Hq with highest weight μ; it is the usual quantization of X = Vμ.\nThen:\nConjecture 1.3 (Nonstandard duality)\n(1) The left action of GH\nq on X⊗r\nq\nand the right action BH\nr (q) on X⊗r\nq\nde-\ntermine each other.\n(2) There is a one-to-one correspondence between the irreducible polynomial\nrepresentations of GH\nq of degree r and the irreducible representations of BH\nr\n3"},{"paragraph_id":"p4","order":4,"text":"so that, as a bimodule,\nX⊗r\nq\n=\nM\nα\nWq,α ⊗Tq,α,\n(3)\nwhere Wq,α runs over the irreducible polynomial representations of GH\nq\nof\ndegree r, and Tq,α denotes the irreducible representation of BH\nr (q) in corre-\nspondence with Wq,α.\nThe irreducible representations Wq,α here need not be q-deformations of\nthe irreducible representations of G, because GH\nq\nis, in general, a nonflat\ndeformation of G. This means the Poincare series of GH\nq need not coincide\nwith that of G. Our first goal is to associate with each Weyl module Vλ of\nG a possibly reducible representation V H\nq,λ of GH\nq , called the q-analogue of\nVλ, so that\nlimq→1V H\nq,λ ∼= Vλ\nas an H-module. In this context:\nConjecture 1.4 (Nonstandard reciprocity) Let λ be a partition of size\nr. Let\nV H\nq,λ =\nM\nα\nmα\nλWq,α,\nwhere mα\nλ denotes the multiplicity of the Specht module Sλ of the symmetric\ngroup Sr in Tq,α(1) = limq→1Tq,α, as defined in Section 6. Then V H\nq,λ is a\nq-analogue of Vλ in the sense defined above.\nThus the multiplicity of the GH\nq -module Wq,α in V H\nq,λ is equal to the mul-\ntiplicity of the Specht module Sλ in the specialization of Tq,α at q = 1.\nA more refined form of this conjecture is given in Section 6. Both duality\nand reciprocity are supported by experimental evidence; cf. Section 7.\nBy the conjectural reciprocity,\naπ\nλ,μ =\nX\nα\nmα\nλnα\nπ,\nwhere nα\nπ is the multiplicity of the irreducible Hq-module Vq,π in Wq,α. Hence\nProblem 1.1 can be decomposed into the following two subproblems:\n(P1): Find a positive (#P-) formula for the multiplicity nα\nπ.\n(P2): Find a positive (#P-) formula for the multiplicity mα\nλ.\n4"},{"paragraph_id":"p5","order":5,"text":"The article [GCT8] gives conjecturally correct algorithms to construct\na canonical basis of the matrix coordinate ring of GH\nq\nwhose conjectural\nproperties would imply a positive formula as needed in the first problem,\nand a canonical basis of BH\nr\nwhose conjectural properties would imply a\npositive formula as needed in the second problem.\nAt present, we cannot prove correctness of these algorithms nor the re-\nquired conjectural properties, because we are unable to deal with the high\ncomplexity of the nonstandard quantum group. Specifically, as we shall see\nin Section 4, the formulae for the minors of the nonstandard group turn out\nto be highly nonelementary in contrast to the elementary formulae for the\nminors of the standard quantum group. The coefficients of these formulae\ndepend on the multiplicative structural constants of canonical bases akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nconstructed by Kashiwara and Lusztig [Kas2, Lu2]. To get explicit formulae\nfor these structural constants, one needs interpretations for them akin to\nthe interpretations for the Kazhdan-Lusztig polynomials and multiplicative\nstructural constants of the canonical basis of the coordinate ring of the stan-\ndard quantum group in terms of perverse sheaves [KL2, Lu1, BBD]. Thus,\nthe linear algebra for the nonstandard quantum group–i.e. the theory of its\nminors–is already highly nonelementary in contrast to the linear algebra for\nthe standard quantum group. This is why its representation theory may turn\nout to be far more complex. In particular, we cannot explicitly construct\nnor classify its irreducible polynomial representations. Of course, all this\nand much more would follow if correctness of the algorithms in [GCT8] for\nconstructing canonical bases and their conjectural properties can be proved.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for helpful discussions,\nespecially for bringing the reference [Ro] to our attention, and for the help\nin explicit computations in Section 7.2 in MATLAB.\n2\nNonstandard quantum group\nWe describe in this section the construction of the nonstandard quantum\ngroup GH\nq\nin Theorem 1.2. The reader may refer to [GCT4] for the full\ndetails in a nontrivial special case of the plethysm problem, called the Kro-\nnecker problem. For the sake of simplicity, we assume here that H = GL(V )\n(type A). Let X = Vμ(H) be its irreducible polynomial representation. The\n5"},{"paragraph_id":"p6","order":6,"text":"goal is to quantize the homomorphism\nH = GL(V ) →G = GL(X).\nLet H and G be the Lie algebras of H and G. We follow the terminology in\n[Kli], which will be our standard reference on quantum groups.\nThe standard quantum group Hq = GLq(V ) associated with GL(V ) can\nbe defined by first constructing the coordinate algebra O(Mq(V )) of the stan-\ndard quantum matrix space Mq(V ) as a suitable FRT-algebra [RTF]. The\ncoordinate ring O(GLq(V )) of GLq(V ) is obtained by localizing O(Mq(V ))\nat the suitably defined quantum determinant. The Drinfeld-Jimbo univer-\nsal enveloping algebra Uq(G) [Dri, Ji] of GLq(V ) can then be defined dually.\nSpecifically, let J be the maximal ideal of the elements in O(Mq(V )) which\nvanish at the identity–i.e. on which ǫ, the counit, vanishes. Then Uq(G) can\nbe identified with the space of linear functions on O(Mq(V )) which vanish\non Jr for some integer r > 0 depending on the linear function.\nAnalogously, we first construct the nonstandard matrix coordinate ring\nO(MH\nq (X)) of the (virtual) nonstandard matrix space MH\nq (X), and then\ndefine the nonstandard quantized universal enveloping algebra U H\nq (G) by\ndualization. We define the nonstandard quantum group GH\nq as the virtual\nobject whose universal enveloping algebra is U H\nq (G). The construction would\nyield natural bialgebra homomorphisms from Uq(H) to U H\nq (G) and from\nO(MH\nq (X)) to O(Mq(V ), thereby giving the desired quantizations of the\nhomomorphisms U(H) →U(G) and O(M(X)) →O(M(V )). This is what\nis meant by the quantization (2) of the map (1). The determinant of GH\nq\nmay vanish, and hence, we cannot, in general, define its coordinate ring\nO(GH\nq ) by localizing O(MH\nq (X)). Fortunately, this will not matter since the\ncoordinate ring O(MH\nq (X)) and the nonstandard quantized algebra BH\nr (q)\n(Section 5) together contain conjecturally all the information that we need\n(cf. Conjecture 1.4), and have properties similar to that of the standard\nmatrix coordinate ring O(Mq(V )) and the Hecke algebra; cf. Theorem 2.1\nbelow.\nThe nonstandard matrix coordinate ring O(MH\nq (X)) is constructed as\nfollows.\nLet ˆRH\nX,X be the ˆR matrix of Xq = Vq,μ considered as an Hq-\nmodule [Kli]. Here and in what follows, we sometimes denote Xq by X; the\nmeaning should be clear from the context. It is well known that ˆRH\nX,X is\ndiagonalizable and that its each eigenvalue is of the form + or −qa/2 for\nsome integer a [Kli]. Let\nI = P +,H\nX,X + P −,H\nX,X ,\n(4)\n6"},{"paragraph_id":"p7","order":7,"text":"be the associated spectral decomposition of the identity, where P +,H\nX,X and\nP −,H\nX,X denote the projections of Xq ⊗Xq on the eigenspaces of ˆRH\nX,X for\neigenvalues with + and −sign, respectively. Let u be a variable matrix\nspecifying a generic transformation from X to X. Let ui\nj denote its variable\nentries. Then O(MH\nq (X)) is defined to be the FRT bialgebra [RTF] associ-\nated with the transformation P +,H\nX,X , or equivalently, P −,H\nX,X . That is, it is the\nquotient of C⟨ui\nj⟩modulo the relations\nP +,H\nX,X (u ⊗u) = (u ⊗u)P +,H\nX,X ,\n(5)\nor equivalently,\nP −,H\nX,X (u ⊗u) = (u ⊗u)P −,H\nX,X .\n(6)\nAn alternative definition of O(MH\nq (X)) is as follows. Let SH\nq (X ⊗X),\nthe symmetric subspace of X ⊗X, be the image of P +,H\nX,X , and AH\nq (X ⊗X),\nthe antisymmetric subspace of X ⊗X, the image of P −,H\nX,X [Kli]. In other\nwords, SH\nq (X ⊗X) is defined by the equation\nP −,H\nX,X x1x2 = 0,\n(7)\nwhere x1 = x ⊗I and x2 = I ⊗x, and AH\nq (X⊗X) is defined by the equation\nP +,H\nX,X x1x2 = 0.\n(8)\nThe braided symmetric algebra [BZ, Ro] CH\nq [X] of X is defined to be\nthe algebra over the entries xi’s of x subject to the relation (7). It will be\ncalled the coordinate ring of the virtual quantum space XH\nsym. Similarly, the\nbraided exterior algebra ∧H\nq [X] of X is defined to be the algebra over the\nentries xi’s of x subject to the relation (8). It will called the coordinate ring\nof the virtual quantum space XH\n∧. Let CH,r\nq\n[X] and ∧H,r\nq\n[X] be the degree\nr components of CH\nq [X] and ∧H\nq [X], respectively.\nIt is known [BZ] that\nthe dimensions of CH,r\nq\n[X] and ∧H,R\nq\nare bounded by the dimensions of the\nclassical Cr[X] and ∧r[X], respectively. But unlike in the standard setting,\nthe dimensions can be strictly less [BZ, Ro]. That is, CH\nq [X] and ∧H\nq [X]\nare, in general, nonflat deformations of the classical symmetric and exterior\nalgebras C[X] and ∧[X]. For example, ∧H,3\nq\n[X] = 0 when H = sl2(C) and\nX is the four dimensional irreducible representation of sl2(C) [BZ].\nThe equation (5) or (6) after reformulation just says that the defining\nrelation (7) of XH\nsym–or equivalently, the defining relation (8) of XH\n∧–is pre-\nserved under the left and right actions of u on x given by x →ux and\nxt →xtu.\n7"},{"paragraph_id":"p8","order":8,"text":"This means CH\nq [X] and ∧H\nq [X] have left and right coactions of O(MH\nq (X)).\nWe define the left and right nonstandard minors of GH\nq to be the matrix co-\nefficients (in a suitable basis specified later) of the left and right coactions\non ∧H\nq [X]. If ∧H,dim(X)\nq\n[X] ̸= 0, then we define the determinant of GH\nq to\nbe the matrix coefficient of the action of O(MH\nq (X)) on ∧H,dim(X)\nq\n[X]. But\nit can vanish, as it does for H = sl2(C), dim(X) = 4. The nonstandard\nminors will be discussed in more detail in Section 4.\nLet J be the ideal of elements in O(MH\nq (X)) on which the counit ǫ van-\nishes. Then the nonstandard universal enveloping algebra U H\nq (G) is defined\nto be the space of linear functions of O(MH\nq (X)) which vanish on Jr for\nsome r > 0 depending on the linear function.\nThe following is a precise form of Theorem 1.2.\nTheorem 2.1 (1) There is a natural bialgebra homomorphism from O(MH\nq (X))\nto O(Mq(V )).\nThis gives the desired quantization of the homomorphism\nO(M(X)) →O(M(V )).\n(2) The matrix coordinate ring O(MH\nq (X)) of GH\nq is cosemisimple. Hence,\nits every finite dimensional corepresentation is completely reducible as a di-\nrect sum of irreducible corepresentations.\n(3) The q-analogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)).\n(4) The nonstandard enveloping algebra U H\nq (G) is a bialgebra with a compact\nreal form (a ∗-structure) such that X⊗r\nq\nis its unitary representation with\nrespect to the Hermitian form on X⊗r\nq\ninduced by the standard Hermitian\nform on Xq. There is a bialgebra homomorphism form Uq(H) to U H\nq (G).\nThis gives a desired quantization of the homomorphism U(H) →U(G).\nHere the standard Hermitian form on Xq is the one that is Uq-invariant,\nwhere Uq ⊆Hq is the compact real form (the unitary subgroup) of Hq in the\nsense of Woronowicz [W]. The special case of this theorem in the context of\nthe Kronecker problem was proved in [GCT4] on the basis of Woronowicz’s\nwork [W]. The latter is no longer applicable in the general context here,\nsince the determinant of GH\nq may vanish, and hence, we cannot, in general,\nconvert O(MH\nq (X)) into a Hopf algebra by localization at the determinant.\n8"},{"paragraph_id":"p9","order":9,"text":"Fortunately, this does not matter since U H\nq (G) still has a compact real form,\nwhose existence can be proved using the spectral properties of ˆRH\nX,X.\nWe also call Wq,α here a polynomial representation of GH\nq . By a poly-\nnomial representation of U H\nq (G) we mean a representation that is induced\nby a (finite dimensional) corepresentation of O(MH\nq (X)). It is completely\nreducible by cosemsimplicity of O(MH\nq (X)).\nIt may be conjectured that\nevery finite dimensional representation of U H\nq (G) is completely reducible (as\nin the standard case), though we shall not need this more general fact.\nThe standard Drinfeld-Jimbo enveloping algebra has an explicit presen-\ntation in the form of explicit generators (ei, fi, Ki) and explicit relations\namong them. It will be interesting to find an analogous explicit presenta-\ntion for U H\nq (G); cf. Section 4 for the problems that arise in this context.\n3\nNonstandard q-Schur algebra\nIn the standard setting, the q-Schur algebra Ar = Ar(q) is defined to be the\ndual O(Mq(V ))∗r of the degree r component O(Mq(V ))r of the standard\nmatrix coordinate algebra O(Mq(V )). Thus Ar(q) acts on V ⊗r from the\nleft. It is known [Kli] that it is the centralizer in End(V ⊗r) of the right\naction of the Hecke algebra Hr(q) on V ⊗r.\nAnalogously, we define the nonstandard q-Schur algebra AH\nr\n= AH\nr (q)\nto be the dual O(MH\nq (X))∗r of the degree r component O(MH\nq (X))r of\nthe nonstandard matrix coordinate algebra O(MH\nq (X)). Thus AH\nr (q) acts\non X⊗r from the left.\nAs per the nonstadard duality conjecture (Con-\njecture 1.3), it is the centralizer in End(X⊗r) of the right action of the\nnonstandard quantized algebra BH\nr (q) (cf. Section 5) on X⊗r.\nEvery irreducible corepresentation Wq,α of O(MH\nq (X)) of degree r can\nalso be considered as a representation of AH\nr (q), and conversely, every irre-\nducible representation of AH\nr (q) arises in this way. Theorem 2.1 now imme-\ndiately implies:\nTheorem 3.1 (1) The nonstandard q-Schur algebra AH\nr (q) is semisimple.\nHence, its every finite dimensional representation is completely reducible as\na direct sum of irreducible representations.\n(2) The q-analogue of the Peter-Weyl theorem in this case is the Wederburn\n9"},{"paragraph_id":"p10","order":10,"text":"structure theorem for AH\nr (q):\nAH\nr (q) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible representations of AH\nr (q).\n(3) The nonstandard q-Schur algebra AH\nr (q) has a compact real form (a ∗-\nstructure) such that X⊗r\nq\nis its unitary representation with respect to the\nHermitian form on X⊗r\nq\ninduced by the standard Hermitian form on Xq.\n4\nNonstandard minors\nIn this section, we give a conjectural formula for the Laplace expansion\nof the minors of GH\nq .\nThe Laplace expansion for the standard quantum\ngroup GLq(V ) is based on the simple relation defining the standard exterior\nalgebra ∧q[V ], namely\nv2\ni = 0\nand\nvivj = −q−1vjvi,\nfor\ni < j.\nThis explains why the Laplace expansion in the standard setting is obtained\nfrom the classical Laplace expansion by simply substituting −q for −1. We\nneed a similar explicit formula for multiplication in CH\nq [X] to get an explicit\nformula for Laplace expansion in the nonstandard setting.\n4.1\nKronecker problem\nWe begin with a special case that arises in the context of the Kronecker\nproblem [GCT4] when H = GL(V ) × GL(W) and X = V ⊗W, with the\nnatural H-action. The article [GCT4] gives a formula for the column or row\nexpansion of the minor of GLH\nq (X) in this special case in terms of fundamen-\ntal Clebsch-Gordon coefficients for the standard quantum groups GLq(V )\nand GLq(W). But this formula cannot be extended for the general Laplace\nexpansion since Clebsch-Gordon coefficients are not well defined when the\nunderlying tensor products do not have multiplicity-free decompositions as\nin the fundamental case. Here we give a formula for general Laplace expan-\nsion of the minors of GLH\nq (X) in this case.\nWe begin by recalling that when V = W ∗the braided symmetric al-\ngebra CH\nq [X] = CH[W ∗⊗W] is isomorphic to the matrix coordinate ring\nO(Mq(W)) of the standard matrix space Mq(W) [GCT4]. For this, we have:\n10"},{"paragraph_id":"p11","order":11,"text":"Theorem 4.1 (Kashiwara and Lusztig [Kas2, Lu2]) The coordinate ring\nO(Mq(W)) has an (upper) canonical basis.\nThis can be naturally and easily extended to:\nTheorem 4.2 The braided symmetric coordinate algebra CH\nq [X] = CH\nq [V ⊗\nW], H = GL(V ) × GL(W), has an (upper) canonical basis.\nThe exterior form of this result is:\nTheorem 4.3 The exterior coordinate algebra ∧H\nq [V ⊗W], H = GL(V ) ×\nGL(W), also has an (upper) canonical basis.\nLusztig [Lu2] has conjectured that the multiplicative and comultiplica-\ntive structural constants of the canonical basis of O(Mq(W)) are polynomials\nin q and q−1 with nonnegative integer coefficients; i.e., belong to N[q, q−1].\nAnalogous conjecture can be made for ∧H\nq [V ⊗W]. Specifically, it can be\nconjectured that for any canonical basis elements b and b′ in ∧H\nq [V ⊗W]:\nbb′ =\nX\nb′′\nǫ(b, b′, b′′)cb′′\nb,b′b′′,\n(9)\nwhere the sign ǫ(b, b′, b′′) is 1 or −1 and the coefficient cb′′\nb,b′ ∈N[q, q−1]. And\nconversely, any b′′ ∈∧H,r′′\nq\n[V ⊗W] can be expressed as:\nb′′ =\nX\nb,b′\nǫ′(b, b′, b′′)db,b′\nb′′ bb′,\n(10)\nwhere b and b′ run over elements of ∧H,r\nq\n[V × W] and ∧H,r′\nq\n[V × W] respec-\ntively with r′′ = r + r′, the sign ǫ′(b, b′, b′′) is 1 or −1, and db,b′\nb′′ ∈N[q, q−1].\nTo prove nonnegativity of the coefficients of cb′′\nb,b′ and db,b′\nb′′ , one needs in-\nterpretations for them in terms of perverse sheaves [BBD] in the spirit of\nKazhdan-Lusztig [KL2] and Lusztig [Lu1].\nWe now define the (left or right) minors of GH\nq\nwith respect to the\ncanonical basis ∧H,r\nq\n[V ⊗W] to be the matrix coefficients of the (left or right)\ncoaction of O(MH\nq (V ⊗W)). We shall call them (left or right) canonical\nminors. Then:\n11"},{"paragraph_id":"p12","order":12,"text":"Theorem 4.4 A canonical minor of degree r′′ of GLH\nq (X), H = GL(V ) ×\nGL(W), admits a Laplace expansion in terms of canonical minors of degree\nr and r′ with r′′ = r + r′. The coefficients of this Laplace expansion are\nquadratic forms in the structural constants cb′′\nb,b′ and db,b′\nb′′\nabove.\nAn explicit formula for Laplace expansion here (omitted) is similar to the\none in Proposition 6.1 of [GCT4] with these structural constants in place of\nthe Clebsch-Gordon coefficients there (which are not well defined for general\nLaplace expansion).\n4.2\nGeneral nonstandard setting\nNow let us turn to the general case. The conjecturally correct algorithm\nin [GCT8] for constructing a canonical basis of O(MH\nq (X)) also yields, as\na byproduct, conjectural canonical bases of ∧H\nq [X] and CH\nq [X] as implicitly\nsought in [BZ]. We define the (left or right) minors of GH\nq in general to be\nthe matrix coefficients of the (left or right) coaction of O(MH\nq (X)) in this\ncanonical basis of ∧H\nq [X]. We call these nonstandard canonical minors, or\nsimply nonstandard minors.\nOne can define structural constants cb′′\nb,b′ and db,b′\nb′′ analogous to the ones\nin (9) and (10) in this case. With this:\nTheorem 4.5 Analogue of Theorem 4.4 holds in general.\nLaplace expansion in the standard setting is used as a straightening\nrelation to construct standard monomial bases of the coordinate ring and\nirreducible representations of GLq(X). In this sense, Laplace expansion is\na mother relation that governs the representation theory of the standard\nquantum group. Similarly, the nonstandard Laplace expansions in Theo-\nrems 4.4 and 4.5 are expected to be mother relations governing the rep-\nresentation theory of the nonstandard quantum group GH\nq . In particular,\nan explicit interpretation for the structural coefficients cb′′\nb,b′ and db,b′\nb′′\nakin\nto the ones based on perverse sheaves for the Kazhdan-Lusztig polynomials\n[KL2] and the multiplicative structural constants of the canonical basis for\nthe standard quantum group [Lu2] is necessary to get fully explicit formulae\nfor the nonstandard minors, and hence, for constructing explicit bases for\nthe irreducible polynomial representations and the matrix coordinate ring\nof GH\nq . In particular, this seems necessary for proving correctness of the\n12"},{"paragraph_id":"p13","order":13,"text":"algorithms in [GCT8] for constructing nonstandard canonical bases for the\npolynomial representations and the matrix coordinate ring of GH\nq . This also\nseems necessary for finding an explicit presentation of the nonstandard uni-\nversal enveloping algebra U H\nq (G) in the spirit of the explicit presentation of\nthe Drinfeld-Jimbo enveloping algebra. Specifically, we expect the coeffi-\ncients occuring in such an explicit presentation to depend on the structural\nconstants such as cb′′\nb,b′ and db,b′\nb′′ above.\n5\nNonstandard quantized algebra\nWe now construct a nonstandard quantization BH\nr (q) of the symmetric\ngroup ring C[Sr] which conjecturally has the same relationship with GH\nq\nthat the Hecke algebra Hr(q), the standard deformation of C[Sr], has with\nthe standard quantum group. For the sake of simplicity, we assume that\nH = GL(V ).\nChoose a standard embedding of X = Vμ(H) in V ⊗d, where d is the size\nof the partition μ. That is, choose a Young symmetrizer cμ ∈C[Sr] such\nthat V ⊗d ·cμ, the image of V ⊗d under the right action of cμ, is isomorphic to\nX = Vμ(H). Let zμ ∈Hd(q) be the quantization of cμ such that V ⊗d\nq\n· zμ ∼=\nXq = Vq,μ. Here Vq denotes the quantization of V and Vq,μ the irreducible\nHq module with highest weight μ.\nAn explicit expression of zμ may be\nfound in [DJ]. Let Zq = V ⊗d\nq\n. Let ˆRH\nZ,Z denote the ˆR-matrix of Zq as an\nHq-module. Let rZ ∈H2d(q), 1 ≤i < r, be the element whose right action\non Zq ⊗Zq = V ⊗2d\nq\ncoincides with the action of ˆRH\nZ,Z. One can easily write\ndown an explicit expression for rZ in terms of the generators of H2d(q).\nNow consider the right action of Hs(q), s = dr, on Z⊗r\nq\n= V ⊗s\nq\n, which\ncommutes with the left action of Hq = GLq(V ). Let rZ,i ∈Hs(q), 1 ≤i < r,\nbe the element whose right action on Z⊗r\nq\ncoincides with the action of ˆRH\nZ,Z\non the product of the i-th and (i + 1)-st factors of Z⊗r\nq . Thus rZ,i is the\nimage of rZ under the obvious embedding of H2d(q) in Hs(q) depending on\ni. One can thus write down an explicit expression for rZ,i in terms of the\ngenerators of Hs(q). Let\nrH\nX,i = zλ,i · zλ,i+1 · rZ,i,\nwhere zλ,i ∈Hs(q) denotes an explicit element whose action on the i-th\nfactor of Z⊗r\nq\ncoincides with the action of zλ on that factor–it is the image of\nzλ under the obvious embedding of Hd(q) in Hs(q) depending on i. Then the\nright action of rH\nX,i on Z⊗r\nq\ncorresponds to the action of ˆRH\nX,X on the product\n13"},{"paragraph_id":"p14","order":14,"text":"of the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq . Let p+,H\nX,i , p−,H\nX,i ∈Hs(q) be\nthe polynomials in rH\nX,i whose actions on Z⊗r\nq\ncorrespond to the actions of\nthe positive and negative projection operators P +,H\nX,X and P −,H\nX,X in eq. (4) on\nthe tensor product of the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq ; one can\nwrite down these polynomials explicitly, using the known explicit spectral\nform of ˆrH\nX,i.\nWe define the nonstandard quantized algebra BH\nr (q) to be the subalgebra\nof Hs(q) generated by the explicit elements p+,H\nX,i , or equivalently, p−,H\nX,i . In\ngeneral, it is a nonflat deformation of C[Sr]. That is, its dimension can be\nlarger than that of C[Sr]. It can be shown to be semisimple. Its right action\non X⊗r\nq\ncommutes with the left action GH\nq by the defining equation (5) of\nGH\nq . Conjecture 1.3 says that its relationship with GH\nq\nis akin to that of\nHr(q) with the standard quantum group Gq = GLq(X).\nThe Hecke algebra has an explicit presentation in terms of explicit re-\nlations among its generators.\nIt will be interesting to find an analogous\nexplicit presentation for BH\nr (q). Its complexity would be much higher than\nthat of the Hecke algebra as indicated by the concrete computations in\n[GCT4].\nSpecifically, we expect an explicit presentation for BH\nr (q) with\ndefining relations whose coefficients are akin to the structural constants cb′′\nb,b′\nand db,b′\nb′′ in Section 4 and have a topological interpretation akin to the one\nfor Kazhdan-Lusztig polynomials. Such an explicit presentation is needed\nto prove correctness of the algorithm in [GCT8] to construct a canonical\nbasis of BH\nr .\nRemark: We can also define a (possibly singular) quantum group ˆGH\nq , in-\nstead of GH\nq , by substituting ˆRH\nX,X in place of P +,H\nX,X in the defining equation\n(5). One can then define a deformation ˆBH\nq (r) of C[Sr] that is conjecturally\npaired with ˆGH\nq , as GH\nq is with BH\nq (r). The main results (semisimplicity, and\nq-analogue of the Peter-Weyl theorem) also hold for these objects. Further-\nmore, variants of the algorithms in [GCT8] can be conjectured to provide\ncanonical bases for these as well.\nHowever, the Poincare series of ˆGH\nq\nis\nmuch smaller than that of GH\nq , and for this and other reasons, it does not\nseem possible to use these objects in the context of the plethysm problem.\nHowever, these may be interesting intermediate quantum objects to study\nnevertheless.\n14"},{"paragraph_id":"p15","order":15,"text":"6\nRefined reciprocity\nWe now describe a refinement of the reciprocity conjecture (Conjecture 1.4)\nthat specifies precisely how the decomposion (3) of X⊗r\nq ,\nX⊗r\nq\n=\nM\nWq,α ⊗Tq,α,\n(11)\nas a GH\nq × BH\nr (q)-bimodule, tends to the decomposition\nX⊗r =\nM\nλ\nVλ ⊗Sλ\n(12)\nof X⊗r as a G × Sr-bimodule, as q →1, and gives an explicit realization\nwithin X⊗r\nq\nof the q-analogue V H\nq,λ of Vλ as in Conjecture 1.4. Here, as usual,\nVλ denotes the Weyl module of G, and Sλ the Specht module of Sr.\nFirst, we have to define the multiplicity mα\nλ of a Specht module Sλ in the\nspecialization Tq,α(1) of Tq,α at q = 1. In this context, it may be remarked\nthat though B = BH\nr (q) is semisimple, its specialization B(1) at q = 1 need\nnot be semisimple; see Section 7.1 for an example. Clearly, every represen-\ntation of Sr is also a representation of B(1), though not always conversely.\nBut it may be conjectured that every irreducible B(1)-representation is also\nan irreducible Sr-representation, i.e., a Specht module. Fix any (maximal)\ncomposition series of Tq,α(1) as a B(1)-module. We define the multiplic-\nity mα\nλ to be the number of factors in this (or any such) series that are\nisomorphic to the specht module Sλ.\nSince B is semisimple (cf. Section 5), it admits a Wederburn structure\ndecomposition of the form\nB =\nM\nU α,\nU α = Tq,α,L ⊗Tq,α,R,\n(13)\nwhere α is as in (11), and Tq,α,L and Tq,α,R denote the left and right irre-\nducible B-modules indexed by α. We call this a complete Wederburn struc-\nture decomposition. Here we are assuming that the base field is C = C(q),\nq complex. This complete decomposition would also hold if the base field\nis instead an appropriate algebraic extension K of Q(q). In the standard\nsetting of Hecke algebras, K = Q(q) suffices. This need not be so in the\nnonstandard setting. That is, an algebraic extension of Q(q) may be ac-\ntually necessary for a complete decomposition of the above form to hold;\nsee Section 7.1 for an example. If the base field is Q(q), each U α in the\nWederburn structure decomposition need not be, in general, of the form\n15"},{"paragraph_id":"p16","order":16,"text":"Tq,α,L ⊗Tq,α,R as above, but rather it would be isomorphic to the endomor-\nphism ring of Tq,α over the division algebra EndB(Tq,α). One has to take\nsimilar variations of the nonstandard q-analogue of the Peter-Weyl theorem\n(Theorem 2.1 (3)) and the duality conjecture (Conjecture 1.3) if the base\nfield is Q(q). However, for the reciprocity conjecture, it is necessary to take\nthe base field as C(q), q complex, or an algebraic extension K of Q(q) as\ndescribed above. We assume this in the rest of this section. See Section 7.1\nfor an example wherein reciprocity fails over Q(q).\nFix any right cell, i.e., an irreducible right B-subrepresentation within\nU α. Let us denote it by Tq,α,R again. Fix a maximal composition series as\na B(1)-module of the specialization Tq,α,R(1) of Tq,α,R at q = 1:\nˆTα,0 ⊂ˆTα,1 ⊂· · · ⊂ˆTα,l(α) = Tq,α,R(1).\nLet {xi} denote the upper canonical basis of Xq as an Hq-module.\nConjecture 6.1 (Nonstandard refined reciprocity)\nThere exists a basis Zα of Tq,α,R for each α with a filtration\nZα,0 ⊂Zα,1 ⊂· · · ⊂Zα,l(α) = Zα,\nsuch that:\n1. The specialization Zα,i(1) of Zα,i at q = 1 is a basis of ˆTα,i.\n2. Let zj\nα,i denote the basis elements in Zα,i \\ Zα,i−1.\nLet λα,i be the\npartition such that ˆTα,i/ ˆTα,i−1 ∼= Sλα,i as a B(1)-module (or equiv-\nalently as an Sr-module).\nFor any α, i, define the left GH\nq -module\nWq,α,i = ∪jX⊗r\nq\n· zj\nα,i. By the duality conjecture (Conjecture 1.3),\nWq,α,i ⊆Wq,α ⊗Tq,α ⊆X⊗r\nq .\n(14)\nWe define its specialization W1,α,i at q = 1, also denoted by Wq,α,i(1),\nas follows. Let a(α, i) be the largest nonnegative integer such that the\nlimit vector\nlimq→1xi1 ⊗· · · ⊗xir.zj\nα,i/(q −1)a(α,i),\nis well defined for any i1, . . . , ir and j. We define W1,α,i to be the span\nof such limits at q = 1. Then, W1,α,i is a left G-module contained\nwithin the component Vλα,i ⊗Sλα,i ⊆X⊗r in (12).\n16"},{"paragraph_id":"p17","order":17,"text":"3. For any fixed partition λ,\nM\nα\nM\ni\nW1,α,i = Vλ ⊗Sλ ⊆X⊗r,\n(15)\nwhere, for a given α, i ranges over all indices such that λα,i = λ.\nFurthermore, it may be conjectured that the canonical basis of Tq,α,R\nin terms of the P-monomials as defined in [GCT8] has this property–this\nwould make everything in the conjecture above explicit.\nThe refined reciprocity conjecture basically says that there is no infor-\nmation loss in the nonstandard setting despite the lack of flatness. In fact,\nit can be thought of as a variant of flatness.\n7\nEvidence for duality and reciprocity\nHere we describe some concrete computations carried out in MATLAB/Maple\nthat support duality and reciprocity conjectures.\nNotation: We denote the q-Weyl module of Gq for a partition λ by Vq,λ(Gq).\nWe denote Vq,λ(GLq(Cn)) by Vq,λ(n).\n7.1\nExample 1\nLet H = GL(C2), H = gl(C2), X = V(3)(H) is its four dimensional irre-\nducible representation, and G = GL(X) = GL(C4). Then Hq = GLq(C2),\nGq = GLq(C4), and Hq = glq(C2). We shall verify duality and reciprocity\nin this case for r = 3. This example is interesting because, as shown in [BZ],\nthe degree three component ∧H,3\nq\n[X] of the braided exterior algebra vanishes\nin this case. We expect that the results in this section can be extended to\nany irreducible representation X of H. But we shall confine ourselves to the\ncase dim(X) = 4, since this seems to be the gist.\nLet ˆR = ˆRH\nX,X be the ˆR-matrix associated with Xq. Let P = P H\nX,X and\nQ = QH\nX,X be the projections on the eigenspaces in Xq ⊗Xq for the positive\nand negative eigenvalues of ˆRH\nX,X, respectively. Let xi = f ix0, where f is\nthe usual operator in Hq, and x0 is the highest weight vector in Xq. Matrices\nof ˆR, P and Q in the basis xi ⊗xj of Xq ⊗Xq can be calculated from the\nknown explicit formulae; cf. [Kass, Kli]. The eigenvalues of ˆR turn out to be\nq9/2, −q−3/2, q−11/2 and −q−15/2. Explicit matrix of P in the basis xi ⊗xj\nof Xq ⊗Xq is given by\n17"},{"paragraph_id":"p18","order":18,"text":"P = 1\nf P,\n(16)\nwhere\nf = (q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n(17)\nand the matrix of P is as specified in Figure 1 with the following sparse\nrepresentation: the entry (j, v) in the i-row in Figure 1 means P(i, j) = v.\nThus the entry (5, (q4 + 1)/q2) in the second row there means P(2, 5) =\n(q4 +1)/q2. The entries of P-matrix not shown in Figure 1 are all zero. The\nscaling factor f here is chosen so that the entries of P-matrix are polynomials\nin q and q−1. Explicit matrix of\nQ = fQ\n(18)\nis similar.\n7.1.1\nExplicit presentation of B\nLet P1 and P2 denote the P operators on the first two and the last two\nfactors X⊗3, respectively; Q1 and Q2 are defined similarly. We have the\ntrivial relations:\nQ2\ni = fQi,\nand P2\ni = fPi.\nThe first nontrivial basic relation among Qi’s, as determined with the help\nof a computer, is:\nX\nσ\naσQσ = 0,\n(19)\nwhere σ ranges over the various strings of 1’s and 2’s as shown in Figure 2,\naσ ∈Q[q, q−1] are as specified there, and, for a string σ = i1i2 · · · , Qσ\ndenotes the monomial Qi1Qi2 · · · .\nThe second relation is obtained from\nthis by simply interchaning Q1 and Q2. Simialrly, the first nontrivial basic\nrelation among Pi’s is\nX\nσ\nbσPσ = 0,\n(20)\nwhere σ ranges over strings of 1’s and 2’s as in Figures 3-4, bσ’s are as shown\nthere, and Pσ is defined similarly. The second relation is obtained from this\nby simply interchanging P1 and P2. All coefficients in Figures 2-4 as well\nas other figures in this section are shown in factored forms, i.e., as products\nof irreducible polynomials. One may ask if these coefficients have a nice\ninterpretation; we shall turn to this question in Section 7.1.7.\n18"},{"paragraph_id":"p19","order":19,"text":"Let B = BH\n3 (q) be the nonstandard algebra in this case, as defined in\nSection 5. It is isomorphic to the algebra generated by Pi’s subject to the\ntwo basic nontrivial relations among Pi’s described above and the trivial\nrelations P2\ni = fPi, or equivalently, to the algebra generated by Qi’s subject\nto the two basic nontrivial relations among Qi’s described above, and the\ntrivial relations Q2\ni = fQi.\nIt is clear from these basic defining relations that {Pσ} or {Qσ}, where σ\nranges over all strings of 1’s and 2’s of length at most 10 without consecutive\n1’s or 2’s, is a basis of B. Its dimension is 21.\n7.1.2\nWederburn structure decomposition\nUnlike for the Hecke algebras, for the complete Wederburn structure decom-\nposition as in (13) to hold for B, the base field has to contain the algebraic\nextension K of Q(q) defined as follows. Let\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(21)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x] be\nthe algebraic extension of Q(q) obtained by adjoining x. We assume that\nB is defined over this base field. It was found by computer that B has one\none-dimensional irreducible representation T0, and five two-dimensional irre-\nducible representations Ti, 1 ≤i ≤5, with a complete Wederburn structure\ndecomposition\nB =\nM\ni\nUi,\nUi = Ti,L ⊗Ti,R,\n(22)\nwhere the basis elements of the various B ⊗B-bimodules Ui and the explicit\nrepresentation matrices of the irreducible B-representations Ti are as follows.\nLet U0 be the K-span of u0 ∈B, where u0 is as specified in Figures 5-6.\nThe coefficients in these and the following figures are in the basis {Qσ}. Let\nUi, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the matrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni"},{"paragraph_id":"p20","order":20,"text":",\nwhere u1\n1 is as specified in Figure 7, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 8-10. Let u2\ni ,\n19"},{"paragraph_id":"p21","order":21,"text":"1 ≤i ≤5, be the element obtained from u1\ni by interchanging Q1 and Q2.\nLet u12\ni\n= u1\ni Q2, and u21\ni\n= Q2u1\ni , for 1 ≤i ≤5.\nThen it can be shown that each Ui has a left and right action of B, and\nas a B ⊗B-bimodule\nB =\nM\ni\nUi.\n(23)\nThe columns of ui correspond to the left cells and the rows to right cells;\ni.e., the span of each column (row) is a left (resp. right) B-module, which\nwe shall denote by Ti,L (resp. Ti,R). Thus,\nB =\nM\ni\nTi,L ⊗Ti,R.\n(24)\nHere T0, the span of u0, is the trivial one dimensional representation of\nB, since it can be verified that:\nQju0 = 0,\nfor j = 1, 2.\nThe representation matrices M1\ni and M2\ni of Q1 and Q2 in the basis {u1\ni , u21\ni }\nof Ti,L, 1 ≤i ≤5, are as follows:\nM1\ni =\n 0\n1\n0\nf"},{"paragraph_id":"p22","order":22,"text":",\nwhere f is the scaling factor in (16),\nM2\ni =\n f\ngi\n0\n0"},{"paragraph_id":"p23","order":23,"text":",\nwhere gi are as shown in Figure 11; g2 is obtained from g1 by substituting\n−x for x.\nLet Ti(1) denote the specialization of Ti at q = 1. It is a representation\nof B(1), the specialization of B at q = 1. Then T0(1) corresponds to the\ntrivial one-dimensional representation of S3. There is no one dimensional\nrepresentation of B that specializes to the alternating (signed) one dimen-\nsional representation of S3. This implies that the degree three component\n∧H,3\nq\n[X] of the braided exterior algebra ∧H\nq [X] in this case is zero–as was\nalready observed and proved by other means in [BZ].\nAt q = 1, the values of f = f(q) and gi = gi(q) are as follows:\nf(1) = g1(1) = g3(1) = g4(1) = g5(1) = 4, and g2(1) = 16.\n20"},{"paragraph_id":"p24","order":24,"text":"Hence the B(1)-modules T1(1), T3(1), T4(2) and T5(2) are all isomorphic, and\nit can be verified that they are isomorphic to the Specht module S(2,1) of the\nsymmetric group S3 for the partition (2, 1). The module T2(1) is reducible.\nBecause it can be verified that it contains an irreducible B(1)-module T 1\n2 (1)\nisomorphic to the trivial one dimensional Specht module S(3) of the sym-\nmetric group S3, and the quotient T 2\n2 (1) = T2(1)/T 1\n2 (1) is isomorphic to\nthe one dimensional signed representation S(1,1,1) of S3. But T2(1) is not\ncompletely reducible as a B(1) module. That is, T2(1) ̸∼= T 1\n2 (1) ⊕T 2\n2 (1),\nsince it does not contain a submodule isomorphic to S(1,1,1). Thus, though\nB is semisimple for generic q, its specialization B(1) is not semisimple.\n7.1.3\nDuality\nPick an element ui from each Ui, 1 ≤i ≤5; say, ui = u1\ni , and u0 is as before.\nFor 0 ≤i ≤5, let Wi = X⊗3\nq\n· ui, which has a left action of the nonstandard\nquantum group GH\nq . These are nonisomorphic irreducible representations\nof GH\nq . Their explicit decompositions as Hq-modules, Hq = GLq(C2), were\ndetermined with the help of computer. They are as follows.\nThe module W0 is isomorphic to the sixteen dimensional degree three\ncomponent CH,3\nq\n[X] of the braided symmetric algebra [BZ] with the following\ndecomposition as an Hq-module:\nW0 = Vq,(9)(2) ⊕Vq,(7,2)(2);\nrecall that Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to\nthe partition λ. This decomposition of CH,3\nq\n[X] in this case agrees with the\none obtained in [BZ] by other means.\nThe modules Wi, i > 0, are distinct irreducible representations of GH\nq\nwith the following decompositions as Hq-modules:\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(25)\nTheir dimensions are 4, 4, 8, 2 and 6, respectively. Though W1 and W2 are\nisomorphic as Hq-modules, they are nonisomorphic as GH\nq -modules; the ma-\ntrix coefficients of W2 are obtained from those for W1 by substituting −x\nfor x.\n21"},{"paragraph_id":"p25","order":25,"text":"It can be verified that, as a GH\nq × B-bimodule,\nX⊗3\nq\n∼= ⊕iWi ⊗Ti,\n(26)\nas per the duality conjecture (Conjecture 1.3).\n7.1.4\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of the symmetric\ngroup S3 in the B(1)-module Ti. Then, we see that\nm0\n(3) = 1,\nm1\n(2,1) = m3\n(2,1) = m4\n(2,1) = m5\n(2,1) = 1,\nm2\n(3) = m2\n(1,1,1) = 1.\nFurthermore, it can be verified that the various Gq-modules, Gq =\nGLq(C4), decompose as follows when considered as Hq-modules:\nVq,(3)(4)\n∼=\nm0\n(3)W0 ⊕m2\n(3)W2,\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2) ⊕Vq,(6,3)(2),\nand\nVq,(2,1)(4)\n∼=\nm1\n(2,1)W1 ⊕m3\n(2,1)W3 ⊕m4\n(2,1)W4 ⊕m5\n(2,1)W5\n∼=\nVq,(6,3)(2) ⊕Vq,(8,1)(2) ⊕Vq,(5,4)(2) ⊕Vq,(7,2)(2).\nThis verifies the nonstandard reciprocity conjecture (Conjecture 1.4) in this\ncase.\n7.1.5\nRefined reciprocity\nFix a right cell within U2 isomorphic to the representation T2,R; say, the one\nspanned by u1\n2 and u12\n2 . We shall denote it by T2,R again. Let z0 ∈T2,R be\nthe element such that z0Q2 = 0. Its coefficients are shown in Figure 12 in\nthe basis {Qσ}. Let z1 = u1\n2. Then the basis Z = {z0, z1} of T2,R admits a\nfiltration\nZ0 = {z0} ⊆Z1 = {z0, z1},\nthat yields at q = 1 a composition series of T2,R(1) as a B(1)-module:\nˆT2,0 ⊂ˆT2,1 = T2,R(1),\n22"},{"paragraph_id":"p26","order":26,"text":"where ˆT2,0, spanned by the specialization z0(1) of z0, is the one dimensional\ntrivial representation of S3, and ˆT2,1/ ˆT2,0 is the one-dimensional signed rep-\nresentation of S3.\nLet W2,1 = X⊗3\nq\n· z1 and W2,0 = X⊗3\nq\n· z0 be the GH\nq -submodules of X⊗3\nq ,\nand W2,1(1), W2,0(1) their specializations at q = 1. It can be verified that\nat q = 1 we get:\nW2,1(1) = ∧3(X) ⊆X⊗3,\nand\nW0(1) ⊕W2,0(1) = Sym3(X) ⊆X⊗3,\n(27)\nwhere ∧3(X) and Sym3(X) are the Weyl modules of G = GL(X) for the\npartitions (1, 1, 1) and (3), respectively, and W0(1) the specialization of W0\nat q = 1.\nFor example, Figures 13-16 show the nonzero coefficients of the elements\na = (x1 ⊗x2 ⊗x0) · z1 and b = (x1 ⊗x2 ⊗x0) · z0 in the monomial basis\n{xi ⊗xj ⊗xk} of X⊗3\nq . It can be verified that the specialization a(1) at\nq = 1 of a indeed belongs to the subspace ∧3(X) ⊆X⊗3. The specialization\nb(1) of b, as it is, just vanishes, since its coefficients are divisible by (q −1)2.\nBut instead we consider the basis element b′ = b/(q −1)2 of W2,0. Then\nits specialization b′(1) at q = 1 indeed belongs to the subspace Sym3(X) of\nX⊗3. The equation (27) can be verified similarly.\nSimilarly it can be verified that\nlim\nq→1\nM\ni=1,3,4,5\n(X⊗3\nq\n· u1\ni ∪X⊗3\nq\n· u12\ni ) = V(2,1) ⊗S(2,1) ⊆X⊗3.\nThis verifies the refined reciprocity conjecture in this case. In particular,\nit explains what happens to the exterior and symmetric algebra components\nhere. Specifically, though the braided exterior algebra component ∧H,3\nq\n[X] =\n0,\nW2,1(1) = ∧H,3[X].\nThus the q-deformation of ∧H\n3 [X] has simply relocated itself as W2,1 in the\ndecomposition\nX⊗3\nq\n= ⊕Wi ⊗Ti.\nSimilarly, the symmetric algebra component CH,3[X] splits in two parts, and\nthe q-deformations of these parts, namely W0 and W2,0, get distributed in\nthis decomposition. The situation for V2,1 is similar. Thus, overall, there\nis no information loss; the information has only been redistributed. As per\nthe refined reciprocity conjecture, this is a general phenomenon.\n23"},{"paragraph_id":"p27","order":27,"text":"7.1.6\nBase field Q(q)\nLet us now see what happens if the base field is Q(q) instead.\nThe B-\nrepresentations T0, T3, T4, T5 are already defined over Q(q).\nBut T1 and\nT2 merge into a four dimensional B-representation T12 defined over Q(q).\nExplicitly, it can be realized within B as the linear span of the elements\nv1\n=\n(u1\n1 + u1\n2)/2,\nv2\n=\n(u1\n1 −u1\n2)/(2x),\nv3\n=\n(u21\n1 + u21\n2 )/2,\nv3\n=\n(u21\n1 −u21\n2 )/(2x).\nRepresentation matrices of left multiplication by Q1 and Q2 in the basis\n{vi} are, respectively,\nM1 ="},{"paragraph_id":"p28","order":28,"text":"q10+q6+q4+1\nq5\n0\na\n1/2 q−20\n0\nq10+q6+q4+1\nq5\nb\na\n0\n0\n0\n0\n0\n0\n0\n0"},{"paragraph_id":"p29","order":29,"text":",\nwith\na\n=\n1/2 3 q16+4 q12−2 q10+10 q8−2 q6+4 q4+3\nq8\n,\nb\n=\n1/2 q4(5 q32 + 8 q28 −4 q26 + 28 q24 −4 q22 + 24 q20 −8 q18 + 46 q16\n−8 q14 + 24 q12 −4 q10 + 28 q8 −4 q6 + 8 q4 + 5),\nand\nM2 ="},{"paragraph_id":"p30","order":30,"text":"0\n0\n0\n0\n0\n0\n0\n0\n1\n0\nq10+q6+q4+1\nq5\n0\n0\n1\n0\nq10+q6+q4+1\nq5"},{"paragraph_id":"p31","order":31,"text":".\nSimilarly, the GH\nq -modules W0, W3, W4, W5 are already defined over Q(q).\nThe modules W1 and W2 merge into an eight-dimensional GH\nq -module W12 ∼=\nX⊗3\nq\n· vi, for any i–this is defined over Q(q). As an Hq-module,\nW1,2 ∼= 2 · Vq,(6,3)(2).\n24"},{"paragraph_id":"p32","order":32,"text":"A variant of the duality also holds. Specifically, the components W1⊗T1 and\nW2⊗T2 in the decomposition (26) of X⊗3\nq\nmerge into one sixteen dimensional\nGH\nq × B-bimodule defined over Q(q). As a GH\nq module, it is a direct sum of\ntwo copies of W12, and as a B-module a direct sum of four copies of T12. But\nfor the reciprocity to hold, the base field has to be K = Q(q)[x] as before or\nlarger. Indeed, it can be seen here that the reciprocity conjecture fails over\nthe base field Q(q). This illustrates the need for base extension in general.\nIt may be illuminating to compare the r = 3 case here with the one for\nthe Kronecker problem treated in [GCT4]. The one here is basically a more\ncomplex version of the one in [GCT4], because the basic defining relations\nhere (Figures 2-4) are more complex versions of the ones in [GCT4].\n7.1.7\nOn r > 3 and positivity\nSimilar symbolic computations for r = 4 seem beyond the reach of desktop\nMATLAB/Maple. Fortunately, this case for the Kronecker problem is within\nthe reach, and will be treated in the next section. The r = 4 case, H =\nGL2(C), X four dimensional, is expected to be its more complex version\njust as for r = 3.\nBut it does not seem possible to progress much beyond r = 3 using the\nbrute force computer-based approach that we are following here. What is\nneeeded is an explicit presentation for BH\nr akin to the explicit presentation\nfor the Hecke algebra, or the one for r = 3 in Section 7.1.1. That is, we need\nan explicit set of generating relations among Qi or Pi’s, each of the form\nX\naσQσ = 0,\n(28)\nor\nX\nbσPσ = 0,\n(29)\nwhere Qσ and Pσ, for a string σ = i1i2 · · · of symbols in {1, · · · , r −1},\ndenote the monomials Qi1Qi2 · · · and Pi1Pi2 · · · , respectively, and each aσ\nand bσ has an explicit interpretation (formula).\nThe coefficients aσ and bσ in Figures 2-4 for the r = 3 case do not\nseem to have any obvious elementary interpretation. Hence, in general, one\ncan only expect nonelementary interpretations for the coefficients aσ and\nbσ in (28)-(29). The following numerical analysis of these coefficients for\nthe r = 3 case suggests that BH\nr , in general, may plausibly have an explicit\npresentation, the coefficients aσ and bσ of whose generating relations have\nnonelementary interpretations in the spirit of the one for Kazhdan-Lusztig\n25"},{"paragraph_id":"p33","order":33,"text":"polynomials. By this we mean that each aσ has an explicit formula of the\nform of an alternating sum\naσ = (−1)d(σ)(q1/2 −q−1/2)d′(σ)(\ns(σ)\nX\nj=0\n(−1)jaj\nσ),\n(30)\nfor some nonnegative integers d(σ), d′(σ), s(σ), where\n1. s(σ) is small, say bounded by a polynomial of a fixed degree in r and\ndim(X) in the present case when H = GL2(C), and in r, the rank of\nH and the size of μ in the general plethysm problem (Problem 1.1),\n2. each aj\nσ is a −-invariant (note that aσ is −-invariant), positive and\nunimodal polynomial in q and q−1; positive means each coefficient of\naj\nσ is nonnegative, and unimodal means, if aj\nσ(−k), . . . , aj\nσ(k) are the\ncoefficients of aj\nσ, then\naj\nσ(−k) ≤aj\nσ(−k + 1) ≤· · · ≤aj\nσ(−1) ≤aj\nσ(0) ≤aj\nσ(1) ≤· · · aj\nσ(k),\n3. each aj\nσ(s) has a topological interpretation akin to that for Kazhdan-\nLusztig polynomials, i.e., as the rank of an appropriate cohomology\ngroup. Then the duality aj\nσ(−s) = aj\nσ(s) as per the −-invariance of aj\nσ\nshould come out as a consequence of some form of Poincare duality\nand the unimodality as a consequence of some form of Hard Lefschetz,\nand each bσ has a similar explicit formula of the form\nbσ = (−1)\n ̄d(σ)(q1/2 −q−1/2)\n ̄d′(σ)(\n ̄s(σ)\nX\nj=0\n(−1)jbj\nσ).\n(31)\nWe shall call such an interpretation for aσ or bσ, if it exists, a positive,\nunimodal, and topological interpretation.\nIdeally speaking, one would like each s(σ) and ̄s(σ) above to be zero, but\nthis may not always be possible for the reasons given below. It is plausible\nthat there exists some notion of cohomological depth that measures the\nextent of nonflatness, and which provides an upper bound on s(σ) and ̄s(σ) in\nsuch a topological interepretation, if it exists. For example, in the Kronecker\nproblem, the braided symmetric and exterior algebras CH\nq [X] and ∧H\nq [X] are\nflat deformations of the classical algebras C[X] and ∧[X]. In this case, one\ncan expect an explicit presentation for BH\nq whose coefficients aσ and bσ have\n26"},{"paragraph_id":"p34","order":34,"text":"positive topological interpretation with s(σ), ̄s(σ) = 0 in (30) and (31). This\nis because aσ and bσ here are akin to the structural constants cb′′\nb,b′, db,b′\nb′′\nin\nTheorem 4.4, which occur in the defining Laplace relations for GH\nq , and\nwhich, in the Kroncker problem, are conjecturally polynomials in q and q−1\nwith nonnegative coefficients for the reasons indicated there. But in general\nwhen CH\nq [X] and ∧H\nq [X] are nonflat deformations, such cohomological depth\nwould not vanish, and hence s(σ) and ̄s(σ) may be nonzero, but still small\nas indicated above.\nWe now turn to the analysis of the coefficients in the r = 3 case men-\ntioned above which suggests that such an interpretation may plausibly exist.\nFirst let us oberve that the scaling factor f in (17) used in the analysis so far\nis formally not the correct scaling factor. To get the latter, we have to look\nat the formal expressions for P and Q in terms of ˆR. Since the eigenvalues\nof ˆR in the present case are\nq1 = q9/2,\nq2 = −q−3/2,\nq3 = q−11/2,\nand\nq4 = −q−15/2,\nwe have\nP = ( ˆR −q2)( ˆR −q3)( ˆR −q4)\n(q1 −q2)(q1 −q3)(q1 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q4)\n(q3 −q1)(q3 −q2)(q3 −q4),\n(32)\nand\nQ = ( ˆR −q1)( ˆR −q3)( ˆR −q4)\n(q2 −q1)(q2 −q3)(q2 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q3)\n(q4 −q1)(q4 −q2)(q4 −q3).\n(33)\nHence, formally we should have defined the rescaled versions P and Q of P\nand Q by the equations\nP = fpP,\nfp = (q1 −q2)(q1 −q3)(q1 −q4)(q3 −q1)(q3 −q2)(q3 −q4), (34)\nand\nQ = fqQ,\nfq = (q2 −q1)(q2 −q3)(q2 −q4)(q4 −q1)(q4 −q3)(q4 −q2), (35)\ninstead of the equations (16) and (18). The scaling factor f in (17) was\nthe smallest factor chosen so that the matrix coefficients of P and Q after\nrescaling become polynomials in q, q−1. But this choice was dependendent on\nthe accidental cancellations in the numerators and denominators in (32) and\n27"},{"paragraph_id":"p35","order":35,"text":"(33). The choice of scaling makes no essential difference in Sections 7.1.1-\n7.1.6. But it does matter in the study of positivity below.\nHence, let us redefine P and Q as per (34) and (35). Let us denote the\ncoefficients of the old defining relations (19) and (20) among Qi’s and Pi’s\nby a′\nσ and b′\nσ, and the coefficients of the defining relations among the new\nQi’s and Pi’s by a′′\nσ and b′′\nσ. Then we have\na′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄aσ,\nwith\n ̄aσ = ( ˆfq)11−l(σ)a′\nσ,\nand\nb′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄bσ,\nwith\n ̄bσ = ( ˆfp)11−l(σ)b′\nσ,\nwhere l(σ) denotes the length of σ,\nˆfp\n=\n−q\n(q−1)2\nfp\nf\n=\n10 + 8 q + 2 q4 + 12 q−1 + 18 q−6 + 6 q2 + 4 q3 + q5\n+14 q−2 + 18 q−4 + 16 q−12 + 16 q−8 + q−27 + 16 q−11\n+16 q−10 + 17 q−7 + 6 q−24 + 18 q−5 + 2 q−26 + 16 q−3\n+16 q−9 + 10 q−22 + 14 q−20 + 4 q−25 + 8 q−23 + 12 q−21 + 18 q−18\n+16 q−19 + 18 q−16 + 18 q−17 + 17 q−15 + 16 q−14 + 16 q−13,\nand\nˆfq\n=\n−q\n(q−1)2\nfq\nf\n=\n10 q−12 + 2 q−8 + q−31 + 8 q−27 + 8 q−11 + 6 q−10 + q−7\n+2 q−30 + 10 q−24 + 6 q−28 + 10 q−26 + 4 q−9 + 6 q−22\n+2 q−20 + 10 q−25 + 8 q−23 + 4 q−29 + 4 q−21 + 2 q−18\n+2 q−19 + 6 q−16 + 4 q−17 + 8 q−15 + 10 q−14 + 10 q−13.\nBoth fp and fq are positive polynomials.\nLet us define\nˆaσ\n=\nˆf 2\nq a′\nσ,\nfor\nσ = 121212121,\n=\nˆfqa′\nσ,\notherwise,\n(36)\nand\nˆbσ\n=\nb′\nσ,\nfor\nσ = ∅, and 2,\n=\nˆfpb′\nσ,\notherwise.\n(37)\n28"},{"paragraph_id":"p36","order":36,"text":"Since fp and fq are positive, the positivity properties of ˆaσ and ̄aσ (also\nˆbσ and ̄bσ) are similar; it turns out that the unimodularity properties are\nalso similar. Hence we shall focus on ˆaσ and ˆbσ in what follows. Since ˆaσ\nis −-invariant, it is of the form ˆaσ(0) + P\nt>0 ˆaσ(t)(qt + q−t). Let ˆAσ be\nthe vector [aσ(0), aσ(1), . . .]; the vector ˆBσ is defined similarly. Figure 17\nshows ˆAσ for the various σ in Figure 2; the vector for each σ is obtained by\nconcatenating the rows in front of that σ. Figures 18-20 similarly show ˆBσ\nfor the various σ in Figures 3-4; only the distinct ˆBσ’s are shown. It may\nbe seen the ˆAσ’s are positive and nonincreasing. Thus all aσ are positive\nand unimodal, and hence, of the form (30) with s(σ) = 0. All ˆBσ’s are\npositive and nonincreasing, except for σ = 121, 1212 and 21212, for which\neach ˆBσ is positive and unimodal except at the tail. Thus all bσ, for σ ̸=\n121, 1212, 21212, are positive and unimodal, and hence of the form (31) with\n ̄s(σ) = 0. For σ = 121, 1212, 21212, bσ seems to be of the form (31) with\n ̄s(σ) = 1, both b0\nσ and b1\nσ being positive and unimodal, b0\nσ being the dominant\npolynomial that accounts for bσ’s mostly positive and unimodal behaviour,\nand b1\nσ the error polynomial that accounts for the deviation at the tail.\nThe (co)multiplicative structural constants cb′′\nb,b′ and db,b′\nb′′ for the canoni-\ncal basis of the braided exterior algebra ∧H,r\nq\n[X], which occur in the Laplace\nrelations for the general nonstandard quantum group GH\nq (cf. Theorem 4.5),\nare akin to the structure constants aσ and bσ in (28) and (29). Hence, we can\nexpect a similar positive topological interpretation for cb′′\nb,b′ and db,b′\nb′′ (but not\nnecessarily unimodality since cb′′\nb,b′ and db,b′\nb′′\nneed not be −-invariant). The\nexperimental evidence in [GCT8] suggests that the structure constants as-\nsociated with the canonical bases of the matrix coordinate ring of GH\nq and\nthe ring BH\nq defined there may also have similar positive topological inter-\npretations (additionally unimodal for BH\nq ).\n7.2\nExample 2\nNow we verify the duality and reciprocity conjectures for the special case of\nthe Kronecker problem (Section 4.1), when H = GL(V )×GL(W), V = W =\nC2 and G = GL(X), X = V ⊗W ∼= C4, and r = 4. Thus Gq = GLq(C4), and\nHq = GLq(C2) × GLq(C2). Let B = BH\nr be the nonstandard algebra in this\ncase and Pi = p+,H\nX,i , Qi = p−,X\nX,i , i < r, the positive and negative projection\noperators as in Section 5. Let Pi and Qi be the rescaled versions of Pi and\nQi as defined in [GCT4]. Then B is generated by Pi, or equivalently, Qi.\nThe explicit generating relations among Pi’s and Qi’s turn out to be very\n29"},{"paragraph_id":"p37","order":37,"text":"1, (q4+1)(q4−q2+1)(q2+1)\nq5\n2, q\n q4 + 1"},{"paragraph_id":"p38","order":38,"text":"5, q4+1\nq2\n3, (q2+1)(q8−q6+q4−q2+1)\nq5\n6,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, q2+1\nq\n4, q8−q6+q4+1\nq3\n7,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n10,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n13, q4+1\nq2\n2, q4+1\nq2\n5, q4+1\nq5\n3, (q2+1)\n2(q−1)(q+1)\nq2\n6, 2 q2+1\nq\n9, −(q2+1)\n2(q−1)(q+1)\nq2\n4,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n7, 2 q8+q6−2 q2+1\nq5\n10, −q8−q6−2 q4−q2+1\nq4\n13, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n8, (q2+1)(q8−q6+q4−q2+1)\nq5\n11, (q2+1)\n2(q−1)(q+1)\nq4\n14, q2+1\nq\n3, q2+1\nq\n6, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, (q2+1)(q8−q6+q4−q2+1)\nq5\n4,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n7, −q8−q6−2 q4−q2+1\nq4\n10, q8−2 q6+q2+2\nq3\n13,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n8,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n11, 2 q2+1\nq\n14, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n12, q\n q4 + 1"},{"paragraph_id":"p39","order":39,"text":"15, q4+1\nq2\n4, q4+1\nq2\n7, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n10,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n13, q8+q4−q2+1\nq5\n8, q2+1\nq\n11, −(q2+1)\n2(q−1)(q+1)\nq4\n14, (q2+1)(q8−q6+q4−q2+1)\nq5\n12, q4+1\nq2\n15, q4+1\nq5\n16, (q4+1)(q4−q2+1)(q2+1)\nq5"},{"paragraph_id":"p40","order":40,"text":"Figure 1: P-matrix\n30"},{"paragraph_id":"p41","order":41,"text":"σ\naσ\n1\n−(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2/q36\n121\n(1 + 7 q4 + q2 + 5 q6 + 18 q8 + 21 q42 −107 q20 + q50 −107 q32 + 73 q14 + 187 q18 −14 q16 + 402 q26\n−197 q28 + 20 q40 + 187 q34 + 73 q38 −197 q24 + 328 q30 + q52 + 328 q22 + 7 q48 + 5 q46 + 18 q44\n+20 q12 + 21 q10 −14 q36)(q2 + 1)2(q4 −q2 + 1)2/q32\n12121\n−(1 + 8 q4 + 3 q2 + 4 q6 + 33 q8 + 12 q42 + 80 q20 + 3 q50 + 80 q32 + 27 q14 + 113 q18 + 115 q16 + 360 q26\n−9 q28 + 76 q40 + 113 q34 + 27 q38 −9 q24 + 253 q30 + q52 + 253 q22 + 8 q48 + 4 q46 + 33 q44 + 76 q12\n+12 q10 + 115 q36)/q26\n1212121\n(3 q36 + 2 q34 + 8 q32 + 5 q30 + 17 q28 + 30 q26 + 11 q24\n+61 q22 −15 q20 + 108 q18 −15 q16 + 61 q14 + 11 q12 + 30 q10 + 17 q8 + 5 q6 + 8 q4 + 2 q2 + 3)/q18\n121212121\n−(q20 + 3 q18 + q16 + 5 q14 −2 q12 + 16 q10 −2 q8 + 5 q6 + q4 + 3 q2 + 1)/q10\n12121212121\n1\nFigure 2: Coefficients of the basic generating relation among Qi’s\n31"},{"paragraph_id":"p42","order":42,"text":"σ\nbσ\n∅\n−(q2 + 1)5(q4 + 1)3(q4 −q2 + 1)6(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)\n×(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)\n×(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q51\n2\n(q2 + 1)4(q4 −q2 + 1)5(q4 + 1)2(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n×(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n×(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\n1\n(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q46\n12\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n21\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n212\n(q2 + 1)2(q4 −q2 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q36\n121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(2 + 2 q4 −4 q2 −2 q6 −2 q8 −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 −3 q12 −9 q10 + q36)/q36\n1212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\nFigure 3: The first eight terms of the basic generating relation among Pi’s\n32"},{"paragraph_id":"p43","order":43,"text":"σ\nbσ\n2121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\n21212\n(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26\n+53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44 −3 q12\n−9 q10 + q36)/q26\n12121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q26\n121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n2121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n1212121\n−(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q18\n12121212\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n21212121\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n212121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n121212121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2/q10\n1212121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n2121212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n21212121212\n1\nFigure 4: The last fourteen terms of the basic generating relation among Pi’s\n33"},{"paragraph_id":"p44","order":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)(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 5: First nine coefficients of u0\n34"},{"paragraph_id":"p45","order":45,"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 6: Last twelve coefficients of u0\n35"},{"paragraph_id":"p46","order":46,"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 7: Coefficients of u1\n1\n36"},{"paragraph_id":"p47","order":47,"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 8: Coefficients of u1\n3\n37"},{"paragraph_id":"p48","order":48,"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 9: Coefficients of u1\n4\n38"},{"paragraph_id":"p49","order":49,"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 10: Coefficients of u1\n5\n39"},{"paragraph_id":"p50","order":50,"text":"g1\n−1/2 −3 q28−4 q24+2 q22−10 q20+2 q18−4 q16−3 q12+x\nq20\ng3\n(q4+1)\n2\nq4\ng4\n(q2+1)\n2(q4−q2+1)\n2\nq6\ng5\n(q2+1)\n2(q8−q6+q4−q2+1)\n2\nq10\nFigure 11: The elements gi\ncomplicated. For example, Figures 21-23 reproduced from [GCT4] shows\na typical generating relation among Qi’s with 74 terms. There are several\ndozen such relations. Because of the nature of these generating relations,\nthere is no good “standard monomial basis” for B as for the Hecke algebra or\nfor the r = 3 case in Section 7.1.1. Fortunately, this makes no difference as\nfar as duality and reciprocity is concerned, as we shall see here, and also as\nfar as existence of a canonical basis is concerned, as we shall see in [GCT8].\nIt was verified by computer that B is of dimension 114 [GCT4]. Since it\nis semisimple, it admits a Wederburn structure decomposition. It turns out\nthat a complete Wederburn structure decomposition of the form (13) works\nover Q(q) itself; i.e., no algebraic extension of Q(q) is necessary here, just\nas in the case of Hecke algebras. This may be conjectured to be the case\nfor the Kronecker problem in general, though it is not so for the plethysm\nproblem in general as we already saw in Section 7.1.\nSo let\nB = ⊗iTi,L ⊗Ti,R,\n(38)\nbe the complete Wederburn structure decomposition of B, where Ti = Ti,L\nranges over all irreducible left B-modules.\n7.2.1\nIrreducible representations\nWe describe these Ti next. There are two distinct irreducible representations\nof B of dimension 1, 2, 3 and 5 each, and one of dimension 6. Since\n114 = 12 + 12 + 22 + 22 + 32 + 32 + 52 + 52 + 62,\nthis is consistent with the Wederburn structure decomposition in (38).\n40"},{"paragraph_id":"p51","order":51,"text":"σ\nCoefficient\n1\n1/2 (q2 + 1)5(q4 −q2 + 1)3(q8 −q6 + q4 −q2 + 1)2(q4 + 1)3(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q45\n12\n−1/2 (q2 + 1)4(q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q40\n121\n−1/2 (q2 + 1)3(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)(q4 + 1)(q4 −q2 + 1)/q41\n1212\n1/2 (q2 + 1)2(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)/q36\n12121\n1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)\n×(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q35\n121212\n−1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)/q30\n1212121\n−1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)(q2 + 1)\n×(q4 + 1)(q4 −q2 + 1)/q25\n12121212\n1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)/q20\n121212121\n−(q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n1212121212\n1\nFigure 12: Nonzero coefficients of z0\n41"},{"paragraph_id":"p52","order":52,"text":"Monomial\nCoefficient\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24 + 11 q48\n−45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8 + 8 xq16\n−7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q51\nx1 ⊗x2 ⊗x0\n1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8\n+8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q46\nx0 ⊗x3 ⊗x0\n−1/2 (q + 1)(q −1)(q4 + 1)(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(3 x + 727 q32 −500 q38\n−330 q22 + 191 q20 + 460 q24 + 59 q48 −359 q42 + 192 q44 −110 q46 −603 q26 −138 q18 + 76 q16 −35 q14\n−20 xq22 + 5 xq24 + 70 xq12 −58 xq10 −77 xq14 + 45 xq8 + 48 xq16 −46 xq6 + 23 xq20 + 30 xq4 −15 xq2\n−32 xq18 + 587 q28 + 402 q40 −780 q34 + 584 q36 −44 q50 + 11 q52 −685 q30 + 7 q12)/q53\nx2 ⊗x0 ⊗x1\n1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20\n+117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34\n+117 q36 −310 q30 + 11 q12)/q48\nx1 ⊗x1 ⊗x1\n−1/2 (q2 + 1)2(q4 + 1)(q −1)(q + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22\n+91 q20 + 117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q47\nx0 ⊗x2 ⊗x1\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(−3 x −1316 q32 + 1364 q38 + 419 q22 −200 q20 −577 q24\n−464 q48 + 957 q42 −617 q44 + 613 q46 + 748 q26 + 5 q30x + 134 q18 −76 q16 + 35 q14 −25 xq28 + 35 xq26\n+67 xq22 −28 xq24 −117 xq12 + 97 xq10 + 110 xq14 −48 xq8 −85 xq16 + 44 xq6 −112 xq20 −30 xq4 + 15 xq2\n+123 xq18 −816 q28 −1224 q40 + 1325 q34 −1132 q36 + 257 q50 + 85 q54 −55 q56 + 11 q58 −108 q52\n+1220 q30 −7 q12)/q50\nFigure 13: First five nonzero coefficients of a ∈X⊗3\nq\n42"},{"paragraph_id":"p53","order":53,"text":"Monomial\nCoefficient\nx0 ⊗x0 ⊗x2\n−1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14\n+19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q46\nx0 ⊗x1 ⊗x2\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(3 x + 951 q32 −1060 q38 −363 q22 + 176 q20 + 449 q24 + 248 q48\n−592 q42 + 395 q44 −451 q46 −532 q26 −97 q18 + 65 q16 −35 q14 + 8 xq28 −27 xq26 −31 xq22 + 16 xq24\n+81 xq12 −85 xq10 −62 xq14 + 40 xq8 + 59 xq16 −27 xq6 + 64 xq20 + 25 xq4 −15 xq2 −97 xq18 + 654 q28\n+797 q40 −898 q34 + 828 q36 −129 q50 −61 q54 + 18 q56 + 52 q52 −998 q30 + 7 q12)/q49\nx0 ⊗x0 ⊗x3\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q50\nFigure 14: Last four nonzero coefficients of a\n43"},{"paragraph_id":"p54","order":54,"text":"Monomial\nCoefficient\nx3 ⊗x0 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(5 x + 1214 q32 −847 q38 −525 q22\n+289 q20 + 714 q24 + 107 q48 −525 q42 + 289 q44 −178 q46 −847 q26 −178 q18 + 107 q16 −55 q14 −25 xq22\n+5 xq24 + 130 xq12 −113 xq10 −113 xq14 + 75 xq8 + 75 xq16 −60 xq6 + 45 xq20 + 45 xq4 −25 xq2 −60 xq18\n+920 q28 + 714 q40 −1139 q34 + 920 q36 −55 q50 + 11 q52 −1139 q30 + 11 q12)/q57\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−5 x −1170 q32 + 977 q38 + 425 q22\n−207 q20 −565 q24 −371 q48 + 636 q42 −451 q44 + 397 q46 + 523 q26 + 5 q30x + 106 q18 −89 q16 + 55 q14\n−20 xq28 + 20 xq26 + 43 xq22 −20 xq24 −119 xq12 + 97 xq10 + 71 xq14 −45 xq8 −59 xq16 + 28 xq6 −95 xq20\n−37 xq4 + 25 xq2 + 87 xq18 −676 q28 −1021 q40 + 891 q34 −849 q36 + 173 q50 + 52 q54 −44 q56 + 11 q58\n−82 q52 + 1002 q30 −11 q12)/q56\nx1 ⊗x2 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−8 x −1447 q32 + 1598 q38 + 590 q22\n−304 q20 −648 q24 −496 q48 + 927 q42 −742 q44 + 696 q46 + 778 q26 + 5 q30x + 153 q18 −89 q16 + 72 q14\n−25 xq28 + 40 xq26 + 58 xq22 −40 xq24 −124 xq12 + 138 xq10 + 96 xq14 −76 xq8 −114 xq16 + 39 xq6 −116 xq20\n−33 xq4 + 32 xq2 + 152 xq18 −1064 q28 −1329 q40 + 1319 q34 −1386 q36 + 244 q50 + 96 q54 −55 q56\n+11 q58 −134 q52 + 1516 q30 −18 q12)/q55\nx0 ⊗x3 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(−3 x −1693 q32 + 1146 q38 + 641 q22 −404 q20\n−1020 q24 −178 q48 + 814 q42 −567 q44 + 296 q46 + 1268 q26 + 266 q18 −155 q16 + 53 q14 + 5 xq26 + 45 xq22\n−25 xq24 −164 xq12 + 123 xq10 + 170 xq14 −108 xq8 −131 xq16 + 96 xq6 −60 xq20 −65 xq4 + 23 xq2\n+78 xq18 −1376 q28 −1006 q40 + 1709 q34 −1491 q36 + 107 q50 + 11 q54 −55 q52 + 1449 q30 −7 q12)/q58\nx2 ⊗x0 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(5 x + 964 q32 −627 q38\n−431 q22 + 224 q20 + 582 q24 + 100 q48 −431 q42 + 224 q44 −143 q46 −627 q26 −143 q18 + 100 q16 −55 q14 −25 xq22\n+5 xq24 + 108 xq12 −87 xq10 −87 xq14 + 50 xq8 + 50 xq16 −45 xq6 + 42 xq20 + 42 xq4 −25 xq2 −45 xq18 + 645 q28\n+582 q40 −860 q34 + 645 q36 −55 q50 + 11 q52 −860 q30 + 11 q12)/q53\nFigure 15: First five nonzero coefficients of b\n44"},{"paragraph_id":"p55","order":55,"text":"Monomial\nCoefficient\nx1 ⊗x1 ⊗x1\n1/2 (q2 + 1)2(q2 −q + 1)(q2 + q + 1)(q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(3 x + 277 q32 −621 q38 −165 q22\n+97 q20 + 83 q24 + 125 q48 −291 q42 + 291 q44 −299 q46 −255 q26 −47 q18 −17 q14 + 5 xq28 −20 xq26 −15 xq22\n+20 xq24 + 5 xq12 −41 xq10 −25 xq14 + 31 xq8 + 55 xq16 −11 xq6 + 21 xq20 −4 xq4 −7 xq2 −65 xq18 + 388 q28\n+308 q40 −428 q34 + 537 q36 −71 q50 −44 q54 + 11 q56 + 52 q52 −514 q30 + 7 q12)/q52\nx0 ⊗x2 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(−3 x −1651 q32 + 1803 q38\n+486 q22 −287 q20 −679 q24 −567 q48 + 1181 q42 −1013 q44 + 814 q46 + 920 q26 + 5 q30x + 147 q18 −72 q16\n+35 q14 −25 xq28 + 45 xq26 + 78 xq22 −60 xq24 −133 xq12 + 122 xq10 + 138 xq14 −87 xq8 −167 xq16\n+49 xq6 −131 xq20 −28 xq4 + 15 xq2 + 170 xq18 −1270 q28 −1556 q40 + 1669 q34 −1825 q36 + 296 q50\n+107 q54 −55 q56 + 11 q58 −178 q52 + 1547 q30 −7 q12)/q55\nx1 ⊗x0 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10\n−15 xq14 + 22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q51\nx0 ⊗x1 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14 + 22 xq8\n+3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q54\nx0 ⊗x0 ⊗x3\n1/2 (q2 + 1)(q4 + 1)2(q4 −q2 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32\n−94 q38 −220 q22 + 132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q55\nFigure 16: Last five nonzero coefficients of b\n45"},{"paragraph_id":"p56","order":56,"text":"Let Sq,λ denote the q-Specht module of the Hecke algebra Hr(q) for\nthe partition λ, and KLλ its Kazhdan-Lusztig basis ordered appropriately.\nSince, in this case, B = BH\nr (q) ⊆Hr(q)⊗Hr(q), the tensor product Sq,λ⊗Sq,μ\nis a representation of B. In particular,\nTq,λ = Sq,λ ⊗Sq,(r) ∼= Sq,(r) ⊗Sq,λ,\nwhere Sq,(r) is the trivial one dimensional q-Specht module, is an irreducible\nB-module, which specializes at q = 1 to the Specht module Sλ of the sym-\nmetric group Sr.\nLet\nT0\n=\nTq,(4),\nT1\n=\nTq,(1,1,1,1),\nT2\n=\nTq,(2,2),\nT3\n=\nTq,(2,1,1),\nT4\n=\nTq,(3,1),\nT5\n=\nSq,(3,1) ⊗Sq,(2,2) ∼= Sq,(2,1,1) ⊗Sq,(2,2).\n(39)\nThese are irreducible B-modules. Their dimensions are 1, 1, 2, 3, 3 and 6\nrespectively.\nTo get the other two dimensional irreducible B-module, we analyze how\nthe tensor product Sq,(2,2) ⊗Sq,(2,2) decomposes as a B-module. It decom-\nposes as:\nSq,(2,2) ⊗Sq,(2,2) ∼= Tq,(4) ⊕Tq,(1,1,1,1) ⊕T6,\nwhere T6 is the other two dimensional irreducible B-module that we were\nlooking for. Explicitly, a basis of T6 in terms of the Kazhdan-Lusztig basis\nKL(2,2) ⊗KL(2,2) of Sq,(2,2) ⊗Sq,(2,2) is given by the rows of the matrix\n\"\n1\n1+q\n2q1/2\n1+q\n2q1/2\n0\n0\n1+q\n2q1/2\n1+q\n2q1/2\n1\n#\n.\nMatrix representations of the right action of the generators Qi’s of B on\nthis basis are:\nQ1 = Q3 =\n (1 + q)2/q\n0\n(1 + q2)/q\n0"},{"paragraph_id":"p57","order":57,"text":"Q2 =\n 0\n(1 + q2)/q\n0\n(1 + q)2/q"},{"paragraph_id":"p58","order":58,"text":"46"},{"paragraph_id":"p59","order":59,"text":"The specialization of T6 at q = 1 is isomorphic to the Specht module\nS(2,2) of S4. But T6 is nonisomorphic to T2, whose specialization at q = 1 is\nthe same.\nTo get the five dimensional irreducible B-modules, we analyze how the\ntensor products Sq,(2,1,1) ⊗Sq,(2,1,1) and Sq,(3,1) ⊗Sq,(2,1,1) decompose as B-\nmodules. We have\nSq,(2,1,1) ⊗Sq,(2,1,1) ∼= Tq,(2,1,1) ⊕Tq,(4) ⊕T7,\nwhere T7 is the first five dimensional irreducible B-representation that we\nwere looking for. Explicitly, its basis in terms of the Kazhdan-Lusztig basis\nKL(2,1,1) ⊗KL(2,1,1) is given by the rows of the matrix:\nw1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n0\n0\n0]\nw2 = [\n−1\n−(1 + q)/(2q1/2)\n0\n0\n(1 + q)/(2q1/2)\n1]\nw3 = [\n0\n0\n−(1 + q)2/(2q)\n0\n−(1 + q)/(2q1/2)\n0]\nv1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n−1\n−(1 + q)/(2q1/2)\n0]\nv2 = [\n1\n(1 + q)/(2q1/2)\n1\n0\n(1 + q)/(2q1/2)\n1]\nMatrix representations of the right action of Qi’s in this basis are:\nQ1 ="},{"paragraph_id":"p60","order":60,"text":"(1 + q)2/q\n0\n0\n0\n0\n(1 + q2)/q\n0\n0\n−(1 + q2)/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n(q −1)2/q\n0\n0\n−(1 + q2)/q\n0"},{"paragraph_id":"p61","order":61,"text":"Q2 ="},{"paragraph_id":"p62","order":62,"text":"0\n(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n(1+q)2\nq\n0\n0\n0\n0\n−(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n0\n0\n0\n−1+q2\nq\n0\n0\n0\n0\n(1+q)2\nq"},{"paragraph_id":"p63","order":63,"text":"Q3 ="},{"paragraph_id":"p64","order":64,"text":"0\n0\n0\n(1 + q)2/q\n0\n0\n0\n−(1 + q2)/q\n(1 + q2)/q\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n(q −1)2/q\n−(1 + q2)/q\n0"},{"paragraph_id":"p65","order":65,"text":"47"},{"paragraph_id":"p66","order":66,"text":"Let V denote the span of the vectors v1 and v2, and V (1) its specialization\nat q = 1. It can be checked that V (1) is isomorphic to the Specht module\nS(2,2) of S4, and the quotient T7(1)/V , where T7(1) denotes the specialization\nof T7 at q = 1, is isomorphic to the Specht module S(3,1) of S4.\nFinally,\nSq,(3,1) ⊗Sq,(2,1,1) ∼= Tq,(3,1) ⊕Tq,(1,1,1,1) ⊕T8,\nwhere T8 ̸∼= T7 is the second five dimnsional irreducible B-representation\nthat we were looking for. Its basis and representation matrices are similar.\nThis specifies all irreducible representations of B.\n7.2.2\nDuality\nUsing the explicit representations Ti above, the Wederburn structure decom-\nposition (38) of B was explicitly determined with the help of a computer.\nThe explicit bases of the structure components Ui = Ti,L ⊗Ti,R in (38) are\nfar too complex to be given here.\nFix any ui ∈Ui, 0 ≤i ≤8, and let Wi = X⊗r\nq\n· ui be the corre-\nsponding left representation of the nonstandard quantum group GH\nq . Com-\nputer experiments indicate that these are nonisomorphic irreducible repre-\nsentations of GH\nq with the following decompositions as Hq-modules, Hq =\nGLq(C2)×GLq(C2). (Recall that Vq,λ(n) is the q-Weyl module of GLq(Cn)).\nW0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nW1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW2\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2),\nW3\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2),\nW4\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2),\nW5\n∼=\nVq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nW6\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW7\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2).\nTheir dimensions are 35, 1, 10, 9, 30, 6, 1 and 9, respectively. The module\nW8 turns out to be zero when dim(V ) = dim(W) = 2, as here; however, it\nwould be nonzero for general dim(V ) and dim(W). Furthermore,\nX⊗4\nq\n=\nM\ni\nWi ⊗Ti,\n48"},{"paragraph_id":"p67","order":67,"text":"in accordance with the duality conjecture.\nRemark: These computations are not final. The main problem is that the\nsymbolic computations needed here are too heavy for MATLAB/Maple to\nhandle. Hence, in some of the computations q was set to a fixed real value\n(such as .5). This introduces floating point errors in various calculations.\nAs far as we can see, this does not affect the decomposition above. But this\nhas to be double checked by other means.\n7.2.3\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of S4 in Ti. Then\nit can be verified that\nm0\n(4) = 1,\nm1\n(1,1,1,1) = 1,\nm2\n(2,2) = 1,\nm3\n(2,1,1) = 1,\nm4\n(3,1) = 1,\nm5\n(3,1) = m5\n(2,1,1) = 1,\nm6\n(2,2) = 1,\nm7\n(3,1) = m7\n(2,2) = 1,\nm8\n(2,1,1) = m8\n(2,2) = 1.\nAll other mi\nμ’s are zero. It can now be seen that, as Hq-modules, Hq =\n49"},{"paragraph_id":"p68","order":68,"text":"GLq(2) × GLq(2), we have\nVq,(4)(4)\n∼=\nm0\n(4)W0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(3,1)(4)\n∼=\nm4\n(3,1)W4 ⊕m5\n(3,1)W5 ⊕m7\n(3,1)W7\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2),\nVq,(2,2)(4)\n∼=\nm2\n(2,2)W2 ⊕m7\n(2,2)W7 ⊕m6\n(2,2)W6\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(2,1,1)(4)\n∼=\nm3\n(2,1,1)W3 ⊕m5\n(2,1,1)W5\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nVq,(1,1,1,1)(4)\n∼=\nm1\n(1,1,1,1)W1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nin accordance with the reciprocity conjecture.\nWe are unable to verify the refined reciprocity conjecture on computer\nsince the necessary symbolic computations turn out to be beyond the reach\nof the desktop MATLAB/Maple.\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[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n50"},{"paragraph_id":"p69","order":69,"text":"σ\nˆAσ\n1\n17738\n17738\n17550\n17362\n16994\n16626\n16114\n15602\n14933\n14264\n13550\n12836\n12008\n11180\n10392\n9604\n8790\n7976\n7226\n6476\n5806\n5136\n4518\n3900\n3418\n2936\n2504\n2072\n1762\n1452\n1202\n952\n776\n600\n482\n364\n280\n196\n152\n108\n77\n46\n34\n22\n14\n6\n4\n2\n1\n121\n20322\n20322\n20083\n19844\n19354\n18864\n18211\n17558\n16668\n15778\n14890\n14002\n12934\n11866\n10916\n9966\n8962\n7958\n7092\n6226\n5470\n4714\n4047\n3380\n2895\n2410\n1982\n1554\n1287\n1020\n804\n588\n463\n338\n253\n168\n122\n76\n53\n30\n19\n8\n5\n2\n1\n12121\n9078\n9078\n8973\n8868\n8623\n8378\n8051\n7724\n7245\n6766\n6335\n5904\n5363\n4822\n4382\n3942\n3443\n2944\n2562\n2180\n1851\n1522\n1267\n1012\n831\n650\n506\n362\n284\n206\n151\n96\n70\n44\n28\n12\n7\n2\n1\n1212121\n1918\n1918\n1913\n1908\n1866\n1824\n1742\n1660\n1523\n1386\n1277\n1168\n1042\n916\n821\n726\n603\n480\n395\n310\n246\n182\n145\n108\n83\n58\n40\n22\n14\n6\n3\n121212121\n25032\n24784\n24124\n23136\n21978\n20808\n19710\n18768\n17934\n17160\n16358\n15440\n14384\n13168\n11849\n10484\n9139\n7880\n6725\n5692\n4765\n3928\n3170\n2480\n1868\n1344\n914\n584\n345\n188\n93\n40\n15\n4\n1\nFigure 17: Positivity and unimodality of ˆAσ’s\n51"},{"paragraph_id":"p70","order":70,"text":"σ\nˆBσ\n∅\n390464\n389128\n385120\n378581\n369652\n358471\n345176\n330055\n313396\n295506\n276692\n257207\n237304\n217316\n197576\n178283\n159636\n141795\n124920\n109152\n94632\n81349\n69292\n58469\n48888\n40497\n33244\n27011\n21680\n17198\n13512\n10505\n8060\n6095\n4528\n3314\n2408\n1731\n1204\n812\n540\n357\n232\n147\n84\n43\n24\n16\n8\n2\n2\n102390\n101996\n100847\n98976\n96425\n93236\n89466\n85172\n80462\n75444\n70190\n64772\n59280\n53804\n48429\n43240\n38271\n33556\n29145\n25088\n21393\n18068\n15099\n12472\n10185\n8236\n6586\n5196\n4040\n3092\n2333\n1744\n1286\n920\n640\n440\n300\n200\n129\n76\n41\n24\n16\n8\n2\n1\n50420\n50420\n49799\n49178\n48066\n46954\n45325\n43696\n41665\n39634\n37420\n35206\n32782\n30358\n27969\n25580\n23303\n21026\n18902\n16778\n14947\n13116\n11553\n9990\n8713\n7436\n6455\n5474\n4724\n3974\n3416\n2858\n2490\n2122\n1831\n1540\n1350\n1160\n1031\n902\n779\n656\n582\n508\n441\n374\n313\n252\n213\n174\n144\n114\n86\n58\n47\n36\n27\n18\n11\n4\n4\n4\n2\n12\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 18: The vectors ˆBσ\n52"},{"paragraph_id":"p71","order":71,"text":"σ\nˆBσ\n212\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\n121\n51252\n51252\n50661\n50070\n48941\n47812\n46219\n44626\n42589\n40552\n38328\n36104\n33645\n31186\n28756\n26326\n23948\n21570\n19376\n17182\n15217\n13252\n11581\n9910\n8522\n7134\n6030\n4926\n4106\n3286\n2664\n2042\n1632\n1222\n941\n660\n497\n334\n233\n132\n89\n46\n25\n4\n−4\n−12\n−10\n−8\n−6\n−4\n−4\n−4\n−2\n1212\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\n21212\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\n12121\n20922\n20922\n20625\n20328\n19815\n19302\n18558\n17814\n16848\n15882\n14868\n13854\n12740\n11626\n10537\n9448\n8430\n7412\n6509\n5606\n4830\n4054\n3439\n2824\n2349\n1874\n1526\n1178\n945\n712\n550\n388\n301\n214\n155\n96\n70\n44\n30\n16\n11\n6\n3\nFigure 19: The vectors ˆBσ (cont.)\n53"},{"paragraph_id":"p72","order":72,"text":"σ\nˆBσ\n121212\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\n2121212\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\n1212121\n3800\n3800\n3735\n3670\n3573\n3476\n3326\n3176\n2974\n2772\n2567\n2362\n2138\n1914\n1692\n1470\n1277\n1084\n921\n758\n624\n490\n396\n302\n238\n174\n131\n88\n65\n42\n29\n16\n11\n6\n3\n12121212\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\n212121212\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\n1212121212\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 ˆBσ (cont)\n54"},{"paragraph_id":"p73","order":73,"text":"[DJ]\nR. Dipper and G. James, Representations of Hecke algebras of\ngeneral linear groups, Proc. London Math. Soc. (3). 52 (1986), 20-\n52.\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\nat: http://ramakrishnadas.cs.uchicago.edu. Revised version to be\navailable here.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups, revised version under\npreparation.\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, 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[Kass]\nC. Kassel, Quantum groups, Springer-Verlag, 1995.\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.\n55"},{"paragraph_id":"p74","order":74,"text":"[Li]\nP. Littelmann: A Littlewood-Richardson rule for symmetrizable\nKac-Moody Lie algebras, Invent. math. 116 (1994), 329-346.\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[Mc]\nI. Macdonald, Symmetric functions and Hall polynomials, Oxford\nScience Publications, 1995.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[Ro]\nO. Rossi-Doria, A Uq(sl(2))-representation with no quantum sym-\nmetric algebra, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.\nRend. Lincei (9), Mat. Appl. 10 10 (1999), no. 1, 5-9.\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.\n56"},{"paragraph_id":"p75","order":75,"text":"Number\nCoefficient\nσ\n1\n20.q0+104.q1+256.q2−113.q3−49.q4−113.q5+256.q6+104.q7+20.q8\n2.q3+12.q4+2.q5\n1\n2\n−16.q0−64.q1−128.q2−192.q3−224.q4−192.q5−128.q6−64.q7−16.q8\n2.q3+12.q4+2.q5\n2\n3\n−4.q0−16.q1−28.q2−32.q3−28.q4−16.q5−4.q6\n2.q3\n3\n4\n1.q0−4.q2+6.q4−4.q6+1.q8\n2.q3+2.q5\n12\n5\n−1.q0−18.q1−65.q2−128.q3−190.q4−220.q5−190.q6−128.q7−65.q8−18.q9−1.q10\n2.q4+12.q5+2.q6\n13\n6\n1.q0+5.q1+17.q2+36.q3+46.q4+46.q5+46.q6+36.q7+17.q8+5.q9+1.q10\n2.q4+2.q6\n21\n7\n7.q0+26.q1+75.q2+152.q3+174.q4+156.q5+174.q6+152.q7+75.q8+26.q9+7.q10\n2.q3+12.q4+4.q5+12.q6+2.q7\n23\n8\n−1.q0−8.q1−20.q2−24.q3−22.q4−24.q5−20.q6−8.q7−1.q8\n2.q3+2.q5\n32\n9\n−22.q0−92.q1−170.q2−200.q3−170.q4−92.q5−22.q6\n2.q2+12.q3+2.q4\n121\n10\n2.q0+2.q1+12.q2+14.q3+4.q4+14.q5+12.q6+2.q7+2.q8\n2.q3+2.q5\n132\n11\n−2.q0−12.q1−40.q2−52.q3−44.q4−52.q5−40.q6−12.q7−2.q8\n2.q3+12.q4+2.q5\n212\n12\n−1.q0−2.q1−12.q2−14.q3−6.q4−14.q5−12.q6−2.q7−1.q8\n2.q3+2.q5\n213\n13\n1.q0+22.q1+88.q2+170.q3+206.q4+170.q5+88.q6+22.q7+1.q8\n2.q3+12.q4+2.q5\n232\n14\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q2\n323\n15\n3.q0+6.q1+5.q2+4.q3+5.q4+6.q5+3.q6\n2.q2+2.q4\n1212\n16\n12.q0+32.q1+40.q2+32.q3+12.q4\n2.q1+12.q2+2.q3\n1213\n17\n−3.q0−2.q1−5.q2−12.q3−5.q4−2.q5−3.q6\n2.q2+2.q4\n1232\n18\n1.q0+4.q1+11.q2+16.q3+11.q4+4.q5+1.q6\n2.q3\n1321\n19\n8.q0+12.q1+24.q2+40.q3+24.q4+12.q5+8.q6\n2.q2+12.q3+2.q4\n1323\n20\n−6.q0−8.q1−4.q2−8.q3−6.q4\n2.q1+2.q3\n2121\n21\n−5.q0−4.q1−44.q2−60.q3−30.q4−60.q5−44.q6−4.q7−5.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n2123\n22\n−1.q0−5.q1−11.q2−14.q3−11.q4−5.q5−1.q6\n2.q3\n2321\n23\n−3.q0−6.q1−5.q2−4.q3−5.q4−6.q5−3.q6\n2.q2+2.q4\n2323\n24\n2.q0+4.q1+4.q2+4.q3+2.q4\n2.q2\n3212\n25\n−1.q0−4.q1−6.q2−4.q3−1.q4\n2.q2\n3213\n26\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q1+2.q3\n3232\n27\n16.q0+32.q1+16.q2\n2.q0+12.q1+2.q2\n12121\n28\n4.q0+8.q1+40.q2+8.q3+4.q4\n2.q1+12.q2+2.q3\n12123\n29\n−3.q0−8.q1−4.q2−8.q3+46.q4−8.q5−4.q6−8.q7−3.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n12132\n30\n−8.q0\n2.q0\n12321\nFigure 21: A relation in BH\n4 from GCT4\n57"},{"paragraph_id":"p76","order":76,"text":"Number\nCoefficient\nσ\n31\n−4.q0−8.q1−40.q2−8.q3−4.q4\n2.q1+12.q2+2.q3\n12323\n32\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n13212\n33\n−9.q0−6.q1−55.q2+12.q3−55.q4−6.q5−9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n13232\n34\n9.q0+6.q1+55.q2−12.q3+55.q4+6.q5+9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n21213\n35\n1.q0−1.q1+3.q2−6.q3+3.q4−1.q5+1.q6\n2.q2+2.q4\n21232\n36\n−1.q0+2.q2−1.q4\n2.q2\n21321\n37\n2.q0+3.q1+6.q2−3.q3−16.q4−3.q5+6.q6+3.q7+2.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n21323\n38\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n23213\n39\n−16.q0−32.q1−16.q2\n2.q0+12.q1+2.q2\n23232\n40\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n32121\n41\n8.q0\n2.q0\n32123\n42\n1.q0−2.q2+1.q4\n2.q2\n32132\n43\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n32321\n44\n−8.q0−16.q1−8.q2\n2.q0+12.q1+2.q2\n121213\n45\n−1.q0−14.q1−15.q2−4.q3−15.q4−14.q5−1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n121232\n46\n−2.q0−4.q1−2.q2\n2.q1\n121321\n47\n−2.q0+4.q2−2.q4\n2.q1+12.q2+2.q3\n123213\n48\n8.q0+16.q1+8.q2\n2.q0+12.q1+2.q2\n123232\n49\n−1.q0−2.q1−1.q2\n2.q1\n132121\n50\n2.q0−4.q2+2.q4\n2.q1+12.q2+2.q3\n132123\n51\n2.q0+8.q1+12.q2+8.q3+2.q4\n2.q1+12.q2+2.q3\n212132\n52\n2.q0+4.q1+2.q2\n2.q1\n212321\n53\n1.q0+14.q1+15.q2+4.q3+15.q4+14.q5+1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n212323\n54\n3.q0+8.q1+10.q2+8.q3+3.q4\n2.q1+12.q2+2.q3\n213212\n55\n−2.q0−8.q1−12.q2−8.q3−2.q4\n2.q1+12.q2+2.q3\n213232\n56\n−1.q0−2.q1−1.q2\n2.q1\n232121\n57\n−3.q0−8.q1−10.q2−8.q3−3.q4\n2.q1+12.q2+2.q3\n232132\n58\n1.q0+2.q1+1.q2\n2.q1\n232321\n59\n−2.q0−4.q1−2.q2\n2.q1\n321232\n60\n2.q0+4.q1+2.q2\n2.q1\n321323\nFigure 22: A relation in BH\n4 from GCT4 continued.\n58"},{"paragraph_id":"p77","order":77,"text":"Number\nCoefficient\nσ\n61\n1.q0+2.q1+1.q2\n2.q1\n323212\n62\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1212132\n63\n2.q0\n2.q0\n1213213\n64\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1213232\n65\n2.q0\n2.q0\n1232121\n66\n2.q0−4.q1+2.q2\n2.q0+12.q1+2.q2\n1232132\n67\n16.q1\n2.q0+12.q1+2.q2\n1321232\n68\n−2.q0\n2.q0\n1321323\n69\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2121323\n70\n−4.q0−8.q1−4.q2\n2.q0+12.q1+2.q2\n2123212\n71\n−16.q1\n2.q0+12.q1+2.q2\n2123213\n72\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2123232\n73\n−2.q0+4.q1−2.q2\n2.q0+12.q1+2.q2\n2132123\n74\n4.q0+8.q1+4.q2\n2.q0+12.q1+2.q2\n2321232\nFigure 23: A relation in BH\n4 from GCT4 continued.\n59"}],"pages":[{"page":1,"text":"arXiv:0709.0749v2 [cs.CC] 1 Sep 2008\nGeometric Complexity Theory VII: Nonstandard\nquantum group for the plethysm problem\n(extended abstract)\nDedicated to Sri Ramakrishna\nKetan D. Mulmuley ∗\nThe University of Chicago\n(Technical Report TR-2007-14\nComputer Science Department\nThe University of Chicago\nSeptember 2007)\nRevised version\nhttp://ramakrishnadas.cs.uchicago.edu\nNovember 3, 2018\nAbstract\nThis article describes a nonstandard quantum group that may be\nused to derive a positive formula for the plethysm problem, just as the\nstandard (Drinfeld-Jimbo) quantum group can be used to derive the\npositive Littlewood-Richardson rule for arbitrary complex semisimple\nLie groups. The sequel [GCT8] gives conjecturally correct algorithms\nto construct canonical bases of the coordinate rings of these nonstan-\ndard quantum groups and canonical bases of the dually paired non-\nstandard deformations of the symmetric group algebra.\nA positive\n#P-formula for the plethysm constant follows from the conjectural\nproperties of these canonical bases and the duality and reciprocity\nconjectures herein.\n∗Part of this work was done while the author was visiting I.I.T. Mumbai\n1"},{"page":2,"text":"1\nIntroduction\nThe following is a fundamental problem in representation theory [GCT6,\nMc, St]:\nProblem 1.1 (Plethysm problem)\nFind an explicit positive (#P-) formula in the spirit of the Littlewood-\nRichardson rule for the plethysm constant aπ\nλ,μ. For given partitions λ, μ\nand π, this is the multiplicity of the irreducible representation Vπ(H) of\nH = GLn(C) in the irreducible representation Vλ(G) of G = GL(X), where\nX = Vμ = Vμ(H) is an irreducible representation of H.\nHere Vλ(G) is\nconsidered an H-module via the representation map ρ : H →G.\n(Generalized plethysm problem):\nThe same as above, letting H be any complex, semisimple (or, more\ngenerally, reductive) classical Lie group, λ a dominant weight of G, π and\nμ dominant weights of H.\nThis article describes a quantum group that may be used to derive such\na positive formula, just as the standard (Drinfeld-Jimbo) quantum group\n[Dri, Ji, RTF] can be used to derive the positive Littlewood-Richardson rule\nfor arbitrary complex semisimple Lie groups [Kas1, Li, Lu2]; the results here\nwere announced in [GCT4] (most of the results here also hold for nonclassical\nH, though we shall only worry about classical H here). For the significance\nof a positive formula in the context of geometric complexity theory, see\n[GCTflip1]. The approach that we wish to follow is:\n1. Find a quantization of the homomorphism\nH →G\n(1)\nof the form\nHq →GH\nq ,\n(2)\nwhere Hq is the standard Drinfeld-Jimbo quantization of H, and GH\nq\nis the new nonstandard quantization of G that we seek.\n2. Develop a theory of canonical (local/global crystal) bases for the rep-\nresentations of GH\nq in the spirit of the canonical bases [Kas1, Lu1] for\nthe representations of the standard quantum group.\n2"},{"page":3,"text":"3. Derive the required explicit positive formula for the plethysm constant\nfrom the properties of the canonical bases.\nThe following addresses the first step.\nTheorem 1.2 (cf.\nSection 2) There exists a possibly singular quantum\ngroup GH\nq\nsuch that the homomorphism (1) can be quantized in the form\n(2).\nFurthermore, all finite dimensional polynomial representations of GH\nq\nare completely reducible, and a quantum analogue of the Peter-Weyl theorem\nholds for the matrix coordinate ring of GH\nq .\nFor the precise meaning of the various terms here, see Section 2. Here and\nin what follows, we assume that the base field is C = C(q), q complex. But\na suitable algebraic extension of Q(q) will also suffice for our purposes; see\nSection 6 for a discussion on the base field.\nWhen H = G, GH\nq specializes to the standard quantum group Hq. When\nH = GL(V ) × GL(W), G = GL(X), X = V ⊗W with natural H-action, it\nreduces to the quantum group in [GCT4] for the Kronecker problem.\nWe call GH\nq\nthe nonstandard quantum group associated with the em-\nbedding (1). It can be singular in general. That is, its determinant may\nvanish, and hence, the antipode need not exist. Strictly speaking, it should\nhence be called a nonstandard quantum semi-group. We still use the term\ngroup, because this object has characteristic features of the standard quan-\ntum group, such as semisimplicity of polynomial representations, Peter-Weyl\ntheorem, and most importantly, conjectural existence of canonical bases for\nits representations and the matrix coordinate ring.\nWe also construct (Section 5) a nonstandard quantization BH\nr = BH\nr (q) of\nthe group algebra C[Sr] of the symmetric group Sr whose relationship with\nGH\nq\nis conjecturally akin to that of the Hecke algebra with the standard\nquantum group. Specifically, let Xq denote the irreducible representation\nVq,μ of Hq with highest weight μ; it is the usual quantization of X = Vμ.\nThen:\nConjecture 1.3 (Nonstandard duality)\n(1) The left action of GH\nq on X⊗r\nq\nand the right action BH\nr (q) on X⊗r\nq\nde-\ntermine each other.\n(2) There is a one-to-one correspondence between the irreducible polynomial\nrepresentations of GH\nq of degree r and the irreducible representations of BH\nr\n3"},{"page":4,"text":"so that, as a bimodule,\nX⊗r\nq\n=\nM\nα\nWq,α ⊗Tq,α,\n(3)\nwhere Wq,α runs over the irreducible polynomial representations of GH\nq\nof\ndegree r, and Tq,α denotes the irreducible representation of BH\nr (q) in corre-\nspondence with Wq,α.\nThe irreducible representations Wq,α here need not be q-deformations of\nthe irreducible representations of G, because GH\nq\nis, in general, a nonflat\ndeformation of G. This means the Poincare series of GH\nq need not coincide\nwith that of G. Our first goal is to associate with each Weyl module Vλ of\nG a possibly reducible representation V H\nq,λ of GH\nq , called the q-analogue of\nVλ, so that\nlimq→1V H\nq,λ ∼= Vλ\nas an H-module. In this context:\nConjecture 1.4 (Nonstandard reciprocity) Let λ be a partition of size\nr. Let\nV H\nq,λ =\nM\nα\nmα\nλWq,α,\nwhere mα\nλ denotes the multiplicity of the Specht module Sλ of the symmetric\ngroup Sr in Tq,α(1) = limq→1Tq,α, as defined in Section 6. Then V H\nq,λ is a\nq-analogue of Vλ in the sense defined above.\nThus the multiplicity of the GH\nq -module Wq,α in V H\nq,λ is equal to the mul-\ntiplicity of the Specht module Sλ in the specialization of Tq,α at q = 1.\nA more refined form of this conjecture is given in Section 6. Both duality\nand reciprocity are supported by experimental evidence; cf. Section 7.\nBy the conjectural reciprocity,\naπ\nλ,μ =\nX\nα\nmα\nλnα\nπ,\nwhere nα\nπ is the multiplicity of the irreducible Hq-module Vq,π in Wq,α. Hence\nProblem 1.1 can be decomposed into the following two subproblems:\n(P1): Find a positive (#P-) formula for the multiplicity nα\nπ.\n(P2): Find a positive (#P-) formula for the multiplicity mα\nλ.\n4"},{"page":5,"text":"The article [GCT8] gives conjecturally correct algorithms to construct\na canonical basis of the matrix coordinate ring of GH\nq\nwhose conjectural\nproperties would imply a positive formula as needed in the first problem,\nand a canonical basis of BH\nr\nwhose conjectural properties would imply a\npositive formula as needed in the second problem.\nAt present, we cannot prove correctness of these algorithms nor the re-\nquired conjectural properties, because we are unable to deal with the high\ncomplexity of the nonstandard quantum group. Specifically, as we shall see\nin Section 4, the formulae for the minors of the nonstandard group turn out\nto be highly nonelementary in contrast to the elementary formulae for the\nminors of the standard quantum group. The coefficients of these formulae\ndepend on the multiplicative structural constants of canonical bases akin to\nthe canonical basis of the coordinate ring of the standard quantum group\nconstructed by Kashiwara and Lusztig [Kas2, Lu2]. To get explicit formulae\nfor these structural constants, one needs interpretations for them akin to\nthe interpretations for the Kazhdan-Lusztig polynomials and multiplicative\nstructural constants of the canonical basis of the coordinate ring of the stan-\ndard quantum group in terms of perverse sheaves [KL2, Lu1, BBD]. Thus,\nthe linear algebra for the nonstandard quantum group–i.e. the theory of its\nminors–is already highly nonelementary in contrast to the linear algebra for\nthe standard quantum group. This is why its representation theory may turn\nout to be far more complex. In particular, we cannot explicitly construct\nnor classify its irreducible polynomial representations. Of course, all this\nand much more would follow if correctness of the algorithms in [GCT8] for\nconstructing canonical bases and their conjectural properties can be proved.\nAcknowledgement: The author is grateful to David Kazhdan for helpful\ndiscussions and comments, and to Milind Sohoni for helpful discussions,\nespecially for bringing the reference [Ro] to our attention, and for the help\nin explicit computations in Section 7.2 in MATLAB.\n2\nNonstandard quantum group\nWe describe in this section the construction of the nonstandard quantum\ngroup GH\nq\nin Theorem 1.2. The reader may refer to [GCT4] for the full\ndetails in a nontrivial special case of the plethysm problem, called the Kro-\nnecker problem. For the sake of simplicity, we assume here that H = GL(V )\n(type A). Let X = Vμ(H) be its irreducible polynomial representation. The\n5"},{"page":6,"text":"goal is to quantize the homomorphism\nH = GL(V ) →G = GL(X).\nLet H and G be the Lie algebras of H and G. We follow the terminology in\n[Kli], which will be our standard reference on quantum groups.\nThe standard quantum group Hq = GLq(V ) associated with GL(V ) can\nbe defined by first constructing the coordinate algebra O(Mq(V )) of the stan-\ndard quantum matrix space Mq(V ) as a suitable FRT-algebra [RTF]. The\ncoordinate ring O(GLq(V )) of GLq(V ) is obtained by localizing O(Mq(V ))\nat the suitably defined quantum determinant. The Drinfeld-Jimbo univer-\nsal enveloping algebra Uq(G) [Dri, Ji] of GLq(V ) can then be defined dually.\nSpecifically, let J be the maximal ideal of the elements in O(Mq(V )) which\nvanish at the identity–i.e. on which ǫ, the counit, vanishes. Then Uq(G) can\nbe identified with the space of linear functions on O(Mq(V )) which vanish\non Jr for some integer r > 0 depending on the linear function.\nAnalogously, we first construct the nonstandard matrix coordinate ring\nO(MH\nq (X)) of the (virtual) nonstandard matrix space MH\nq (X), and then\ndefine the nonstandard quantized universal enveloping algebra U H\nq (G) by\ndualization. We define the nonstandard quantum group GH\nq as the virtual\nobject whose universal enveloping algebra is U H\nq (G). The construction would\nyield natural bialgebra homomorphisms from Uq(H) to U H\nq (G) and from\nO(MH\nq (X)) to O(Mq(V ), thereby giving the desired quantizations of the\nhomomorphisms U(H) →U(G) and O(M(X)) →O(M(V )). This is what\nis meant by the quantization (2) of the map (1). The determinant of GH\nq\nmay vanish, and hence, we cannot, in general, define its coordinate ring\nO(GH\nq ) by localizing O(MH\nq (X)). Fortunately, this will not matter since the\ncoordinate ring O(MH\nq (X)) and the nonstandard quantized algebra BH\nr (q)\n(Section 5) together contain conjecturally all the information that we need\n(cf. Conjecture 1.4), and have properties similar to that of the standard\nmatrix coordinate ring O(Mq(V )) and the Hecke algebra; cf. Theorem 2.1\nbelow.\nThe nonstandard matrix coordinate ring O(MH\nq (X)) is constructed as\nfollows.\nLet ˆRH\nX,X be the ˆR matrix of Xq = Vq,μ considered as an Hq-\nmodule [Kli]. Here and in what follows, we sometimes denote Xq by X; the\nmeaning should be clear from the context. It is well known that ˆRH\nX,X is\ndiagonalizable and that its each eigenvalue is of the form + or −qa/2 for\nsome integer a [Kli]. Let\nI = P +,H\nX,X + P −,H\nX,X ,\n(4)\n6"},{"page":7,"text":"be the associated spectral decomposition of the identity, where P +,H\nX,X and\nP −,H\nX,X denote the projections of Xq ⊗Xq on the eigenspaces of ˆRH\nX,X for\neigenvalues with + and −sign, respectively. Let u be a variable matrix\nspecifying a generic transformation from X to X. Let ui\nj denote its variable\nentries. Then O(MH\nq (X)) is defined to be the FRT bialgebra [RTF] associ-\nated with the transformation P +,H\nX,X , or equivalently, P −,H\nX,X . That is, it is the\nquotient of C⟨ui\nj⟩modulo the relations\nP +,H\nX,X (u ⊗u) = (u ⊗u)P +,H\nX,X ,\n(5)\nor equivalently,\nP −,H\nX,X (u ⊗u) = (u ⊗u)P −,H\nX,X .\n(6)\nAn alternative definition of O(MH\nq (X)) is as follows. Let SH\nq (X ⊗X),\nthe symmetric subspace of X ⊗X, be the image of P +,H\nX,X , and AH\nq (X ⊗X),\nthe antisymmetric subspace of X ⊗X, the image of P −,H\nX,X [Kli]. In other\nwords, SH\nq (X ⊗X) is defined by the equation\nP −,H\nX,X x1x2 = 0,\n(7)\nwhere x1 = x ⊗I and x2 = I ⊗x, and AH\nq (X⊗X) is defined by the equation\nP +,H\nX,X x1x2 = 0.\n(8)\nThe braided symmetric algebra [BZ, Ro] CH\nq [X] of X is defined to be\nthe algebra over the entries xi’s of x subject to the relation (7). It will be\ncalled the coordinate ring of the virtual quantum space XH\nsym. Similarly, the\nbraided exterior algebra ∧H\nq [X] of X is defined to be the algebra over the\nentries xi’s of x subject to the relation (8). It will called the coordinate ring\nof the virtual quantum space XH\n∧. Let CH,r\nq\n[X] and ∧H,r\nq\n[X] be the degree\nr components of CH\nq [X] and ∧H\nq [X], respectively.\nIt is known [BZ] that\nthe dimensions of CH,r\nq\n[X] and ∧H,R\nq\nare bounded by the dimensions of the\nclassical Cr[X] and ∧r[X], respectively. But unlike in the standard setting,\nthe dimensions can be strictly less [BZ, Ro]. That is, CH\nq [X] and ∧H\nq [X]\nare, in general, nonflat deformations of the classical symmetric and exterior\nalgebras C[X] and ∧[X]. For example, ∧H,3\nq\n[X] = 0 when H = sl2(C) and\nX is the four dimensional irreducible representation of sl2(C) [BZ].\nThe equation (5) or (6) after reformulation just says that the defining\nrelation (7) of XH\nsym–or equivalently, the defining relation (8) of XH\n∧–is pre-\nserved under the left and right actions of u on x given by x →ux and\nxt →xtu.\n7"},{"page":8,"text":"This means CH\nq [X] and ∧H\nq [X] have left and right coactions of O(MH\nq (X)).\nWe define the left and right nonstandard minors of GH\nq to be the matrix co-\nefficients (in a suitable basis specified later) of the left and right coactions\non ∧H\nq [X]. If ∧H,dim(X)\nq\n[X] ̸= 0, then we define the determinant of GH\nq to\nbe the matrix coefficient of the action of O(MH\nq (X)) on ∧H,dim(X)\nq\n[X]. But\nit can vanish, as it does for H = sl2(C), dim(X) = 4. The nonstandard\nminors will be discussed in more detail in Section 4.\nLet J be the ideal of elements in O(MH\nq (X)) on which the counit ǫ van-\nishes. Then the nonstandard universal enveloping algebra U H\nq (G) is defined\nto be the space of linear functions of O(MH\nq (X)) which vanish on Jr for\nsome r > 0 depending on the linear function.\nThe following is a precise form of Theorem 1.2.\nTheorem 2.1 (1) There is a natural bialgebra homomorphism from O(MH\nq (X))\nto O(Mq(V )).\nThis gives the desired quantization of the homomorphism\nO(M(X)) →O(M(V )).\n(2) The matrix coordinate ring O(MH\nq (X)) of GH\nq is cosemisimple. Hence,\nits every finite dimensional corepresentation is completely reducible as a di-\nrect sum of irreducible corepresentations.\n(3) The q-analogue of the Peter-Weyl theorem holds: i.e.,\nO(MH\nq (X)) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible corepresentations of O(MH\nq (X)).\n(4) The nonstandard enveloping algebra U H\nq (G) is a bialgebra with a compact\nreal form (a ∗-structure) such that X⊗r\nq\nis its unitary representation with\nrespect to the Hermitian form on X⊗r\nq\ninduced by the standard Hermitian\nform on Xq. There is a bialgebra homomorphism form Uq(H) to U H\nq (G).\nThis gives a desired quantization of the homomorphism U(H) →U(G).\nHere the standard Hermitian form on Xq is the one that is Uq-invariant,\nwhere Uq ⊆Hq is the compact real form (the unitary subgroup) of Hq in the\nsense of Woronowicz [W]. The special case of this theorem in the context of\nthe Kronecker problem was proved in [GCT4] on the basis of Woronowicz’s\nwork [W]. The latter is no longer applicable in the general context here,\nsince the determinant of GH\nq may vanish, and hence, we cannot, in general,\nconvert O(MH\nq (X)) into a Hopf algebra by localization at the determinant.\n8"},{"page":9,"text":"Fortunately, this does not matter since U H\nq (G) still has a compact real form,\nwhose existence can be proved using the spectral properties of ˆRH\nX,X.\nWe also call Wq,α here a polynomial representation of GH\nq . By a poly-\nnomial representation of U H\nq (G) we mean a representation that is induced\nby a (finite dimensional) corepresentation of O(MH\nq (X)). It is completely\nreducible by cosemsimplicity of O(MH\nq (X)).\nIt may be conjectured that\nevery finite dimensional representation of U H\nq (G) is completely reducible (as\nin the standard case), though we shall not need this more general fact.\nThe standard Drinfeld-Jimbo enveloping algebra has an explicit presen-\ntation in the form of explicit generators (ei, fi, Ki) and explicit relations\namong them. It will be interesting to find an analogous explicit presenta-\ntion for U H\nq (G); cf. Section 4 for the problems that arise in this context.\n3\nNonstandard q-Schur algebra\nIn the standard setting, the q-Schur algebra Ar = Ar(q) is defined to be the\ndual O(Mq(V ))∗r of the degree r component O(Mq(V ))r of the standard\nmatrix coordinate algebra O(Mq(V )). Thus Ar(q) acts on V ⊗r from the\nleft. It is known [Kli] that it is the centralizer in End(V ⊗r) of the right\naction of the Hecke algebra Hr(q) on V ⊗r.\nAnalogously, we define the nonstandard q-Schur algebra AH\nr\n= AH\nr (q)\nto be the dual O(MH\nq (X))∗r of the degree r component O(MH\nq (X))r of\nthe nonstandard matrix coordinate algebra O(MH\nq (X)). Thus AH\nr (q) acts\non X⊗r from the left.\nAs per the nonstadard duality conjecture (Con-\njecture 1.3), it is the centralizer in End(X⊗r) of the right action of the\nnonstandard quantized algebra BH\nr (q) (cf. Section 5) on X⊗r.\nEvery irreducible corepresentation Wq,α of O(MH\nq (X)) of degree r can\nalso be considered as a representation of AH\nr (q), and conversely, every irre-\nducible representation of AH\nr (q) arises in this way. Theorem 2.1 now imme-\ndiately implies:\nTheorem 3.1 (1) The nonstandard q-Schur algebra AH\nr (q) is semisimple.\nHence, its every finite dimensional representation is completely reducible as\na direct sum of irreducible representations.\n(2) The q-analogue of the Peter-Weyl theorem in this case is the Wederburn\n9"},{"page":10,"text":"structure theorem for AH\nr (q):\nAH\nr (q) =\nM\nα\nW ∗\nq,α ⊗Wq,α,\nwhere Wq,α runs over all irreducible representations of AH\nr (q).\n(3) The nonstandard q-Schur algebra AH\nr (q) has a compact real form (a ∗-\nstructure) such that X⊗r\nq\nis its unitary representation with respect to the\nHermitian form on X⊗r\nq\ninduced by the standard Hermitian form on Xq.\n4\nNonstandard minors\nIn this section, we give a conjectural formula for the Laplace expansion\nof the minors of GH\nq .\nThe Laplace expansion for the standard quantum\ngroup GLq(V ) is based on the simple relation defining the standard exterior\nalgebra ∧q[V ], namely\nv2\ni = 0\nand\nvivj = −q−1vjvi,\nfor\ni < j.\nThis explains why the Laplace expansion in the standard setting is obtained\nfrom the classical Laplace expansion by simply substituting −q for −1. We\nneed a similar explicit formula for multiplication in CH\nq [X] to get an explicit\nformula for Laplace expansion in the nonstandard setting.\n4.1\nKronecker problem\nWe begin with a special case that arises in the context of the Kronecker\nproblem [GCT4] when H = GL(V ) × GL(W) and X = V ⊗W, with the\nnatural H-action. The article [GCT4] gives a formula for the column or row\nexpansion of the minor of GLH\nq (X) in this special case in terms of fundamen-\ntal Clebsch-Gordon coefficients for the standard quantum groups GLq(V )\nand GLq(W). But this formula cannot be extended for the general Laplace\nexpansion since Clebsch-Gordon coefficients are not well defined when the\nunderlying tensor products do not have multiplicity-free decompositions as\nin the fundamental case. Here we give a formula for general Laplace expan-\nsion of the minors of GLH\nq (X) in this case.\nWe begin by recalling that when V = W ∗the braided symmetric al-\ngebra CH\nq [X] = CH[W ∗⊗W] is isomorphic to the matrix coordinate ring\nO(Mq(W)) of the standard matrix space Mq(W) [GCT4]. For this, we have:\n10"},{"page":11,"text":"Theorem 4.1 (Kashiwara and Lusztig [Kas2, Lu2]) The coordinate ring\nO(Mq(W)) has an (upper) canonical basis.\nThis can be naturally and easily extended to:\nTheorem 4.2 The braided symmetric coordinate algebra CH\nq [X] = CH\nq [V ⊗\nW], H = GL(V ) × GL(W), has an (upper) canonical basis.\nThe exterior form of this result is:\nTheorem 4.3 The exterior coordinate algebra ∧H\nq [V ⊗W], H = GL(V ) ×\nGL(W), also has an (upper) canonical basis.\nLusztig [Lu2] has conjectured that the multiplicative and comultiplica-\ntive structural constants of the canonical basis of O(Mq(W)) are polynomials\nin q and q−1 with nonnegative integer coefficients; i.e., belong to N[q, q−1].\nAnalogous conjecture can be made for ∧H\nq [V ⊗W]. Specifically, it can be\nconjectured that for any canonical basis elements b and b′ in ∧H\nq [V ⊗W]:\nbb′ =\nX\nb′′\nǫ(b, b′, b′′)cb′′\nb,b′b′′,\n(9)\nwhere the sign ǫ(b, b′, b′′) is 1 or −1 and the coefficient cb′′\nb,b′ ∈N[q, q−1]. And\nconversely, any b′′ ∈∧H,r′′\nq\n[V ⊗W] can be expressed as:\nb′′ =\nX\nb,b′\nǫ′(b, b′, b′′)db,b′\nb′′ bb′,\n(10)\nwhere b and b′ run over elements of ∧H,r\nq\n[V × W] and ∧H,r′\nq\n[V × W] respec-\ntively with r′′ = r + r′, the sign ǫ′(b, b′, b′′) is 1 or −1, and db,b′\nb′′ ∈N[q, q−1].\nTo prove nonnegativity of the coefficients of cb′′\nb,b′ and db,b′\nb′′ , one needs in-\nterpretations for them in terms of perverse sheaves [BBD] in the spirit of\nKazhdan-Lusztig [KL2] and Lusztig [Lu1].\nWe now define the (left or right) minors of GH\nq\nwith respect to the\ncanonical basis ∧H,r\nq\n[V ⊗W] to be the matrix coefficients of the (left or right)\ncoaction of O(MH\nq (V ⊗W)). We shall call them (left or right) canonical\nminors. Then:\n11"},{"page":12,"text":"Theorem 4.4 A canonical minor of degree r′′ of GLH\nq (X), H = GL(V ) ×\nGL(W), admits a Laplace expansion in terms of canonical minors of degree\nr and r′ with r′′ = r + r′. The coefficients of this Laplace expansion are\nquadratic forms in the structural constants cb′′\nb,b′ and db,b′\nb′′\nabove.\nAn explicit formula for Laplace expansion here (omitted) is similar to the\none in Proposition 6.1 of [GCT4] with these structural constants in place of\nthe Clebsch-Gordon coefficients there (which are not well defined for general\nLaplace expansion).\n4.2\nGeneral nonstandard setting\nNow let us turn to the general case. The conjecturally correct algorithm\nin [GCT8] for constructing a canonical basis of O(MH\nq (X)) also yields, as\na byproduct, conjectural canonical bases of ∧H\nq [X] and CH\nq [X] as implicitly\nsought in [BZ]. We define the (left or right) minors of GH\nq in general to be\nthe matrix coefficients of the (left or right) coaction of O(MH\nq (X)) in this\ncanonical basis of ∧H\nq [X]. We call these nonstandard canonical minors, or\nsimply nonstandard minors.\nOne can define structural constants cb′′\nb,b′ and db,b′\nb′′ analogous to the ones\nin (9) and (10) in this case. With this:\nTheorem 4.5 Analogue of Theorem 4.4 holds in general.\nLaplace expansion in the standard setting is used as a straightening\nrelation to construct standard monomial bases of the coordinate ring and\nirreducible representations of GLq(X). In this sense, Laplace expansion is\na mother relation that governs the representation theory of the standard\nquantum group. Similarly, the nonstandard Laplace expansions in Theo-\nrems 4.4 and 4.5 are expected to be mother relations governing the rep-\nresentation theory of the nonstandard quantum group GH\nq . In particular,\nan explicit interpretation for the structural coefficients cb′′\nb,b′ and db,b′\nb′′\nakin\nto the ones based on perverse sheaves for the Kazhdan-Lusztig polynomials\n[KL2] and the multiplicative structural constants of the canonical basis for\nthe standard quantum group [Lu2] is necessary to get fully explicit formulae\nfor the nonstandard minors, and hence, for constructing explicit bases for\nthe irreducible polynomial representations and the matrix coordinate ring\nof GH\nq . In particular, this seems necessary for proving correctness of the\n12"},{"page":13,"text":"algorithms in [GCT8] for constructing nonstandard canonical bases for the\npolynomial representations and the matrix coordinate ring of GH\nq . This also\nseems necessary for finding an explicit presentation of the nonstandard uni-\nversal enveloping algebra U H\nq (G) in the spirit of the explicit presentation of\nthe Drinfeld-Jimbo enveloping algebra. Specifically, we expect the coeffi-\ncients occuring in such an explicit presentation to depend on the structural\nconstants such as cb′′\nb,b′ and db,b′\nb′′ above.\n5\nNonstandard quantized algebra\nWe now construct a nonstandard quantization BH\nr (q) of the symmetric\ngroup ring C[Sr] which conjecturally has the same relationship with GH\nq\nthat the Hecke algebra Hr(q), the standard deformation of C[Sr], has with\nthe standard quantum group. For the sake of simplicity, we assume that\nH = GL(V ).\nChoose a standard embedding of X = Vμ(H) in V ⊗d, where d is the size\nof the partition μ. That is, choose a Young symmetrizer cμ ∈C[Sr] such\nthat V ⊗d ·cμ, the image of V ⊗d under the right action of cμ, is isomorphic to\nX = Vμ(H). Let zμ ∈Hd(q) be the quantization of cμ such that V ⊗d\nq\n· zμ ∼=\nXq = Vq,μ. Here Vq denotes the quantization of V and Vq,μ the irreducible\nHq module with highest weight μ.\nAn explicit expression of zμ may be\nfound in [DJ]. Let Zq = V ⊗d\nq\n. Let ˆRH\nZ,Z denote the ˆR-matrix of Zq as an\nHq-module. Let rZ ∈H2d(q), 1 ≤i < r, be the element whose right action\non Zq ⊗Zq = V ⊗2d\nq\ncoincides with the action of ˆRH\nZ,Z. One can easily write\ndown an explicit expression for rZ in terms of the generators of H2d(q).\nNow consider the right action of Hs(q), s = dr, on Z⊗r\nq\n= V ⊗s\nq\n, which\ncommutes with the left action of Hq = GLq(V ). Let rZ,i ∈Hs(q), 1 ≤i < r,\nbe the element whose right action on Z⊗r\nq\ncoincides with the action of ˆRH\nZ,Z\non the product of the i-th and (i + 1)-st factors of Z⊗r\nq . Thus rZ,i is the\nimage of rZ under the obvious embedding of H2d(q) in Hs(q) depending on\ni. One can thus write down an explicit expression for rZ,i in terms of the\ngenerators of Hs(q). Let\nrH\nX,i = zλ,i · zλ,i+1 · rZ,i,\nwhere zλ,i ∈Hs(q) denotes an explicit element whose action on the i-th\nfactor of Z⊗r\nq\ncoincides with the action of zλ on that factor–it is the image of\nzλ under the obvious embedding of Hd(q) in Hs(q) depending on i. Then the\nright action of rH\nX,i on Z⊗r\nq\ncorresponds to the action of ˆRH\nX,X on the product\n13"},{"page":14,"text":"of the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq . Let p+,H\nX,i , p−,H\nX,i ∈Hs(q) be\nthe polynomials in rH\nX,i whose actions on Z⊗r\nq\ncorrespond to the actions of\nthe positive and negative projection operators P +,H\nX,X and P −,H\nX,X in eq. (4) on\nthe tensor product of the i-th and (i + 1)-st factors of X⊗d\nq\n⊆Z⊗d\nq ; one can\nwrite down these polynomials explicitly, using the known explicit spectral\nform of ˆrH\nX,i.\nWe define the nonstandard quantized algebra BH\nr (q) to be the subalgebra\nof Hs(q) generated by the explicit elements p+,H\nX,i , or equivalently, p−,H\nX,i . In\ngeneral, it is a nonflat deformation of C[Sr]. That is, its dimension can be\nlarger than that of C[Sr]. It can be shown to be semisimple. Its right action\non X⊗r\nq\ncommutes with the left action GH\nq by the defining equation (5) of\nGH\nq . Conjecture 1.3 says that its relationship with GH\nq\nis akin to that of\nHr(q) with the standard quantum group Gq = GLq(X).\nThe Hecke algebra has an explicit presentation in terms of explicit re-\nlations among its generators.\nIt will be interesting to find an analogous\nexplicit presentation for BH\nr (q). Its complexity would be much higher than\nthat of the Hecke algebra as indicated by the concrete computations in\n[GCT4].\nSpecifically, we expect an explicit presentation for BH\nr (q) with\ndefining relations whose coefficients are akin to the structural constants cb′′\nb,b′\nand db,b′\nb′′ in Section 4 and have a topological interpretation akin to the one\nfor Kazhdan-Lusztig polynomials. Such an explicit presentation is needed\nto prove correctness of the algorithm in [GCT8] to construct a canonical\nbasis of BH\nr .\nRemark: We can also define a (possibly singular) quantum group ˆGH\nq , in-\nstead of GH\nq , by substituting ˆRH\nX,X in place of P +,H\nX,X in the defining equation\n(5). One can then define a deformation ˆBH\nq (r) of C[Sr] that is conjecturally\npaired with ˆGH\nq , as GH\nq is with BH\nq (r). The main results (semisimplicity, and\nq-analogue of the Peter-Weyl theorem) also hold for these objects. Further-\nmore, variants of the algorithms in [GCT8] can be conjectured to provide\ncanonical bases for these as well.\nHowever, the Poincare series of ˆGH\nq\nis\nmuch smaller than that of GH\nq , and for this and other reasons, it does not\nseem possible to use these objects in the context of the plethysm problem.\nHowever, these may be interesting intermediate quantum objects to study\nnevertheless.\n14"},{"page":15,"text":"6\nRefined reciprocity\nWe now describe a refinement of the reciprocity conjecture (Conjecture 1.4)\nthat specifies precisely how the decomposion (3) of X⊗r\nq ,\nX⊗r\nq\n=\nM\nWq,α ⊗Tq,α,\n(11)\nas a GH\nq × BH\nr (q)-bimodule, tends to the decomposition\nX⊗r =\nM\nλ\nVλ ⊗Sλ\n(12)\nof X⊗r as a G × Sr-bimodule, as q →1, and gives an explicit realization\nwithin X⊗r\nq\nof the q-analogue V H\nq,λ of Vλ as in Conjecture 1.4. Here, as usual,\nVλ denotes the Weyl module of G, and Sλ the Specht module of Sr.\nFirst, we have to define the multiplicity mα\nλ of a Specht module Sλ in the\nspecialization Tq,α(1) of Tq,α at q = 1. In this context, it may be remarked\nthat though B = BH\nr (q) is semisimple, its specialization B(1) at q = 1 need\nnot be semisimple; see Section 7.1 for an example. Clearly, every represen-\ntation of Sr is also a representation of B(1), though not always conversely.\nBut it may be conjectured that every irreducible B(1)-representation is also\nan irreducible Sr-representation, i.e., a Specht module. Fix any (maximal)\ncomposition series of Tq,α(1) as a B(1)-module. We define the multiplic-\nity mα\nλ to be the number of factors in this (or any such) series that are\nisomorphic to the specht module Sλ.\nSince B is semisimple (cf. Section 5), it admits a Wederburn structure\ndecomposition of the form\nB =\nM\nU α,\nU α = Tq,α,L ⊗Tq,α,R,\n(13)\nwhere α is as in (11), and Tq,α,L and Tq,α,R denote the left and right irre-\nducible B-modules indexed by α. We call this a complete Wederburn struc-\nture decomposition. Here we are assuming that the base field is C = C(q),\nq complex. This complete decomposition would also hold if the base field\nis instead an appropriate algebraic extension K of Q(q). In the standard\nsetting of Hecke algebras, K = Q(q) suffices. This need not be so in the\nnonstandard setting. That is, an algebraic extension of Q(q) may be ac-\ntually necessary for a complete decomposition of the above form to hold;\nsee Section 7.1 for an example. If the base field is Q(q), each U α in the\nWederburn structure decomposition need not be, in general, of the form\n15"},{"page":16,"text":"Tq,α,L ⊗Tq,α,R as above, but rather it would be isomorphic to the endomor-\nphism ring of Tq,α over the division algebra EndB(Tq,α). One has to take\nsimilar variations of the nonstandard q-analogue of the Peter-Weyl theorem\n(Theorem 2.1 (3)) and the duality conjecture (Conjecture 1.3) if the base\nfield is Q(q). However, for the reciprocity conjecture, it is necessary to take\nthe base field as C(q), q complex, or an algebraic extension K of Q(q) as\ndescribed above. We assume this in the rest of this section. See Section 7.1\nfor an example wherein reciprocity fails over Q(q).\nFix any right cell, i.e., an irreducible right B-subrepresentation within\nU α. Let us denote it by Tq,α,R again. Fix a maximal composition series as\na B(1)-module of the specialization Tq,α,R(1) of Tq,α,R at q = 1:\nˆTα,0 ⊂ˆTα,1 ⊂· · · ⊂ˆTα,l(α) = Tq,α,R(1).\nLet {xi} denote the upper canonical basis of Xq as an Hq-module.\nConjecture 6.1 (Nonstandard refined reciprocity)\nThere exists a basis Zα of Tq,α,R for each α with a filtration\nZα,0 ⊂Zα,1 ⊂· · · ⊂Zα,l(α) = Zα,\nsuch that:\n1. The specialization Zα,i(1) of Zα,i at q = 1 is a basis of ˆTα,i.\n2. Let zj\nα,i denote the basis elements in Zα,i \\ Zα,i−1.\nLet λα,i be the\npartition such that ˆTα,i/ ˆTα,i−1 ∼= Sλα,i as a B(1)-module (or equiv-\nalently as an Sr-module).\nFor any α, i, define the left GH\nq -module\nWq,α,i = ∪jX⊗r\nq\n· zj\nα,i. By the duality conjecture (Conjecture 1.3),\nWq,α,i ⊆Wq,α ⊗Tq,α ⊆X⊗r\nq .\n(14)\nWe define its specialization W1,α,i at q = 1, also denoted by Wq,α,i(1),\nas follows. Let a(α, i) be the largest nonnegative integer such that the\nlimit vector\nlimq→1xi1 ⊗· · · ⊗xir.zj\nα,i/(q −1)a(α,i),\nis well defined for any i1, . . . , ir and j. We define W1,α,i to be the span\nof such limits at q = 1. Then, W1,α,i is a left G-module contained\nwithin the component Vλα,i ⊗Sλα,i ⊆X⊗r in (12).\n16"},{"page":17,"text":"3. For any fixed partition λ,\nM\nα\nM\ni\nW1,α,i = Vλ ⊗Sλ ⊆X⊗r,\n(15)\nwhere, for a given α, i ranges over all indices such that λα,i = λ.\nFurthermore, it may be conjectured that the canonical basis of Tq,α,R\nin terms of the P-monomials as defined in [GCT8] has this property–this\nwould make everything in the conjecture above explicit.\nThe refined reciprocity conjecture basically says that there is no infor-\nmation loss in the nonstandard setting despite the lack of flatness. In fact,\nit can be thought of as a variant of flatness.\n7\nEvidence for duality and reciprocity\nHere we describe some concrete computations carried out in MATLAB/Maple\nthat support duality and reciprocity conjectures.\nNotation: We denote the q-Weyl module of Gq for a partition λ by Vq,λ(Gq).\nWe denote Vq,λ(GLq(Cn)) by Vq,λ(n).\n7.1\nExample 1\nLet H = GL(C2), H = gl(C2), X = V(3)(H) is its four dimensional irre-\nducible representation, and G = GL(X) = GL(C4). Then Hq = GLq(C2),\nGq = GLq(C4), and Hq = glq(C2). We shall verify duality and reciprocity\nin this case for r = 3. This example is interesting because, as shown in [BZ],\nthe degree three component ∧H,3\nq\n[X] of the braided exterior algebra vanishes\nin this case. We expect that the results in this section can be extended to\nany irreducible representation X of H. But we shall confine ourselves to the\ncase dim(X) = 4, since this seems to be the gist.\nLet ˆR = ˆRH\nX,X be the ˆR-matrix associated with Xq. Let P = P H\nX,X and\nQ = QH\nX,X be the projections on the eigenspaces in Xq ⊗Xq for the positive\nand negative eigenvalues of ˆRH\nX,X, respectively. Let xi = f ix0, where f is\nthe usual operator in Hq, and x0 is the highest weight vector in Xq. Matrices\nof ˆR, P and Q in the basis xi ⊗xj of Xq ⊗Xq can be calculated from the\nknown explicit formulae; cf. [Kass, Kli]. The eigenvalues of ˆR turn out to be\nq9/2, −q−3/2, q−11/2 and −q−15/2. Explicit matrix of P in the basis xi ⊗xj\nof Xq ⊗Xq is given by\n17"},{"page":18,"text":"P = 1\nf P,\n(16)\nwhere\nf = (q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n(17)\nand the matrix of P is as specified in Figure 1 with the following sparse\nrepresentation: the entry (j, v) in the i-row in Figure 1 means P(i, j) = v.\nThus the entry (5, (q4 + 1)/q2) in the second row there means P(2, 5) =\n(q4 +1)/q2. The entries of P-matrix not shown in Figure 1 are all zero. The\nscaling factor f here is chosen so that the entries of P-matrix are polynomials\nin q and q−1. Explicit matrix of\nQ = fQ\n(18)\nis similar.\n7.1.1\nExplicit presentation of B\nLet P1 and P2 denote the P operators on the first two and the last two\nfactors X⊗3, respectively; Q1 and Q2 are defined similarly. We have the\ntrivial relations:\nQ2\ni = fQi,\nand P2\ni = fPi.\nThe first nontrivial basic relation among Qi’s, as determined with the help\nof a computer, is:\nX\nσ\naσQσ = 0,\n(19)\nwhere σ ranges over the various strings of 1’s and 2’s as shown in Figure 2,\naσ ∈Q[q, q−1] are as specified there, and, for a string σ = i1i2 · · · , Qσ\ndenotes the monomial Qi1Qi2 · · · .\nThe second relation is obtained from\nthis by simply interchaning Q1 and Q2. Simialrly, the first nontrivial basic\nrelation among Pi’s is\nX\nσ\nbσPσ = 0,\n(20)\nwhere σ ranges over strings of 1’s and 2’s as in Figures 3-4, bσ’s are as shown\nthere, and Pσ is defined similarly. The second relation is obtained from this\nby simply interchanging P1 and P2. All coefficients in Figures 2-4 as well\nas other figures in this section are shown in factored forms, i.e., as products\nof irreducible polynomials. One may ask if these coefficients have a nice\ninterpretation; we shall turn to this question in Section 7.1.7.\n18"},{"page":19,"text":"Let B = BH\n3 (q) be the nonstandard algebra in this case, as defined in\nSection 5. It is isomorphic to the algebra generated by Pi’s subject to the\ntwo basic nontrivial relations among Pi’s described above and the trivial\nrelations P2\ni = fPi, or equivalently, to the algebra generated by Qi’s subject\nto the two basic nontrivial relations among Qi’s described above, and the\ntrivial relations Q2\ni = fQi.\nIt is clear from these basic defining relations that {Pσ} or {Qσ}, where σ\nranges over all strings of 1’s and 2’s of length at most 10 without consecutive\n1’s or 2’s, is a basis of B. Its dimension is 21.\n7.1.2\nWederburn structure decomposition\nUnlike for the Hecke algebras, for the complete Wederburn structure decom-\nposition as in (13) to hold for B, the base field has to contain the algebraic\nextension K of Q(q) defined as follows. Let\ndisc =\n 5 q16 + 8 q12 −4 q10 + 18 q8 −4 q6 + 8 q4 + 5\n q8 + 1\n 2 q24,\n(21)\nand\nx = disc1/2.\nSince disc is not a square, x does not belong to Q(q). Let K = Q(q)[x] be\nthe algebraic extension of Q(q) obtained by adjoining x. We assume that\nB is defined over this base field. It was found by computer that B has one\none-dimensional irreducible representation T0, and five two-dimensional irre-\nducible representations Ti, 1 ≤i ≤5, with a complete Wederburn structure\ndecomposition\nB =\nM\ni\nUi,\nUi = Ti,L ⊗Ti,R,\n(22)\nwhere the basis elements of the various B ⊗B-bimodules Ui and the explicit\nrepresentation matrices of the irreducible B-representations Ti are as follows.\nLet U0 be the K-span of u0 ∈B, where u0 is as specified in Figures 5-6.\nThe coefficients in these and the following figures are in the basis {Qσ}. Let\nUi, 1 ≤i ≤5, be the K-span of the entries u1\ni , u12\ni , u21\ni , u2\ni ∈B of the matrix\nui =\n u1\ni\nu12\ni\nu21\ni\nu2\ni\n \n,\nwhere u1\n1 is as specified in Figure 7, u1\n2 the element obtained from u1\n1 by\nsubstituting −x for x, and u1\n3, u1\n4, u1\n5 as specified in Figures 8-10. Let u2\ni ,\n19"},{"page":20,"text":"1 ≤i ≤5, be the element obtained from u1\ni by interchanging Q1 and Q2.\nLet u12\ni\n= u1\ni Q2, and u21\ni\n= Q2u1\ni , for 1 ≤i ≤5.\nThen it can be shown that each Ui has a left and right action of B, and\nas a B ⊗B-bimodule\nB =\nM\ni\nUi.\n(23)\nThe columns of ui correspond to the left cells and the rows to right cells;\ni.e., the span of each column (row) is a left (resp. right) B-module, which\nwe shall denote by Ti,L (resp. Ti,R). Thus,\nB =\nM\ni\nTi,L ⊗Ti,R.\n(24)\nHere T0, the span of u0, is the trivial one dimensional representation of\nB, since it can be verified that:\nQju0 = 0,\nfor j = 1, 2.\nThe representation matrices M1\ni and M2\ni of Q1 and Q2 in the basis {u1\ni , u21\ni }\nof Ti,L, 1 ≤i ≤5, are as follows:\nM1\ni =\n 0\n1\n0\nf\n \n,\nwhere f is the scaling factor in (16),\nM2\ni =\n f\ngi\n0\n0\n \n,\nwhere gi are as shown in Figure 11; g2 is obtained from g1 by substituting\n−x for x.\nLet Ti(1) denote the specialization of Ti at q = 1. It is a representation\nof B(1), the specialization of B at q = 1. Then T0(1) corresponds to the\ntrivial one-dimensional representation of S3. There is no one dimensional\nrepresentation of B that specializes to the alternating (signed) one dimen-\nsional representation of S3. This implies that the degree three component\n∧H,3\nq\n[X] of the braided exterior algebra ∧H\nq [X] in this case is zero–as was\nalready observed and proved by other means in [BZ].\nAt q = 1, the values of f = f(q) and gi = gi(q) are as follows:\nf(1) = g1(1) = g3(1) = g4(1) = g5(1) = 4, and g2(1) = 16.\n20"},{"page":21,"text":"Hence the B(1)-modules T1(1), T3(1), T4(2) and T5(2) are all isomorphic, and\nit can be verified that they are isomorphic to the Specht module S(2,1) of the\nsymmetric group S3 for the partition (2, 1). The module T2(1) is reducible.\nBecause it can be verified that it contains an irreducible B(1)-module T 1\n2 (1)\nisomorphic to the trivial one dimensional Specht module S(3) of the sym-\nmetric group S3, and the quotient T 2\n2 (1) = T2(1)/T 1\n2 (1) is isomorphic to\nthe one dimensional signed representation S(1,1,1) of S3. But T2(1) is not\ncompletely reducible as a B(1) module. That is, T2(1) ̸∼= T 1\n2 (1) ⊕T 2\n2 (1),\nsince it does not contain a submodule isomorphic to S(1,1,1). Thus, though\nB is semisimple for generic q, its specialization B(1) is not semisimple.\n7.1.3\nDuality\nPick an element ui from each Ui, 1 ≤i ≤5; say, ui = u1\ni , and u0 is as before.\nFor 0 ≤i ≤5, let Wi = X⊗3\nq\n· ui, which has a left action of the nonstandard\nquantum group GH\nq . These are nonisomorphic irreducible representations\nof GH\nq . Their explicit decompositions as Hq-modules, Hq = GLq(C2), were\ndetermined with the help of computer. They are as follows.\nThe module W0 is isomorphic to the sixteen dimensional degree three\ncomponent CH,3\nq\n[X] of the braided symmetric algebra [BZ] with the following\ndecomposition as an Hq-module:\nW0 = Vq,(9)(2) ⊕Vq,(7,2)(2);\nrecall that Vq,λ(n) denotes the q-Weyl module of GLq(n) corresponding to\nthe partition λ. This decomposition of CH,3\nq\n[X] in this case agrees with the\none obtained in [BZ] by other means.\nThe modules Wi, i > 0, are distinct irreducible representations of GH\nq\nwith the following decompositions as Hq-modules:\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(25)\nTheir dimensions are 4, 4, 8, 2 and 6, respectively. Though W1 and W2 are\nisomorphic as Hq-modules, they are nonisomorphic as GH\nq -modules; the ma-\ntrix coefficients of W2 are obtained from those for W1 by substituting −x\nfor x.\n21"},{"page":22,"text":"It can be verified that, as a GH\nq × B-bimodule,\nX⊗3\nq\n∼= ⊕iWi ⊗Ti,\n(26)\nas per the duality conjecture (Conjecture 1.3).\n7.1.4\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of the symmetric\ngroup S3 in the B(1)-module Ti. Then, we see that\nm0\n(3) = 1,\nm1\n(2,1) = m3\n(2,1) = m4\n(2,1) = m5\n(2,1) = 1,\nm2\n(3) = m2\n(1,1,1) = 1.\nFurthermore, it can be verified that the various Gq-modules, Gq =\nGLq(C4), decompose as follows when considered as Hq-modules:\nVq,(3)(4)\n∼=\nm0\n(3)W0 ⊕m2\n(3)W2,\n∼=\nVq,(9)(2) ⊕Vq,(7,2)(2) ⊕Vq,(6,3)(2),\nand\nVq,(2,1)(4)\n∼=\nm1\n(2,1)W1 ⊕m3\n(2,1)W3 ⊕m4\n(2,1)W4 ⊕m5\n(2,1)W5\n∼=\nVq,(6,3)(2) ⊕Vq,(8,1)(2) ⊕Vq,(5,4)(2) ⊕Vq,(7,2)(2).\nThis verifies the nonstandard reciprocity conjecture (Conjecture 1.4) in this\ncase.\n7.1.5\nRefined reciprocity\nFix a right cell within U2 isomorphic to the representation T2,R; say, the one\nspanned by u1\n2 and u12\n2 . We shall denote it by T2,R again. Let z0 ∈T2,R be\nthe element such that z0Q2 = 0. Its coefficients are shown in Figure 12 in\nthe basis {Qσ}. Let z1 = u1\n2. Then the basis Z = {z0, z1} of T2,R admits a\nfiltration\nZ0 = {z0} ⊆Z1 = {z0, z1},\nthat yields at q = 1 a composition series of T2,R(1) as a B(1)-module:\nˆT2,0 ⊂ˆT2,1 = T2,R(1),\n22"},{"page":23,"text":"where ˆT2,0, spanned by the specialization z0(1) of z0, is the one dimensional\ntrivial representation of S3, and ˆT2,1/ ˆT2,0 is the one-dimensional signed rep-\nresentation of S3.\nLet W2,1 = X⊗3\nq\n· z1 and W2,0 = X⊗3\nq\n· z0 be the GH\nq -submodules of X⊗3\nq ,\nand W2,1(1), W2,0(1) their specializations at q = 1. It can be verified that\nat q = 1 we get:\nW2,1(1) = ∧3(X) ⊆X⊗3,\nand\nW0(1) ⊕W2,0(1) = Sym3(X) ⊆X⊗3,\n(27)\nwhere ∧3(X) and Sym3(X) are the Weyl modules of G = GL(X) for the\npartitions (1, 1, 1) and (3), respectively, and W0(1) the specialization of W0\nat q = 1.\nFor example, Figures 13-16 show the nonzero coefficients of the elements\na = (x1 ⊗x2 ⊗x0) · z1 and b = (x1 ⊗x2 ⊗x0) · z0 in the monomial basis\n{xi ⊗xj ⊗xk} of X⊗3\nq . It can be verified that the specialization a(1) at\nq = 1 of a indeed belongs to the subspace ∧3(X) ⊆X⊗3. The specialization\nb(1) of b, as it is, just vanishes, since its coefficients are divisible by (q −1)2.\nBut instead we consider the basis element b′ = b/(q −1)2 of W2,0. Then\nits specialization b′(1) at q = 1 indeed belongs to the subspace Sym3(X) of\nX⊗3. The equation (27) can be verified similarly.\nSimilarly it can be verified that\nlim\nq→1\nM\ni=1,3,4,5\n(X⊗3\nq\n· u1\ni ∪X⊗3\nq\n· u12\ni ) = V(2,1) ⊗S(2,1) ⊆X⊗3.\nThis verifies the refined reciprocity conjecture in this case. In particular,\nit explains what happens to the exterior and symmetric algebra components\nhere. Specifically, though the braided exterior algebra component ∧H,3\nq\n[X] =\n0,\nW2,1(1) = ∧H,3[X].\nThus the q-deformation of ∧H\n3 [X] has simply relocated itself as W2,1 in the\ndecomposition\nX⊗3\nq\n= ⊕Wi ⊗Ti.\nSimilarly, the symmetric algebra component CH,3[X] splits in two parts, and\nthe q-deformations of these parts, namely W0 and W2,0, get distributed in\nthis decomposition. The situation for V2,1 is similar. Thus, overall, there\nis no information loss; the information has only been redistributed. As per\nthe refined reciprocity conjecture, this is a general phenomenon.\n23"},{"page":24,"text":"7.1.6\nBase field Q(q)\nLet us now see what happens if the base field is Q(q) instead.\nThe B-\nrepresentations T0, T3, T4, T5 are already defined over Q(q).\nBut T1 and\nT2 merge into a four dimensional B-representation T12 defined over Q(q).\nExplicitly, it can be realized within B as the linear span of the elements\nv1\n=\n(u1\n1 + u1\n2)/2,\nv2\n=\n(u1\n1 −u1\n2)/(2x),\nv3\n=\n(u21\n1 + u21\n2 )/2,\nv3\n=\n(u21\n1 −u21\n2 )/(2x).\nRepresentation matrices of left multiplication by Q1 and Q2 in the basis\n{vi} are, respectively,\nM1 =\n \n \nq10+q6+q4+1\nq5\n0\na\n1/2 q−20\n0\nq10+q6+q4+1\nq5\nb\na\n0\n0\n0\n0\n0\n0\n0\n0\n \n \n,\nwith\na\n=\n1/2 3 q16+4 q12−2 q10+10 q8−2 q6+4 q4+3\nq8\n,\nb\n=\n1/2 q4(5 q32 + 8 q28 −4 q26 + 28 q24 −4 q22 + 24 q20 −8 q18 + 46 q16\n−8 q14 + 24 q12 −4 q10 + 28 q8 −4 q6 + 8 q4 + 5),\nand\nM2 =\n \n \n0\n0\n0\n0\n0\n0\n0\n0\n1\n0\nq10+q6+q4+1\nq5\n0\n0\n1\n0\nq10+q6+q4+1\nq5\n \n \n.\nSimilarly, the GH\nq -modules W0, W3, W4, W5 are already defined over Q(q).\nThe modules W1 and W2 merge into an eight-dimensional GH\nq -module W12 ∼=\nX⊗3\nq\n· vi, for any i–this is defined over Q(q). As an Hq-module,\nW1,2 ∼= 2 · Vq,(6,3)(2).\n24"},{"page":25,"text":"A variant of the duality also holds. Specifically, the components W1⊗T1 and\nW2⊗T2 in the decomposition (26) of X⊗3\nq\nmerge into one sixteen dimensional\nGH\nq × B-bimodule defined over Q(q). As a GH\nq module, it is a direct sum of\ntwo copies of W12, and as a B-module a direct sum of four copies of T12. But\nfor the reciprocity to hold, the base field has to be K = Q(q)[x] as before or\nlarger. Indeed, it can be seen here that the reciprocity conjecture fails over\nthe base field Q(q). This illustrates the need for base extension in general.\nIt may be illuminating to compare the r = 3 case here with the one for\nthe Kronecker problem treated in [GCT4]. The one here is basically a more\ncomplex version of the one in [GCT4], because the basic defining relations\nhere (Figures 2-4) are more complex versions of the ones in [GCT4].\n7.1.7\nOn r > 3 and positivity\nSimilar symbolic computations for r = 4 seem beyond the reach of desktop\nMATLAB/Maple. Fortunately, this case for the Kronecker problem is within\nthe reach, and will be treated in the next section. The r = 4 case, H =\nGL2(C), X four dimensional, is expected to be its more complex version\njust as for r = 3.\nBut it does not seem possible to progress much beyond r = 3 using the\nbrute force computer-based approach that we are following here. What is\nneeeded is an explicit presentation for BH\nr akin to the explicit presentation\nfor the Hecke algebra, or the one for r = 3 in Section 7.1.1. That is, we need\nan explicit set of generating relations among Qi or Pi’s, each of the form\nX\naσQσ = 0,\n(28)\nor\nX\nbσPσ = 0,\n(29)\nwhere Qσ and Pσ, for a string σ = i1i2 · · · of symbols in {1, · · · , r −1},\ndenote the monomials Qi1Qi2 · · · and Pi1Pi2 · · · , respectively, and each aσ\nand bσ has an explicit interpretation (formula).\nThe coefficients aσ and bσ in Figures 2-4 for the r = 3 case do not\nseem to have any obvious elementary interpretation. Hence, in general, one\ncan only expect nonelementary interpretations for the coefficients aσ and\nbσ in (28)-(29). The following numerical analysis of these coefficients for\nthe r = 3 case suggests that BH\nr , in general, may plausibly have an explicit\npresentation, the coefficients aσ and bσ of whose generating relations have\nnonelementary interpretations in the spirit of the one for Kazhdan-Lusztig\n25"},{"page":26,"text":"polynomials. By this we mean that each aσ has an explicit formula of the\nform of an alternating sum\naσ = (−1)d(σ)(q1/2 −q−1/2)d′(σ)(\ns(σ)\nX\nj=0\n(−1)jaj\nσ),\n(30)\nfor some nonnegative integers d(σ), d′(σ), s(σ), where\n1. s(σ) is small, say bounded by a polynomial of a fixed degree in r and\ndim(X) in the present case when H = GL2(C), and in r, the rank of\nH and the size of μ in the general plethysm problem (Problem 1.1),\n2. each aj\nσ is a −-invariant (note that aσ is −-invariant), positive and\nunimodal polynomial in q and q−1; positive means each coefficient of\naj\nσ is nonnegative, and unimodal means, if aj\nσ(−k), . . . , aj\nσ(k) are the\ncoefficients of aj\nσ, then\naj\nσ(−k) ≤aj\nσ(−k + 1) ≤· · · ≤aj\nσ(−1) ≤aj\nσ(0) ≤aj\nσ(1) ≤· · · aj\nσ(k),\n3. each aj\nσ(s) has a topological interpretation akin to that for Kazhdan-\nLusztig polynomials, i.e., as the rank of an appropriate cohomology\ngroup. Then the duality aj\nσ(−s) = aj\nσ(s) as per the −-invariance of aj\nσ\nshould come out as a consequence of some form of Poincare duality\nand the unimodality as a consequence of some form of Hard Lefschetz,\nand each bσ has a similar explicit formula of the form\nbσ = (−1)\n ̄d(σ)(q1/2 −q−1/2)\n ̄d′(σ)(\n ̄s(σ)\nX\nj=0\n(−1)jbj\nσ).\n(31)\nWe shall call such an interpretation for aσ or bσ, if it exists, a positive,\nunimodal, and topological interpretation.\nIdeally speaking, one would like each s(σ) and ̄s(σ) above to be zero, but\nthis may not always be possible for the reasons given below. It is plausible\nthat there exists some notion of cohomological depth that measures the\nextent of nonflatness, and which provides an upper bound on s(σ) and ̄s(σ) in\nsuch a topological interepretation, if it exists. For example, in the Kronecker\nproblem, the braided symmetric and exterior algebras CH\nq [X] and ∧H\nq [X] are\nflat deformations of the classical algebras C[X] and ∧[X]. In this case, one\ncan expect an explicit presentation for BH\nq whose coefficients aσ and bσ have\n26"},{"page":27,"text":"positive topological interpretation with s(σ), ̄s(σ) = 0 in (30) and (31). This\nis because aσ and bσ here are akin to the structural constants cb′′\nb,b′, db,b′\nb′′\nin\nTheorem 4.4, which occur in the defining Laplace relations for GH\nq , and\nwhich, in the Kroncker problem, are conjecturally polynomials in q and q−1\nwith nonnegative coefficients for the reasons indicated there. But in general\nwhen CH\nq [X] and ∧H\nq [X] are nonflat deformations, such cohomological depth\nwould not vanish, and hence s(σ) and ̄s(σ) may be nonzero, but still small\nas indicated above.\nWe now turn to the analysis of the coefficients in the r = 3 case men-\ntioned above which suggests that such an interpretation may plausibly exist.\nFirst let us oberve that the scaling factor f in (17) used in the analysis so far\nis formally not the correct scaling factor. To get the latter, we have to look\nat the formal expressions for P and Q in terms of ˆR. Since the eigenvalues\nof ˆR in the present case are\nq1 = q9/2,\nq2 = −q−3/2,\nq3 = q−11/2,\nand\nq4 = −q−15/2,\nwe have\nP = ( ˆR −q2)( ˆR −q3)( ˆR −q4)\n(q1 −q2)(q1 −q3)(q1 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q4)\n(q3 −q1)(q3 −q2)(q3 −q4),\n(32)\nand\nQ = ( ˆR −q1)( ˆR −q3)( ˆR −q4)\n(q2 −q1)(q2 −q3)(q2 −q4) + ( ˆR −q1)( ˆR −q2)( ˆR −q3)\n(q4 −q1)(q4 −q2)(q4 −q3).\n(33)\nHence, formally we should have defined the rescaled versions P and Q of P\nand Q by the equations\nP = fpP,\nfp = (q1 −q2)(q1 −q3)(q1 −q4)(q3 −q1)(q3 −q2)(q3 −q4), (34)\nand\nQ = fqQ,\nfq = (q2 −q1)(q2 −q3)(q2 −q4)(q4 −q1)(q4 −q3)(q4 −q2), (35)\ninstead of the equations (16) and (18). The scaling factor f in (17) was\nthe smallest factor chosen so that the matrix coefficients of P and Q after\nrescaling become polynomials in q, q−1. But this choice was dependendent on\nthe accidental cancellations in the numerators and denominators in (32) and\n27"},{"page":28,"text":"(33). The choice of scaling makes no essential difference in Sections 7.1.1-\n7.1.6. But it does matter in the study of positivity below.\nHence, let us redefine P and Q as per (34) and (35). Let us denote the\ncoefficients of the old defining relations (19) and (20) among Qi’s and Pi’s\nby a′\nσ and b′\nσ, and the coefficients of the defining relations among the new\nQi’s and Pi’s by a′′\nσ and b′′\nσ. Then we have\na′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄aσ,\nwith\n ̄aσ = ( ˆfq)11−l(σ)a′\nσ,\nand\nb′′\nσ = (−(q −1)2\nq\n)11−l(σ) ̄bσ,\nwith\n ̄bσ = ( ˆfp)11−l(σ)b′\nσ,\nwhere l(σ) denotes the length of σ,\nˆfp\n=\n−q\n(q−1)2\nfp\nf\n=\n10 + 8 q + 2 q4 + 12 q−1 + 18 q−6 + 6 q2 + 4 q3 + q5\n+14 q−2 + 18 q−4 + 16 q−12 + 16 q−8 + q−27 + 16 q−11\n+16 q−10 + 17 q−7 + 6 q−24 + 18 q−5 + 2 q−26 + 16 q−3\n+16 q−9 + 10 q−22 + 14 q−20 + 4 q−25 + 8 q−23 + 12 q−21 + 18 q−18\n+16 q−19 + 18 q−16 + 18 q−17 + 17 q−15 + 16 q−14 + 16 q−13,\nand\nˆfq\n=\n−q\n(q−1)2\nfq\nf\n=\n10 q−12 + 2 q−8 + q−31 + 8 q−27 + 8 q−11 + 6 q−10 + q−7\n+2 q−30 + 10 q−24 + 6 q−28 + 10 q−26 + 4 q−9 + 6 q−22\n+2 q−20 + 10 q−25 + 8 q−23 + 4 q−29 + 4 q−21 + 2 q−18\n+2 q−19 + 6 q−16 + 4 q−17 + 8 q−15 + 10 q−14 + 10 q−13.\nBoth fp and fq are positive polynomials.\nLet us define\nˆaσ\n=\nˆf 2\nq a′\nσ,\nfor\nσ = 121212121,\n=\nˆfqa′\nσ,\notherwise,\n(36)\nand\nˆbσ\n=\nb′\nσ,\nfor\nσ = ∅, and 2,\n=\nˆfpb′\nσ,\notherwise.\n(37)\n28"},{"page":29,"text":"Since fp and fq are positive, the positivity properties of ˆaσ and ̄aσ (also\nˆbσ and ̄bσ) are similar; it turns out that the unimodularity properties are\nalso similar. Hence we shall focus on ˆaσ and ˆbσ in what follows. Since ˆaσ\nis −-invariant, it is of the form ˆaσ(0) + P\nt>0 ˆaσ(t)(qt + q−t). Let ˆAσ be\nthe vector [aσ(0), aσ(1), . . .]; the vector ˆBσ is defined similarly. Figure 17\nshows ˆAσ for the various σ in Figure 2; the vector for each σ is obtained by\nconcatenating the rows in front of that σ. Figures 18-20 similarly show ˆBσ\nfor the various σ in Figures 3-4; only the distinct ˆBσ’s are shown. It may\nbe seen the ˆAσ’s are positive and nonincreasing. Thus all aσ are positive\nand unimodal, and hence, of the form (30) with s(σ) = 0. All ˆBσ’s are\npositive and nonincreasing, except for σ = 121, 1212 and 21212, for which\neach ˆBσ is positive and unimodal except at the tail. Thus all bσ, for σ ̸=\n121, 1212, 21212, are positive and unimodal, and hence of the form (31) with\n ̄s(σ) = 0. For σ = 121, 1212, 21212, bσ seems to be of the form (31) with\n ̄s(σ) = 1, both b0\nσ and b1\nσ being positive and unimodal, b0\nσ being the dominant\npolynomial that accounts for bσ’s mostly positive and unimodal behaviour,\nand b1\nσ the error polynomial that accounts for the deviation at the tail.\nThe (co)multiplicative structural constants cb′′\nb,b′ and db,b′\nb′′ for the canoni-\ncal basis of the braided exterior algebra ∧H,r\nq\n[X], which occur in the Laplace\nrelations for the general nonstandard quantum group GH\nq (cf. Theorem 4.5),\nare akin to the structure constants aσ and bσ in (28) and (29). Hence, we can\nexpect a similar positive topological interpretation for cb′′\nb,b′ and db,b′\nb′′ (but not\nnecessarily unimodality since cb′′\nb,b′ and db,b′\nb′′\nneed not be −-invariant). The\nexperimental evidence in [GCT8] suggests that the structure constants as-\nsociated with the canonical bases of the matrix coordinate ring of GH\nq and\nthe ring BH\nq defined there may also have similar positive topological inter-\npretations (additionally unimodal for BH\nq ).\n7.2\nExample 2\nNow we verify the duality and reciprocity conjectures for the special case of\nthe Kronecker problem (Section 4.1), when H = GL(V )×GL(W), V = W =\nC2 and G = GL(X), X = V ⊗W ∼= C4, and r = 4. Thus Gq = GLq(C4), and\nHq = GLq(C2) × GLq(C2). Let B = BH\nr be the nonstandard algebra in this\ncase and Pi = p+,H\nX,i , Qi = p−,X\nX,i , i < r, the positive and negative projection\noperators as in Section 5. Let Pi and Qi be the rescaled versions of Pi and\nQi as defined in [GCT4]. Then B is generated by Pi, or equivalently, Qi.\nThe explicit generating relations among Pi’s and Qi’s turn out to be very\n29"},{"page":30,"text":"1, (q4+1)(q4−q2+1)(q2+1)\nq5\n2, q\n q4 + 1\n \n5, q4+1\nq2\n3, (q2+1)(q8−q6+q4−q2+1)\nq5\n6,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, q2+1\nq\n4, q8−q6+q4+1\nq3\n7,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n10,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n13, q4+1\nq2\n2, q4+1\nq2\n5, q4+1\nq5\n3, (q2+1)\n2(q−1)(q+1)\nq2\n6, 2 q2+1\nq\n9, −(q2+1)\n2(q−1)(q+1)\nq2\n4,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n7, 2 q8+q6−2 q2+1\nq5\n10, −q8−q6−2 q4−q2+1\nq4\n13, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n8, (q2+1)(q8−q6+q4−q2+1)\nq5\n11, (q2+1)\n2(q−1)(q+1)\nq4\n14, q2+1\nq\n3, q2+1\nq\n6, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq4\n9, (q2+1)(q8−q6+q4−q2+1)\nq5\n4,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq\n7, −q8−q6−2 q4−q2+1\nq4\n10, q8−2 q6+q2+2\nq3\n13,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq2\n8,\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n11, 2 q2+1\nq\n14, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq2\n12, q\n q4 + 1\n \n15, q4+1\nq2\n4, q4+1\nq2\n7, −\n(q−1)(q+1)(q2+q+1)(q2−q+1)\nq5\n10,\n(q−1)2(q+1)2(q2+q+1)(q2−q+1)\nq6\n13, q8+q4−q2+1\nq5\n8, q2+1\nq\n11, −(q2+1)\n2(q−1)(q+1)\nq4\n14, (q2+1)(q8−q6+q4−q2+1)\nq5\n12, q4+1\nq2\n15, q4+1\nq5\n16, (q4+1)(q4−q2+1)(q2+1)\nq5\n \n \nFigure 1: P-matrix\n30"},{"page":31,"text":"σ\naσ\n1\n−(q12 + q10 + 2 q8 + 2 q4 + q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2/q36\n121\n(1 + 7 q4 + q2 + 5 q6 + 18 q8 + 21 q42 −107 q20 + q50 −107 q32 + 73 q14 + 187 q18 −14 q16 + 402 q26\n−197 q28 + 20 q40 + 187 q34 + 73 q38 −197 q24 + 328 q30 + q52 + 328 q22 + 7 q48 + 5 q46 + 18 q44\n+20 q12 + 21 q10 −14 q36)(q2 + 1)2(q4 −q2 + 1)2/q32\n12121\n−(1 + 8 q4 + 3 q2 + 4 q6 + 33 q8 + 12 q42 + 80 q20 + 3 q50 + 80 q32 + 27 q14 + 113 q18 + 115 q16 + 360 q26\n−9 q28 + 76 q40 + 113 q34 + 27 q38 −9 q24 + 253 q30 + q52 + 253 q22 + 8 q48 + 4 q46 + 33 q44 + 76 q12\n+12 q10 + 115 q36)/q26\n1212121\n(3 q36 + 2 q34 + 8 q32 + 5 q30 + 17 q28 + 30 q26 + 11 q24\n+61 q22 −15 q20 + 108 q18 −15 q16 + 61 q14 + 11 q12 + 30 q10 + 17 q8 + 5 q6 + 8 q4 + 2 q2 + 3)/q18\n121212121\n−(q20 + 3 q18 + q16 + 5 q14 −2 q12 + 16 q10 −2 q8 + 5 q6 + q4 + 3 q2 + 1)/q10\n12121212121\n1\nFigure 2: Coefficients of the basic generating relation among Qi’s\n31"},{"page":32,"text":"σ\nbσ\n∅\n−(q2 + 1)5(q4 + 1)3(q4 −q2 + 1)6(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)(q6 −q3 + 1)\n×(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)(2 q8 −2 q6 + 3 q4 −2 q2 + 2)\n×(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q51\n2\n(q2 + 1)4(q4 −q2 + 1)5(q4 + 1)2(q4 + q3 + q2 + q + 1)(q4 −q3 + q2 −q + 1)\n×(q6 −q3 + 1)(q6 + q3 + 1)(q6 −q5 + q4 −q3 + q2 −q + 1)(q6 + q5 + q4 + q3 + q2 + q + 1)\n×(2 q8 −2 q6 + 3 q4 −2 q2 + 2)(q2 + q + 1)2(q2 −q + 1)2(q −1)4(q + 1)4/q46\n1\n(q2 + 1)4(q4 −q2 + 1)4(q4 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q46\n12\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n21\n−(q2 + 1)3(q4 + 1)(q4 −q2 + 1)3(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54\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 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q41\n212\n(q2 + 1)2(q4 −q2 + 1)2(2 + 7 q4 −4 q2 −9 q6 + 11 q8 −10 q42 −4 q58 + 7 q56 −9 q54 + 12 q20\n−12 q50 −11 q32 −12 q14 −10 q18 + 13 q16 + 16 q26 −11 q28 + 12 q40 + 16 q34 + q38 + q24 + 28 q30 + 11 q52\n+q22 + 13 q48 −12 q46 + 2 q60 + 13 q44 + 13 q12 −12 q10 + q36)/q36\n121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(2 + 2 q4 −4 q2 −2 q6 −2 q8 −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 −3 q12 −9 q10 + q36)/q36\n1212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\nFigure 3: The first eight terms of the basic generating relation among Pi’s\n32"},{"page":33,"text":"σ\nbσ\n2121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32\n−13 q14 −48 q18 + q16 −110 q26 + 53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22\n+2 q48 −2 q46 −2 q44 −3 q12 −9 q10 + q36)/q31\n21212\n(2 + 2 q4 −4 q2 −2 q6 −2 q8 −9 q42 + 27 q20 −4 q50 + 27 q32 −13 q14 −48 q18 + q16 −110 q26\n+53 q28 −3 q40 −48 q34 −13 q38 + 53 q24 −77 q30 + 2 q52 −77 q22 + 2 q48 −2 q46 −2 q44 −3 q12\n−9 q10 + q36)/q26\n12121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q26\n121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q21\n2121212\n(3 q16 + 2 q14 + 14 q8 + 2 q2 + 3)(q8 + q4 −q2 + 1)(q8 −q6 + q4 + 1)/q16\n1212121\n−(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q18\n12121212\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n21212121\n(q2 + 1)(q4 + 1)(q4 −q2 + 1)(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q13\n212121212\n−(3 q16 −q14 + 3 q12 −3 q10 + 12 q8 −3 q6 + 3 q4 −q2 + 3)/q8\n121212121\n(q2 + 1)2(q4 + 1)2(q4 −q2 + 1)2/q10\n1212121212\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n2121212121\n−(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q5\n21212121212\n1\nFigure 4: The last fourteen terms of the basic generating relation among Pi’s\n33"},{"page":34,"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 5: First nine coefficients of u0\n34"},{"page":35,"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 6: Last twelve coefficients of u0\n35"},{"page":36,"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 7: Coefficients of u1\n1\n36"},{"page":37,"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 8: Coefficients of u1\n3\n37"},{"page":38,"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 9: Coefficients of u1\n4\n38"},{"page":39,"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 10: Coefficients of u1\n5\n39"},{"page":40,"text":"g1\n−1/2 −3 q28−4 q24+2 q22−10 q20+2 q18−4 q16−3 q12+x\nq20\ng3\n(q4+1)\n2\nq4\ng4\n(q2+1)\n2(q4−q2+1)\n2\nq6\ng5\n(q2+1)\n2(q8−q6+q4−q2+1)\n2\nq10\nFigure 11: The elements gi\ncomplicated. For example, Figures 21-23 reproduced from [GCT4] shows\na typical generating relation among Qi’s with 74 terms. There are several\ndozen such relations. Because of the nature of these generating relations,\nthere is no good “standard monomial basis” for B as for the Hecke algebra or\nfor the r = 3 case in Section 7.1.1. Fortunately, this makes no difference as\nfar as duality and reciprocity is concerned, as we shall see here, and also as\nfar as existence of a canonical basis is concerned, as we shall see in [GCT8].\nIt was verified by computer that B is of dimension 114 [GCT4]. Since it\nis semisimple, it admits a Wederburn structure decomposition. It turns out\nthat a complete Wederburn structure decomposition of the form (13) works\nover Q(q) itself; i.e., no algebraic extension of Q(q) is necessary here, just\nas in the case of Hecke algebras. This may be conjectured to be the case\nfor the Kronecker problem in general, though it is not so for the plethysm\nproblem in general as we already saw in Section 7.1.\nSo let\nB = ⊗iTi,L ⊗Ti,R,\n(38)\nbe the complete Wederburn structure decomposition of B, where Ti = Ti,L\nranges over all irreducible left B-modules.\n7.2.1\nIrreducible representations\nWe describe these Ti next. There are two distinct irreducible representations\nof B of dimension 1, 2, 3 and 5 each, and one of dimension 6. Since\n114 = 12 + 12 + 22 + 22 + 32 + 32 + 52 + 52 + 62,\nthis is consistent with the Wederburn structure decomposition in (38).\n40"},{"page":41,"text":"σ\nCoefficient\n1\n1/2 (q2 + 1)5(q4 −q2 + 1)3(q8 −q6 + q4 −q2 + 1)2(q4 + 1)3(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q45\n12\n−1/2 (q2 + 1)4(q4 −q2 + 1)2(q8 −q6 + q4 −q2 + 1)2(q4 + 1)2(−3 q28 −4 q24 + 2 q22\n−10 q20 + 2 q18 −4 q16 −3 q12 + x)/q40\n121\n−1/2 (q2 + 1)3(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)(q4 + 1)(q4 −q2 + 1)/q41\n1212\n1/2 (q2 + 1)2(x + 516 q32 −462 q38 −47 q22 + q20 + 84 q24 + q48 −210 q42 + 84 q44 −47 q46 −210 q26\n−2 q18 −9 q16 + 3 q14 + xq28 −xq26 + 4 xq22 + xq24 −24 xq12 + 19 xq10 + 30 xq14 −9 xq8 −24 xq16\n+4 xq6 −9 xq20 + xq4 −xq2 + 19 xq18 + 295 q28 + 295 q40 −604 q34 + 516 q36 −2 q50 + 3 q54 −3 q56\n−9 q52 −462 q30 −3 q12)/q36\n12121\n1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)\n×(q2 + 1)(q4 + 1)(q4 −q2 + 1)/q35\n121212\n−1/2 (x + 2 q32 −18 q38 −18 q22 −24 q20 −22 q24 −3 q48 −q42 −9 q44 −2 q46 −45 q26 −q18 −9 q16\n−2 q14 + 6 xq10 + xq14 + xq16 + xq6 + xq20 + xq4 + 2 q28 −24 q40 −45 q34 −22 q36 −92 q30 −3 q12)/q30\n1212121\n−1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)(q2 + 1)\n×(q4 + 1)(q4 −q2 + 1)/q25\n12121212\n1/2 (−2 q10 −22 q20 + 2 q18 −2 q14 −3 q12 −6 q16 −2 q30 + 2 q22 −6 q24 −3 q28 −2 q26 + x)/q20\n121212121\n−(q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5\n1212121212\n1\nFigure 12: Nonzero coefficients of z0\n41"},{"page":42,"text":"Monomial\nCoefficient\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24 + 11 q48\n−45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8 + 8 xq16\n−7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q51\nx1 ⊗x2 ⊗x0\n1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14 + 19 xq8\n+8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36 −310 q30 + 11 q12)/q46\nx0 ⊗x3 ⊗x0\n−1/2 (q + 1)(q −1)(q4 + 1)(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(3 x + 727 q32 −500 q38\n−330 q22 + 191 q20 + 460 q24 + 59 q48 −359 q42 + 192 q44 −110 q46 −603 q26 −138 q18 + 76 q16 −35 q14\n−20 xq22 + 5 xq24 + 70 xq12 −58 xq10 −77 xq14 + 45 xq8 + 48 xq16 −46 xq6 + 23 xq20 + 30 xq4 −15 xq2\n−32 xq18 + 587 q28 + 402 q40 −780 q34 + 584 q36 −44 q50 + 11 q52 −685 q30 + 7 q12)/q53\nx2 ⊗x0 ⊗x1\n1/2 (q4 + 1)2(q2 −q + 1)(q2 + q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20\n+117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34\n+117 q36 −310 q30 + 11 q12)/q48\nx1 ⊗x1 ⊗x1\n−1/2 (q2 + 1)2(q4 + 1)(q −1)(q + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22\n+91 q20 + 117 q24 + 11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10\n−7 xq14 + 19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q47\nx0 ⊗x2 ⊗x1\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(−3 x −1316 q32 + 1364 q38 + 419 q22 −200 q20 −577 q24\n−464 q48 + 957 q42 −617 q44 + 613 q46 + 748 q26 + 5 q30x + 134 q18 −76 q16 + 35 q14 −25 xq28 + 35 xq26\n+67 xq22 −28 xq24 −117 xq12 + 97 xq10 + 110 xq14 −48 xq8 −85 xq16 + 44 xq6 −112 xq20 −30 xq4 + 15 xq2\n+123 xq18 −816 q28 −1224 q40 + 1325 q34 −1132 q36 + 257 q50 + 85 q54 −55 q56 + 11 q58 −108 q52\n+1220 q30 −7 q12)/q50\nFigure 13: First five nonzero coefficients of a ∈X⊗3\nq\n42"},{"page":43,"text":"Monomial\nCoefficient\nx0 ⊗x0 ⊗x2\n−1/2 (q2 + 1)(q2 + q + 1)(q2 −q + 1)(q4 + 1)(q8 + 1)2(5 x + 173 q32 −192 q38 −192 q22 + 91 q20 + 117 q24\n+11 q48 −45 q42 + 24 q44 −37 q46 −131 q26 −45 q18 + 24 q16 −37 q14 + 19 xq12 −40 xq10 −7 xq14\n+19 xq8 + 8 xq16 −7 xq6 + 5 xq20 + 8 xq4 −17 xq2 −17 xq18 + 173 q28 + 91 q40 −131 q34 + 117 q36\n−310 q30 + 11 q12)/q46\nx0 ⊗x1 ⊗x2\n1/2 (q4 + 1)(q2 + q + 1)(q2 −q + 1)(q8 + 1)2(3 x + 951 q32 −1060 q38 −363 q22 + 176 q20 + 449 q24 + 248 q48\n−592 q42 + 395 q44 −451 q46 −532 q26 −97 q18 + 65 q16 −35 q14 + 8 xq28 −27 xq26 −31 xq22 + 16 xq24\n+81 xq12 −85 xq10 −62 xq14 + 40 xq8 + 59 xq16 −27 xq6 + 64 xq20 + 25 xq4 −15 xq2 −97 xq18 + 654 q28\n+797 q40 −898 q34 + 828 q36 −129 q50 −61 q54 + 18 q56 + 52 q52 −998 q30 + 7 q12)/q49\nx0 ⊗x0 ⊗x3\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q50\nFigure 14: Last four nonzero coefficients of a\n43"},{"page":44,"text":"Monomial\nCoefficient\nx3 ⊗x0 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 + q + 1)3(q2 −q + 1)3(5 x + 1214 q32 −847 q38 −525 q22\n+289 q20 + 714 q24 + 107 q48 −525 q42 + 289 q44 −178 q46 −847 q26 −178 q18 + 107 q16 −55 q14 −25 xq22\n+5 xq24 + 130 xq12 −113 xq10 −113 xq14 + 75 xq8 + 75 xq16 −60 xq6 + 45 xq20 + 45 xq4 −25 xq2 −60 xq18\n+920 q28 + 714 q40 −1139 q34 + 920 q36 −55 q50 + 11 q52 −1139 q30 + 11 q12)/q57\nx2 ⊗x1 ⊗x0\n−1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−5 x −1170 q32 + 977 q38 + 425 q22\n−207 q20 −565 q24 −371 q48 + 636 q42 −451 q44 + 397 q46 + 523 q26 + 5 q30x + 106 q18 −89 q16 + 55 q14\n−20 xq28 + 20 xq26 + 43 xq22 −20 xq24 −119 xq12 + 97 xq10 + 71 xq14 −45 xq8 −59 xq16 + 28 xq6 −95 xq20\n−37 xq4 + 25 xq2 + 87 xq18 −676 q28 −1021 q40 + 891 q34 −849 q36 + 173 q50 + 52 q54 −44 q56 + 11 q58\n−82 q52 + 1002 q30 −11 q12)/q56\nx1 ⊗x2 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q2 + q + 1)2(q2 −q + 1)2(q8 + 1)2(−8 x −1447 q32 + 1598 q38 + 590 q22\n−304 q20 −648 q24 −496 q48 + 927 q42 −742 q44 + 696 q46 + 778 q26 + 5 q30x + 153 q18 −89 q16 + 72 q14\n−25 xq28 + 40 xq26 + 58 xq22 −40 xq24 −124 xq12 + 138 xq10 + 96 xq14 −76 xq8 −114 xq16 + 39 xq6 −116 xq20\n−33 xq4 + 32 xq2 + 152 xq18 −1064 q28 −1329 q40 + 1319 q34 −1386 q36 + 244 q50 + 96 q54 −55 q56\n+11 q58 −134 q52 + 1516 q30 −18 q12)/q55\nx0 ⊗x3 ⊗x0\n1/2 (q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(−3 x −1693 q32 + 1146 q38 + 641 q22 −404 q20\n−1020 q24 −178 q48 + 814 q42 −567 q44 + 296 q46 + 1268 q26 + 266 q18 −155 q16 + 53 q14 + 5 xq26 + 45 xq22\n−25 xq24 −164 xq12 + 123 xq10 + 170 xq14 −108 xq8 −131 xq16 + 96 xq6 −60 xq20 −65 xq4 + 23 xq2\n+78 xq18 −1376 q28 −1006 q40 + 1709 q34 −1491 q36 + 107 q50 + 11 q54 −55 q52 + 1449 q30 −7 q12)/q58\nx2 ⊗x0 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(5 x + 964 q32 −627 q38\n−431 q22 + 224 q20 + 582 q24 + 100 q48 −431 q42 + 224 q44 −143 q46 −627 q26 −143 q18 + 100 q16 −55 q14 −25 xq22\n+5 xq24 + 108 xq12 −87 xq10 −87 xq14 + 50 xq8 + 50 xq16 −45 xq6 + 42 xq20 + 42 xq4 −25 xq2 −45 xq18 + 645 q28\n+582 q40 −860 q34 + 645 q36 −55 q50 + 11 q52 −860 q30 + 11 q12)/q53\nFigure 15: First five nonzero coefficients of b\n44"},{"page":45,"text":"Monomial\nCoefficient\nx1 ⊗x1 ⊗x1\n1/2 (q2 + 1)2(q2 −q + 1)(q2 + q + 1)(q4 + 1)(q −1)2(q + 1)2(q8 + 1)2(3 x + 277 q32 −621 q38 −165 q22\n+97 q20 + 83 q24 + 125 q48 −291 q42 + 291 q44 −299 q46 −255 q26 −47 q18 −17 q14 + 5 xq28 −20 xq26 −15 xq22\n+20 xq24 + 5 xq12 −41 xq10 −25 xq14 + 31 xq8 + 55 xq16 −11 xq6 + 21 xq20 −4 xq4 −7 xq2 −65 xq18 + 388 q28\n+308 q40 −428 q34 + 537 q36 −71 q50 −44 q54 + 11 q56 + 52 q52 −514 q30 + 7 q12)/q52\nx0 ⊗x2 ⊗x1\n−1/2 (q2 + 1)(q4 + 1)(q −1)2(q + 1)2(q2 −q + 1)2(q2 + q + 1)2(q8 + 1)2(−3 x −1651 q32 + 1803 q38\n+486 q22 −287 q20 −679 q24 −567 q48 + 1181 q42 −1013 q44 + 814 q46 + 920 q26 + 5 q30x + 147 q18 −72 q16\n+35 q14 −25 xq28 + 45 xq26 + 78 xq22 −60 xq24 −133 xq12 + 122 xq10 + 138 xq14 −87 xq8 −167 xq16\n+49 xq6 −131 xq20 −28 xq4 + 15 xq2 + 170 xq18 −1270 q28 −1556 q40 + 1669 q34 −1825 q36 + 296 q50\n+107 q54 −55 q56 + 11 q58 −178 q52 + 1547 q30 −7 q12)/q55\nx1 ⊗x0 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10\n−15 xq14 + 22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q51\nx0 ⊗x1 ⊗x2\n−1/2 (q4 + 1)2(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32 −94 q38 −220 q22\n+132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14 + 22 xq8\n+3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q54\nx0 ⊗x0 ⊗x3\n1/2 (q2 + 1)(q4 + 1)2(q4 −q2 + 1)(q −1)2(q + 1)2(q8 + 1)2(q2 −q + 1)3(q2 + q + 1)3(3 x + 275 q32\n−94 q38 −220 q22 + 132 q20 + 275 q24 −35 q42 + 7 q44 −279 q26 −94 q18 + 65 q16 −35 q14 + 25 xq12 −26 xq10 −15 xq14\n+22 xq8 + 3 xq16 −26 xq6 + 25 xq4 −15 xq2 + 250 q28 + 65 q40 −220 q34 + 132 q36 −279 q30 + 7 q12)/q55\nFigure 16: Last five nonzero coefficients of b\n45"},{"page":46,"text":"Let Sq,λ denote the q-Specht module of the Hecke algebra Hr(q) for\nthe partition λ, and KLλ its Kazhdan-Lusztig basis ordered appropriately.\nSince, in this case, B = BH\nr (q) ⊆Hr(q)⊗Hr(q), the tensor product Sq,λ⊗Sq,μ\nis a representation of B. In particular,\nTq,λ = Sq,λ ⊗Sq,(r) ∼= Sq,(r) ⊗Sq,λ,\nwhere Sq,(r) is the trivial one dimensional q-Specht module, is an irreducible\nB-module, which specializes at q = 1 to the Specht module Sλ of the sym-\nmetric group Sr.\nLet\nT0\n=\nTq,(4),\nT1\n=\nTq,(1,1,1,1),\nT2\n=\nTq,(2,2),\nT3\n=\nTq,(2,1,1),\nT4\n=\nTq,(3,1),\nT5\n=\nSq,(3,1) ⊗Sq,(2,2) ∼= Sq,(2,1,1) ⊗Sq,(2,2).\n(39)\nThese are irreducible B-modules. Their dimensions are 1, 1, 2, 3, 3 and 6\nrespectively.\nTo get the other two dimensional irreducible B-module, we analyze how\nthe tensor product Sq,(2,2) ⊗Sq,(2,2) decomposes as a B-module. It decom-\nposes as:\nSq,(2,2) ⊗Sq,(2,2) ∼= Tq,(4) ⊕Tq,(1,1,1,1) ⊕T6,\nwhere T6 is the other two dimensional irreducible B-module that we were\nlooking for. Explicitly, a basis of T6 in terms of the Kazhdan-Lusztig basis\nKL(2,2) ⊗KL(2,2) of Sq,(2,2) ⊗Sq,(2,2) is given by the rows of the matrix\n\"\n1\n1+q\n2q1/2\n1+q\n2q1/2\n0\n0\n1+q\n2q1/2\n1+q\n2q1/2\n1\n#\n.\nMatrix representations of the right action of the generators Qi’s of B on\nthis basis are:\nQ1 = Q3 =\n (1 + q)2/q\n0\n(1 + q2)/q\n0\n \nQ2 =\n 0\n(1 + q2)/q\n0\n(1 + q)2/q\n \n46"},{"page":47,"text":"The specialization of T6 at q = 1 is isomorphic to the Specht module\nS(2,2) of S4. But T6 is nonisomorphic to T2, whose specialization at q = 1 is\nthe same.\nTo get the five dimensional irreducible B-modules, we analyze how the\ntensor products Sq,(2,1,1) ⊗Sq,(2,1,1) and Sq,(3,1) ⊗Sq,(2,1,1) decompose as B-\nmodules. We have\nSq,(2,1,1) ⊗Sq,(2,1,1) ∼= Tq,(2,1,1) ⊕Tq,(4) ⊕T7,\nwhere T7 is the first five dimensional irreducible B-representation that we\nwere looking for. Explicitly, its basis in terms of the Kazhdan-Lusztig basis\nKL(2,1,1) ⊗KL(2,1,1) is given by the rows of the matrix:\nw1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n0\n0\n0]\nw2 = [\n−1\n−(1 + q)/(2q1/2)\n0\n0\n(1 + q)/(2q1/2)\n1]\nw3 = [\n0\n0\n−(1 + q)2/(2q)\n0\n−(1 + q)/(2q1/2)\n0]\nv1 = [\n0\n−(1 + q)/(2q1/2)\n−(1 + q)2/(2q)\n−1\n−(1 + q)/(2q1/2)\n0]\nv2 = [\n1\n(1 + q)/(2q1/2)\n1\n0\n(1 + q)/(2q1/2)\n1]\nMatrix representations of the right action of Qi’s in this basis are:\nQ1 =\n \n \n(1 + q)2/q\n0\n0\n0\n0\n(1 + q2)/q\n0\n0\n−(1 + q2)/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n(1 + q)2/q\n0\n(q −1)2/q\n0\n0\n−(1 + q2)/q\n0\n \n \nQ2 =\n \n \n0\n(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n(1+q)2\nq\n0\n0\n0\n0\n−(1+q)2\n2q\n0\n0\n−(1+q)2\n2q\n0\n0\n0\n0\n−1+q2\nq\n0\n0\n0\n0\n(1+q)2\nq\n \n \nQ3 =\n \n \n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n−(1 + q2)/q\n(1 + q2)/q\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n0\n0\n(1 + q)2/q\n0\n0\n0\n(q −1)2/q\n−(1 + q2)/q\n0\n \n \n47"},{"page":48,"text":"Let V denote the span of the vectors v1 and v2, and V (1) its specialization\nat q = 1. It can be checked that V (1) is isomorphic to the Specht module\nS(2,2) of S4, and the quotient T7(1)/V , where T7(1) denotes the specialization\nof T7 at q = 1, is isomorphic to the Specht module S(3,1) of S4.\nFinally,\nSq,(3,1) ⊗Sq,(2,1,1) ∼= Tq,(3,1) ⊕Tq,(1,1,1,1) ⊕T8,\nwhere T8 ̸∼= T7 is the second five dimnsional irreducible B-representation\nthat we were looking for. Its basis and representation matrices are similar.\nThis specifies all irreducible representations of B.\n7.2.2\nDuality\nUsing the explicit representations Ti above, the Wederburn structure decom-\nposition (38) of B was explicitly determined with the help of a computer.\nThe explicit bases of the structure components Ui = Ti,L ⊗Ti,R in (38) are\nfar too complex to be given here.\nFix any ui ∈Ui, 0 ≤i ≤8, and let Wi = X⊗r\nq\n· ui be the corre-\nsponding left representation of the nonstandard quantum group GH\nq . Com-\nputer experiments indicate that these are nonisomorphic irreducible repre-\nsentations of GH\nq with the following decompositions as Hq-modules, Hq =\nGLq(C2)×GLq(C2). (Recall that Vq,λ(n) is the q-Weyl module of GLq(Cn)).\nW0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nW1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW2\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2),\nW3\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2),\nW4\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2),\nW5\n∼=\nVq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nW6\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nW7\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2).\nTheir dimensions are 35, 1, 10, 9, 30, 6, 1 and 9, respectively. The module\nW8 turns out to be zero when dim(V ) = dim(W) = 2, as here; however, it\nwould be nonzero for general dim(V ) and dim(W). Furthermore,\nX⊗4\nq\n=\nM\ni\nWi ⊗Ti,\n48"},{"page":49,"text":"in accordance with the duality conjecture.\nRemark: These computations are not final. The main problem is that the\nsymbolic computations needed here are too heavy for MATLAB/Maple to\nhandle. Hence, in some of the computations q was set to a fixed real value\n(such as .5). This introduces floating point errors in various calculations.\nAs far as we can see, this does not affect the decomposition above. But this\nhas to be double checked by other means.\n7.2.3\nReciprocity\nLet mi\nμ denote the multiplicity of the Specht module Sμ of S4 in Ti. Then\nit can be verified that\nm0\n(4) = 1,\nm1\n(1,1,1,1) = 1,\nm2\n(2,2) = 1,\nm3\n(2,1,1) = 1,\nm4\n(3,1) = 1,\nm5\n(3,1) = m5\n(2,1,1) = 1,\nm6\n(2,2) = 1,\nm7\n(3,1) = m7\n(2,2) = 1,\nm8\n(2,1,1) = m8\n(2,2) = 1.\nAll other mi\nμ’s are zero. It can now be seen that, as Hq-modules, Hq =\n49"},{"page":50,"text":"GLq(2) × GLq(2), we have\nVq,(4)(4)\n∼=\nm0\n(4)W0\n∼=\nVq,(4)(2) ⊗Vq,(4)(2) ⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(3,1)(4)\n∼=\nm4\n(3,1)W4 ⊕m5\n(3,1)W5 ⊕m7\n(3,1)W7\n∼=\nVq,(3,1)(2) ⊗Vq,(4)(2) ⊕Vq,(4)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2) ⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2),\nVq,(2,2)(4)\n∼=\nm2\n(2,2)W2 ⊕m7\n(2,2)W7 ⊕m6\n(2,2)W6\n∼=\nVq,(4)(2) ⊗Vq,(2,2)(2) ⊕Vq,(2,2)(2) ⊗Vq,(4)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(2,2)(2),\nVq,(2,1,1)(4)\n∼=\nm3\n(2,1,1)W3 ⊕m5\n(2,1,1)W5\n∼=\nVq,(3,1)(2) ⊗Vq,(3,1)(2) ⊕Vq,(2,2)(2) ⊗Vq,(3,1)(2)\n⊕Vq,(3,1)(2) ⊗Vq,(2,2)(2),\nVq,(1,1,1,1)(4)\n∼=\nm1\n(1,1,1,1)W1\n∼=\nVq,(2,2)(2) ⊗Vq,(2,2)(2),\nin accordance with the reciprocity conjecture.\nWe are unable to verify the refined reciprocity conjecture on computer\nsince the necessary symbolic computations turn out to be beyond the reach\nof the desktop MATLAB/Maple.\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[Dl2]\nP. Deligne, La conjecture de Weil II, Publ. Math. Inst. Haut. ́Etud.\nSci. 52, (1980) 137-252.\n50"},{"page":51,"text":"σ\nˆAσ\n1\n17738\n17738\n17550\n17362\n16994\n16626\n16114\n15602\n14933\n14264\n13550\n12836\n12008\n11180\n10392\n9604\n8790\n7976\n7226\n6476\n5806\n5136\n4518\n3900\n3418\n2936\n2504\n2072\n1762\n1452\n1202\n952\n776\n600\n482\n364\n280\n196\n152\n108\n77\n46\n34\n22\n14\n6\n4\n2\n1\n121\n20322\n20322\n20083\n19844\n19354\n18864\n18211\n17558\n16668\n15778\n14890\n14002\n12934\n11866\n10916\n9966\n8962\n7958\n7092\n6226\n5470\n4714\n4047\n3380\n2895\n2410\n1982\n1554\n1287\n1020\n804\n588\n463\n338\n253\n168\n122\n76\n53\n30\n19\n8\n5\n2\n1\n12121\n9078\n9078\n8973\n8868\n8623\n8378\n8051\n7724\n7245\n6766\n6335\n5904\n5363\n4822\n4382\n3942\n3443\n2944\n2562\n2180\n1851\n1522\n1267\n1012\n831\n650\n506\n362\n284\n206\n151\n96\n70\n44\n28\n12\n7\n2\n1\n1212121\n1918\n1918\n1913\n1908\n1866\n1824\n1742\n1660\n1523\n1386\n1277\n1168\n1042\n916\n821\n726\n603\n480\n395\n310\n246\n182\n145\n108\n83\n58\n40\n22\n14\n6\n3\n121212121\n25032\n24784\n24124\n23136\n21978\n20808\n19710\n18768\n17934\n17160\n16358\n15440\n14384\n13168\n11849\n10484\n9139\n7880\n6725\n5692\n4765\n3928\n3170\n2480\n1868\n1344\n914\n584\n345\n188\n93\n40\n15\n4\n1\nFigure 17: Positivity and unimodality of ˆAσ’s\n51"},{"page":52,"text":"σ\nˆBσ\n∅\n390464\n389128\n385120\n378581\n369652\n358471\n345176\n330055\n313396\n295506\n276692\n257207\n237304\n217316\n197576\n178283\n159636\n141795\n124920\n109152\n94632\n81349\n69292\n58469\n48888\n40497\n33244\n27011\n21680\n17198\n13512\n10505\n8060\n6095\n4528\n3314\n2408\n1731\n1204\n812\n540\n357\n232\n147\n84\n43\n24\n16\n8\n2\n2\n102390\n101996\n100847\n98976\n96425\n93236\n89466\n85172\n80462\n75444\n70190\n64772\n59280\n53804\n48429\n43240\n38271\n33556\n29145\n25088\n21393\n18068\n15099\n12472\n10185\n8236\n6586\n5196\n4040\n3092\n2333\n1744\n1286\n920\n640\n440\n300\n200\n129\n76\n41\n24\n16\n8\n2\n1\n50420\n50420\n49799\n49178\n48066\n46954\n45325\n43696\n41665\n39634\n37420\n35206\n32782\n30358\n27969\n25580\n23303\n21026\n18902\n16778\n14947\n13116\n11553\n9990\n8713\n7436\n6455\n5474\n4724\n3974\n3416\n2858\n2490\n2122\n1831\n1540\n1350\n1160\n1031\n902\n779\n656\n582\n508\n441\n374\n313\n252\n213\n174\n144\n114\n86\n58\n47\n36\n27\n18\n11\n4\n4\n4\n2\n12\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 18: The vectors ˆBσ\n52"},{"page":53,"text":"σ\nˆBσ\n212\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\n121\n51252\n51252\n50661\n50070\n48941\n47812\n46219\n44626\n42589\n40552\n38328\n36104\n33645\n31186\n28756\n26326\n23948\n21570\n19376\n17182\n15217\n13252\n11581\n9910\n8522\n7134\n6030\n4926\n4106\n3286\n2664\n2042\n1632\n1222\n941\n660\n497\n334\n233\n132\n89\n46\n25\n4\n−4\n−12\n−10\n−8\n−6\n−4\n−4\n−4\n−2\n1212\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\n21212\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\n12121\n20922\n20922\n20625\n20328\n19815\n19302\n18558\n17814\n16848\n15882\n14868\n13854\n12740\n11626\n10537\n9448\n8430\n7412\n6509\n5606\n4830\n4054\n3439\n2824\n2349\n1874\n1526\n1178\n945\n712\n550\n388\n301\n214\n155\n96\n70\n44\n30\n16\n11\n6\n3\nFigure 19: The vectors ˆBσ (cont.)\n53"},{"page":54,"text":"σ\nˆBσ\n121212\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\n2121212\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\n1212121\n3800\n3800\n3735\n3670\n3573\n3476\n3326\n3176\n2974\n2772\n2567\n2362\n2138\n1914\n1692\n1470\n1277\n1084\n921\n758\n624\n490\n396\n302\n238\n174\n131\n88\n65\n42\n29\n16\n11\n6\n3\n12121212\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\n212121212\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\n1212121212\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 ˆBσ (cont)\n54"},{"page":55,"text":"[DJ]\nR. Dipper and G. James, Representations of Hecke algebras of\ngeneral linear groups, Proc. London Math. Soc. (3). 52 (1986), 20-\n52.\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\nat: http://ramakrishnadas.cs.uchicago.edu. Revised version to be\navailable here.\n[GCT8] K. Mulmuley, Geometric complexity theory VIII: On canonical\nbases for the nonstandard quantum groups, revised version under\npreparation.\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, 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[Kass]\nC. Kassel, Quantum groups, Springer-Verlag, 1995.\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.\n55"},{"page":56,"text":"[Li]\nP. Littelmann: A Littlewood-Richardson rule for symmetrizable\nKac-Moody Lie algebras, Invent. math. 116 (1994), 329-346.\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[Mc]\nI. Macdonald, Symmetric functions and Hall polynomials, Oxford\nScience Publications, 1995.\n[RTF]\nN. Reshetikhin, L. Takhtajan, L. Faddeev, Quantization of Lie\ngroups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225.\n[Ro]\nO. Rossi-Doria, A Uq(sl(2))-representation with no quantum sym-\nmetric algebra, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.\nRend. Lincei (9), Mat. Appl. 10 10 (1999), no. 1, 5-9.\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.\n56"},{"page":57,"text":"Number\nCoefficient\nσ\n1\n20.q0+104.q1+256.q2−113.q3−49.q4−113.q5+256.q6+104.q7+20.q8\n2.q3+12.q4+2.q5\n1\n2\n−16.q0−64.q1−128.q2−192.q3−224.q4−192.q5−128.q6−64.q7−16.q8\n2.q3+12.q4+2.q5\n2\n3\n−4.q0−16.q1−28.q2−32.q3−28.q4−16.q5−4.q6\n2.q3\n3\n4\n1.q0−4.q2+6.q4−4.q6+1.q8\n2.q3+2.q5\n12\n5\n−1.q0−18.q1−65.q2−128.q3−190.q4−220.q5−190.q6−128.q7−65.q8−18.q9−1.q10\n2.q4+12.q5+2.q6\n13\n6\n1.q0+5.q1+17.q2+36.q3+46.q4+46.q5+46.q6+36.q7+17.q8+5.q9+1.q10\n2.q4+2.q6\n21\n7\n7.q0+26.q1+75.q2+152.q3+174.q4+156.q5+174.q6+152.q7+75.q8+26.q9+7.q10\n2.q3+12.q4+4.q5+12.q6+2.q7\n23\n8\n−1.q0−8.q1−20.q2−24.q3−22.q4−24.q5−20.q6−8.q7−1.q8\n2.q3+2.q5\n32\n9\n−22.q0−92.q1−170.q2−200.q3−170.q4−92.q5−22.q6\n2.q2+12.q3+2.q4\n121\n10\n2.q0+2.q1+12.q2+14.q3+4.q4+14.q5+12.q6+2.q7+2.q8\n2.q3+2.q5\n132\n11\n−2.q0−12.q1−40.q2−52.q3−44.q4−52.q5−40.q6−12.q7−2.q8\n2.q3+12.q4+2.q5\n212\n12\n−1.q0−2.q1−12.q2−14.q3−6.q4−14.q5−12.q6−2.q7−1.q8\n2.q3+2.q5\n213\n13\n1.q0+22.q1+88.q2+170.q3+206.q4+170.q5+88.q6+22.q7+1.q8\n2.q3+12.q4+2.q5\n232\n14\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q2\n323\n15\n3.q0+6.q1+5.q2+4.q3+5.q4+6.q5+3.q6\n2.q2+2.q4\n1212\n16\n12.q0+32.q1+40.q2+32.q3+12.q4\n2.q1+12.q2+2.q3\n1213\n17\n−3.q0−2.q1−5.q2−12.q3−5.q4−2.q5−3.q6\n2.q2+2.q4\n1232\n18\n1.q0+4.q1+11.q2+16.q3+11.q4+4.q5+1.q6\n2.q3\n1321\n19\n8.q0+12.q1+24.q2+40.q3+24.q4+12.q5+8.q6\n2.q2+12.q3+2.q4\n1323\n20\n−6.q0−8.q1−4.q2−8.q3−6.q4\n2.q1+2.q3\n2121\n21\n−5.q0−4.q1−44.q2−60.q3−30.q4−60.q5−44.q6−4.q7−5.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n2123\n22\n−1.q0−5.q1−11.q2−14.q3−11.q4−5.q5−1.q6\n2.q3\n2321\n23\n−3.q0−6.q1−5.q2−4.q3−5.q4−6.q5−3.q6\n2.q2+2.q4\n2323\n24\n2.q0+4.q1+4.q2+4.q3+2.q4\n2.q2\n3212\n25\n−1.q0−4.q1−6.q2−4.q3−1.q4\n2.q2\n3213\n26\n6.q0+8.q1+4.q2+8.q3+6.q4\n2.q1+2.q3\n3232\n27\n16.q0+32.q1+16.q2\n2.q0+12.q1+2.q2\n12121\n28\n4.q0+8.q1+40.q2+8.q3+4.q4\n2.q1+12.q2+2.q3\n12123\n29\n−3.q0−8.q1−4.q2−8.q3+46.q4−8.q5−4.q6−8.q7−3.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n12132\n30\n−8.q0\n2.q0\n12321\nFigure 21: A relation in BH\n4 from GCT4\n57"},{"page":58,"text":"Number\nCoefficient\nσ\n31\n−4.q0−8.q1−40.q2−8.q3−4.q4\n2.q1+12.q2+2.q3\n12323\n32\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n13212\n33\n−9.q0−6.q1−55.q2+12.q3−55.q4−6.q5−9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n13232\n34\n9.q0+6.q1+55.q2−12.q3+55.q4+6.q5+9.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n21213\n35\n1.q0−1.q1+3.q2−6.q3+3.q4−1.q5+1.q6\n2.q2+2.q4\n21232\n36\n−1.q0+2.q2−1.q4\n2.q2\n21321\n37\n2.q0+3.q1+6.q2−3.q3−16.q4−3.q5+6.q6+3.q7+2.q8\n2.q2+12.q3+4.q4+12.q5+2.q6\n21323\n38\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n23213\n39\n−16.q0−32.q1−16.q2\n2.q0+12.q1+2.q2\n23232\n40\n3.q0+4.q1+2.q2+4.q3+3.q4\n2.q1+2.q3\n32121\n41\n8.q0\n2.q0\n32123\n42\n1.q0−2.q2+1.q4\n2.q2\n32132\n43\n−3.q0−4.q1−2.q2−4.q3−3.q4\n2.q1+2.q3\n32321\n44\n−8.q0−16.q1−8.q2\n2.q0+12.q1+2.q2\n121213\n45\n−1.q0−14.q1−15.q2−4.q3−15.q4−14.q5−1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n121232\n46\n−2.q0−4.q1−2.q2\n2.q1\n121321\n47\n−2.q0+4.q2−2.q4\n2.q1+12.q2+2.q3\n123213\n48\n8.q0+16.q1+8.q2\n2.q0+12.q1+2.q2\n123232\n49\n−1.q0−2.q1−1.q2\n2.q1\n132121\n50\n2.q0−4.q2+2.q4\n2.q1+12.q2+2.q3\n132123\n51\n2.q0+8.q1+12.q2+8.q3+2.q4\n2.q1+12.q2+2.q3\n212132\n52\n2.q0+4.q1+2.q2\n2.q1\n212321\n53\n1.q0+14.q1+15.q2+4.q3+15.q4+14.q5+1.q6\n2.q1+12.q2+4.q3+12.q4+2.q5\n212323\n54\n3.q0+8.q1+10.q2+8.q3+3.q4\n2.q1+12.q2+2.q3\n213212\n55\n−2.q0−8.q1−12.q2−8.q3−2.q4\n2.q1+12.q2+2.q3\n213232\n56\n−1.q0−2.q1−1.q2\n2.q1\n232121\n57\n−3.q0−8.q1−10.q2−8.q3−3.q4\n2.q1+12.q2+2.q3\n232132\n58\n1.q0+2.q1+1.q2\n2.q1\n232321\n59\n−2.q0−4.q1−2.q2\n2.q1\n321232\n60\n2.q0+4.q1+2.q2\n2.q1\n321323\nFigure 22: A relation in BH\n4 from GCT4 continued.\n58"},{"page":59,"text":"Number\nCoefficient\nσ\n61\n1.q0+2.q1+1.q2\n2.q1\n323212\n62\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1212132\n63\n2.q0\n2.q0\n1213213\n64\n1.q0−2.q1+1.q2\n2.q0+2.q2\n1213232\n65\n2.q0\n2.q0\n1232121\n66\n2.q0−4.q1+2.q2\n2.q0+12.q1+2.q2\n1232132\n67\n16.q1\n2.q0+12.q1+2.q2\n1321232\n68\n−2.q0\n2.q0\n1321323\n69\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2121323\n70\n−4.q0−8.q1−4.q2\n2.q0+12.q1+2.q2\n2123212\n71\n−16.q1\n2.q0+12.q1+2.q2\n2123213\n72\n−1.q0+2.q1−1.q2\n2.q0+2.q2\n2123232\n73\n−2.q0+4.q1−2.q2\n2.q0+12.q1+2.q2\n2132123\n74\n4.q0+8.q1+4.q2\n2.q0+12.q1+2.q2\n2321232\nFigure 23: A relation in BH\n4 from GCT4 continued.\n59"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"H = GLn(C) in the irreducible representation Vλ(G) of G = GL(X), where","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"X = Vμ = Vμ(H) is an irreducible representation of H.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"in what follows, we assume that the base field is C = C(q), q complex. But","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"When H = G, GH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"H = GL(V ) × GL(W), G = GL(X), X = V ⊗W with natural H-action, it","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"r = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"Vq,μ of Hq with highest weight μ; it is the usual quantization of X = Vμ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"q,λ ∼= Vλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"q,λ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"group Sr in Tq,α(1) = limq→1Tq,α, as defined in Section 6. Then V H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"tiplicity of the Specht module Sλ in the specialization of Tq,α at q = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"necker problem. For the sake of simplicity, we assume here that H = GL(V )","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"(type A). Let X = Vμ(H) be its irreducible polynomial representation. The","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"H = GL(V ) →G = GL(X).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"The standard quantum group Hq = GLq(V ) associated with GL(V ) can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"X,X be the ˆR matrix of Xq = Vq,μ considered as an Hq-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"I = P +,H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"X,X (u ⊗u) = (u ⊗u)P +,H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"X,X (u ⊗u) = (u ⊗u)P −,H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"X,X x1x2 = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"where x1 = x ⊗I and x2 = I ⊗x, and AH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"X,X x1x2 = 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"[X] = 0 when H = sl2(C) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"[X] ̸= 0, then we define the determinant of GH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"it can vanish, as it does for H = sl2(C), dim(X) = 4. The nonstandard","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"q (X)) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"In the standard setting, the q-Schur algebra Ar = Ar(q) is defined to be the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"= AH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"r (q) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"i = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"vivj = −q−1vjvi,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"problem [GCT4] when H = GL(V ) × GL(W) and X = V ⊗W, with the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"We begin by recalling that when V = W ∗the braided symmetric al-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"q [X] = CH[W ∗⊗W] is isomorphic to the matrix coordinate ring","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"q [X] = CH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"W], H = GL(V ) × GL(W), has an (upper) canonical basis.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"q [V ⊗W], H = GL(V ) ×","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"bb′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"b′′ =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"tively with r′′ = r + r′, the sign ǫ′(b, b′, b′′) is 1 or −1, and db,b′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"q (X), H = GL(V ) ×","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"r and r′ with r′′ = r + r′. The coefficients of this Laplace expansion are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"H = GL(V ).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"Choose a standard embedding of X = Vμ(H) in V ⊗d, where d is the size","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"X = Vμ(H). Let zμ ∈Hd(q) be the quantization of cμ such that V ⊗d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"· zμ ∼=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"Xq = Vq,μ. Here Vq denotes the quantization of V and Vq,μ the irreducible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"found in [DJ]. Let Zq = V ⊗d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"on Zq ⊗Zq = V ⊗2d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"Now consider the right action of Hs(q), s = dr, on Z⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"= V ⊗s","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"commutes with the left action of Hq = GLq(V ). Let rZ,i ∈Hs(q), 1 ≤i < r,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"X,i = zλ,i · zλ,i+1 · rZ,i,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"Hr(q) with the standard quantum group Gq = GLq(X).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"X⊗r =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"specialization Tq,α(1) of Tq,α at q = 1. In this context, it may be remarked","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"that though B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"r (q) is semisimple, its specialization B(1) at q = 1 need","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"B =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"U α = Tq,α,L ⊗Tq,α,R,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"ture decomposition. Here we are assuming that the base field is C = C(q),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"setting of Hecke algebras, K = Q(q) suffices. This need not be so in the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"a B(1)-module of the specialization Tq,α,R(1) of Tq,α,R at q = 1:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"ˆTα,0 ⊂ˆTα,1 ⊂· · · ⊂ˆTα,l(α) = Tq,α,R(1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"Zα,0 ⊂Zα,1 ⊂· · · ⊂Zα,l(α) = Zα,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"1. The specialization Zα,i(1) of Zα,i at q = 1 is a basis of ˆTα,i.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"partition such that ˆTα,i/ ˆTα,i−1 ∼= Sλα,i as a B(1)-module (or equiv-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"Wq,α,i = ∪jX⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"We define its specialization W1,α,i at q = 1, also denoted by Wq,α,i(1),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"of such limits at q = 1. Then, W1,α,i is a left G-module contained","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"W1,α,i = Vλ ⊗Sλ ⊆X⊗r,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"where, for a given α, i ranges over all indices such that λα,i = λ.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"Let H = GL(C2), H = gl(C2), X = V(3)(H) is its four dimensional irre-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"ducible representation, and G = GL(X) = GL(C4). Then Hq = GLq(C2),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"Gq = GLq(C4), and Hq = glq(C2). We shall verify duality and reciprocity","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"in this case for r = 3. This example is interesting because, as shown in [BZ],","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"case dim(X) = 4, since this seems to be the gist.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"Let ˆR = ˆRH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"X,X be the ˆR-matrix associated with Xq. Let P = P H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"Q = QH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"X,X, respectively. Let xi = f ix0, where f is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"P = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"f = (q4 + 1)(q4 −q2 + 1)(q2 + 1)/q5","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"representation: the entry (j, v) in the i-row in Figure 1 means P(i, j) = v.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"Thus the entry (5, (q4 + 1)/q2) in the second row there means P(2, 5) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"Q = fQ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"i = fQi,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"i = fPi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"aσQσ = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"aσ ∈Q[q, q−1] are as specified there, and, for a string σ = i1i2 · · · , Qσ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"bσPσ = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"Let B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"i = fPi, or equivalently, to the algebra generated by Qi’s subject","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"i = fQi.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"disc =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"x = disc1/2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"Since disc is not a square, x does not belong to Q(q). Let K = Q(q)[x] be","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"B =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"Ui = Ti,L ⊗Ti,R,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"ui =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"= u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"= Q2u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"B =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"B =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"Qju0 = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"for j = 1, 2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"i =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"Let Ti(1) denote the specialization of Ti at q = 1. It is a representation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"of B(1), the specialization of B at q = 1. Then T0(1) corresponds to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"At q = 1, the values of f = f(q) and gi = gi(q) are as follows:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"f(1) = g1(1) = g3(1) = g4(1) = g5(1) = 4, and g2(1) = 16.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"2 (1) = T2(1)/T 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"completely reducible as a B(1) module. That is, T2(1) ̸∼= T 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"Pick an element ui from each Ui, 1 ≤i ≤5; say, ui = u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"For 0 ≤i ≤5, let Wi = X⊗3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"q . Their explicit decompositions as Hq-modules, Hq = GLq(C2), were","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"W0 = Vq,(9)(2) ⊕Vq,(7,2)(2);","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"∼= ⊕iWi ⊗Ti,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"(2,1) = m3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"(2,1) = m4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"(2,1) = m5","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"(3) = m2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"Furthermore, it can be verified that the various Gq-modules, Gq =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"the element such that z0Q2 = 0. Its coefficients are shown in Figure 12 in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"the basis {Qσ}. Let z1 = u1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"2. Then the basis Z = {z0, z1} of T2,R admits a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"Z0 = {z0} ⊆Z1 = {z0, z1},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"that yields at q = 1 a composition series of T2,R(1) as a B(1)-module:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"ˆT2,0 ⊂ˆT2,1 = T2,R(1),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"Let W2,1 = X⊗3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"· z1 and W2,0 = X⊗3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"and W2,1(1), W2,0(1) their specializations at q = 1. It can be verified that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"at q = 1 we get:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"W2,1(1) = ∧3(X) ⊆X⊗3,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"W0(1) ⊕W2,0(1) = Sym3(X) ⊆X⊗3,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"where ∧3(X) and Sym3(X) are the Weyl modules of G = GL(X) for the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"at q = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"a = (x1 ⊗x2 ⊗x0) · z1 and b = (x1 ⊗x2 ⊗x0) · z0 in the monomial basis","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"q = 1 of a indeed belongs to the subspace ∧3(X) ⊆X⊗3. The specialization","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"But instead we consider the basis element b′ = b/(q −1)2 of W2,0. Then","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"its specialization b′(1) at q = 1 indeed belongs to the subspace Sym3(X) of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"i=1,3,4,5","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"i ) = V(2,1) ⊗S(2,1) ⊆X⊗3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"[X] =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"W2,1(1) = ∧H,3[X].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"= ⊕Wi ⊗Ti.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"M1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"M2 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"q -module W12 ∼=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"W1,2 ∼= 2 · Vq,(6,3)(2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"for the reciprocity to hold, the base field has to be K = Q(q)[x] as before or","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"It may be illuminating to compare the r = 3 case here with the one for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"Similar symbolic computations for r = 4 seem beyond the reach of desktop","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"the reach, and will be treated in the next section. The r = 4 case, H =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"just as for r = 3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"But it does not seem possible to progress much beyond r = 3 using the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"for the Hecke algebra, or the one for r = 3 in Section 7.1.1. That is, we need","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"aσQσ = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"bσPσ = 0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"where Qσ and Pσ, for a string σ = i1i2 · · · of symbols in {1, · · · , r −1},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"The coefficients aσ and bσ in Figures 2-4 for the r = 3 case do not","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"the r = 3 case suggests that BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"aσ = (−1)d(σ)(q1/2 −q−1/2)d′(σ)(","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"dim(X) in the present case when H = GL2(C), and in r, the rank of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"σ(−s) = aj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"bσ = (−1)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"j=0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"positive topological interpretation with s(σ), ̄s(σ) = 0 in (30) and (31). This","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"We now turn to the analysis of the coefficients in the r = 3 case men-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"q1 = q9/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"q2 = −q−3/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"q3 = q−11/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"q4 = −q−15/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"P = ( ˆR −q2)( ˆR −q3)( ˆR −q4)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"Q = ( ˆR −q1)( ˆR −q3)( ˆR −q4)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"P = fpP,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"fp = (q1 −q2)(q1 −q3)(q1 −q4)(q3 −q1)(q3 −q2)(q3 −q4), (34)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"Q = fqQ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"fq = (q2 −q1)(q2 −q3)(q2 −q4)(q4 −q1)(q4 −q3)(q4 −q2), (35)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"σ = (−(q −1)2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"̄aσ = ( ˆfq)11−l(σ)a′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"σ = (−(q −1)2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"̄bσ = ( ˆfp)11−l(σ)b′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq186","equation_number":null,"raw_text":"σ = ∅, and 2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq187","equation_number":null,"raw_text":"and unimodal, and hence, of the form (30) with s(σ) = 0. All ˆBσ’s are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq188","equation_number":null,"raw_text":"positive and nonincreasing, except for σ = 121, 1212 and 21212, for which","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq189","equation_number":null,"raw_text":"each ˆBσ is positive and unimodal except at the tail. Thus all bσ, for σ ̸=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq190","equation_number":null,"raw_text":"̄s(σ) = 0. For σ = 121, 1212, 21212, bσ seems to be of the form (31) with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq191","equation_number":null,"raw_text":"̄s(σ) = 1, both b0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq192","equation_number":null,"raw_text":"the Kronecker problem (Section 4.1), when H = GL(V )×GL(W), V = W =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq193","equation_number":null,"raw_text":"C2 and G = GL(X), X = V ⊗W ∼= C4, and r = 4. Thus Gq = GLq(C4), and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq194","equation_number":null,"raw_text":"Hq = GLq(C2) × GLq(C2). Let B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq195","equation_number":null,"raw_text":"case and Pi = p+,H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq196","equation_number":null,"raw_text":"X,i , Qi = p−,X","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq197","equation_number":null,"raw_text":"for the r = 3 case in Section 7.1.1. Fortunately, this makes no difference as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq198","equation_number":null,"raw_text":"B = ⊗iTi,L ⊗Ti,R,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq199","equation_number":null,"raw_text":"be the complete Wederburn structure decomposition of B, where Ti = Ti,L","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq200","equation_number":null,"raw_text":"Since, in this case, B = BH","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq201","equation_number":null,"raw_text":"Tq,λ = Sq,λ ⊗Sq,(r) ∼= Sq,(r) ⊗Sq,λ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq202","equation_number":null,"raw_text":"B-module, which specializes at q = 1 to the Specht module Sλ of the sym-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq203","equation_number":null,"raw_text":"Sq,(3,1) ⊗Sq,(2,2) ∼= Sq,(2,1,1) ⊗Sq,(2,2).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq204","equation_number":null,"raw_text":"Sq,(2,2) ⊗Sq,(2,2) ∼= Tq,(4) ⊕Tq,(1,1,1,1) ⊕T6,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq205","equation_number":null,"raw_text":"Q1 = Q3 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq206","equation_number":null,"raw_text":"Q2 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq207","equation_number":null,"raw_text":"The specialization of T6 at q = 1 is isomorphic to the Specht module","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq208","equation_number":null,"raw_text":"S(2,2) of S4. But T6 is nonisomorphic to T2, whose specialization at q = 1 is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq209","equation_number":null,"raw_text":"Sq,(2,1,1) ⊗Sq,(2,1,1) ∼= Tq,(2,1,1) ⊕Tq,(4) ⊕T7,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq210","equation_number":null,"raw_text":"w1 = [","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq211","equation_number":null,"raw_text":"w2 = [","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq212","equation_number":null,"raw_text":"w3 = [","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq213","equation_number":null,"raw_text":"v1 = [","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq214","equation_number":null,"raw_text":"v2 = [","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq215","equation_number":null,"raw_text":"Q1 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq216","equation_number":null,"raw_text":"Q2 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq217","equation_number":null,"raw_text":"Q3 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq218","equation_number":null,"raw_text":"at q = 1. It can be checked that V (1) is isomorphic to the Specht module","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq219","equation_number":null,"raw_text":"of T7 at q = 1, is isomorphic to the Specht module S(3,1) of S4.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq220","equation_number":null,"raw_text":"Sq,(3,1) ⊗Sq,(2,1,1) ∼= Tq,(3,1) ⊕Tq,(1,1,1,1) ⊕T8,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq221","equation_number":null,"raw_text":"where T8 ̸∼= T7 is the second five dimnsional irreducible B-representation","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq222","equation_number":null,"raw_text":"The explicit bases of the structure components Ui = Ti,L ⊗Ti,R in (38) are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq223","equation_number":null,"raw_text":"Fix any ui ∈Ui, 0 ≤i ≤8, and let Wi = X⊗r","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq224","equation_number":null,"raw_text":"q with the following decompositions as Hq-modules, Hq =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq225","equation_number":null,"raw_text":"W8 turns out to be zero when dim(V ) = dim(W) = 2, as here; however, it","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq226","equation_number":null,"raw_text":"(3,1) = m5","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq227","equation_number":null,"raw_text":"(3,1) = m7","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq228","equation_number":null,"raw_text":"(2,1,1) = m8","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq229","equation_number":null,"raw_text":"μ’s are zero. It can now be seen that, as Hq-modules, Hq =","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":95495,"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}}