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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A prism tableau is a set of reverse semistandard tableaux, each positioned within an ambient grid. Prism tableaux were introduced in [the author and \textit{A. Yong}, J. Comb. Theory, Ser. A 154, 551--582 (2018; Zbl 1373.05219)] to provide a formula for the Schubert polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)]. This formula directly generalizes the well known expression for Schur polynomials as a sum over semistandard tableaux. Alternating sign matrix varieties generalize the matrix Schubert varieties of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)]. We use prism tableaux to give a formula for the multidegree of an alternating sign matrix variety. Schubert polynomials; alternating sign matrices; prism tableaux Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Prism tableaux for alternating sign matrix varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that Lapointe-Lascoux-Morse \(k\)-Schur functions (at \(t=1\)) and Fomin-Gelfand-Postnikov quantum Schubert polynomials can be obtained from each other by a rational substitution. This is based on Kostant's solution of the Toda lattice and Peterson's work on quantum Schubert calculus. Lapointe-Lascoux-Morse \(k\)-Schur functions; Fomin-Gelfand-Postnikov quantum Schubert polynomials Lam, Thomas; Shimozono, Mark, From double quantum Schubert polynomials to \(k\)-double Schur functions via the Toda lattice, (2011) Symmetric functions and generalizations, Classical problems, Schubert calculus From quantum Schubert polynomials to \(k\)-Schur functions via the Toda lattice | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is concerned with the statement and the proof of a Giambelli formula holding in the integral cohomology ring \(H^*(G/P)\) of \(G/P\), where \(G\) is any classical linear algebraic group (e.g. the special linear group or any group of automorphisms of a vector space equipped with some symmetric or skew symmetric non degenerate bilinear form) and \(P\) is a parabolic subgroup of it, which means that the quotient \(G/P\) is a (homogeneous) projective variety. The homogeneous variety \(G/P\) is often called a \textsl{generalized flag variety}, because it is a natural generalization of the variety parameterizing (possibly non complete) flags of subspaces of \({\mathbb C}^n\). Among them, one should mention the orthogonal or symplectic Grassmannians or, more generally, Grassmannians of subspaces which are isotropic with respect to some non-degenerate symmetric or skew symmetric bilinear form.
To explain the undoubted relevance of the result, the author spends the first few pages of the introduction to tell where does such a problem come from. The origin is the classical Schubert calculus for the usual familiar Grassmannian variety \(\mathrm{Gr}(k,n)\), which parameterizes \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). The latter is of the form \(G/P\), where \(G:=\mathrm{GL}_n({\mathbb C})\) is the general linear group and \(P\) is the maximal parabolic subgroup of \(G\), obtained as the stabilizer in \(G\) of \(k\) elements of a basis of \({\mathbb C}^n\). By obvious normalizations, \(\mathrm{Gr}(k,n)\) can be also written as \(\mathrm{SL}_n({\mathbb C})/P\). It is classically known, as recalled in the introduction, that the cohomology \(H^*(\mathrm{Gr}(k,n))\) is a \({\mathbb Z}\)-module generated by the classes of the closures of the so-called Schubert cells, named Schubert cycles: the Giambelli formula expresses all Schubert cycles as explicit polynomials in certain distinguished ones, that in this case are the Chern classes of the universal quotient bundle.
In the more general setting (\(G\) any classical group), the homogeneous variety \(G/P\) admits a cellular decomposition via the so-called Bruhat-cells, and its integral cohomology is generated by the cohomology classes of their closures. A first achievement of the investigation pursued in the paper under review, is the individuation of special Schubert cycles: in most cases they are, as in the usual Grassmannian issue, the Chern classes of the universal quotient bundle, but it is not so in all the cases. Using these generalized special cycles, the author is finally able to prove an analogue of Giambelli's formula for \(G/P\), where \(G\) is any classical group.
What makes the job a little bit tricky is that, as it was noticed in previous literature, Giambelli formulas for non-maximal isotropic Grassmannians are not determinantal in nature (like e.g. those for the usual Grassmannians). One must substitute classical Schur polynomials by other kind of polynomials, whose definition supplied in the paper is, as it stands, a further relevant progress in the subject.
Just to give a more precise flavor of the paper, the author studies the variety \(\mathfrak{X}({\mathfrak d})\) parameterizing certain partial flag of isotropic subspaces in \({\mathbb C}^{2n}\) equipped with a non degenerate skew-symmetric bilinear form. Natural Schubert varieties, indexed Weyl group of \(\mathrm{Sp}_{2n}({\mathbb C})\)), can be defined there. One of the main results expresses the classes of such Schubert varieties of \(\mathfrak{X}({\mathfrak d})\) in terms of the substitutes of Schubert polynomials alluded to above. This central result, which is in fact a Giambelli formula for the considered situation, is not just isolated but amazingly embedded in a paper that is rich of intriguing propositions, which are interesting in their own. It is worth to add that the nice Section 1, devoted to the preliminaries, is very helpful and is a concrete attempt to keep the paper fairly self-contained.
Another nice feature, which enlarges the perspective of the results, is that the aforementioned main theorem extends also to the case of torus-equivariant cohomology ring \(G/P\) and to the setting of symplectic and orthogonal degeneracy loci.
The paper ends with a huge and probably complete reference list, which is very useful not just for the interested reader, but fundamental for all researchers aiming to draw their own path for studying and possibly working on this hot subject. homogeneous varieties; classical and equivariant Schubert Calculus; Giambelli formula; Schubert polynomials; bi-Tableaux; mixed Stanley functions Tamvakis, H., \textit{A Giambelli formula for classical G/P spaces}, J. Algebraic Geom., 23, 245-278, (2014) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A Giambelli formula for classical \(G/P\) spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we study a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials \(f_{i,j}\) were introduced by \textit{T. Abe} et al. [``Hessenberg varieties and hyperplane arrangements'', Preprint, \url{arXiv:1611.00269}]. We show that every polynomial \(f_{i,j}\) is an alternating sum of certain Schubert polynomials. flag varieties; Hessenberg varieties; Schubert polynomials Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Extending results of \textit{B. J. Wyser} [Transform. Groups 18, No. 2, 557--594 (2013; Zbl 1284.14068)], we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of the general linear group on the flag variety. Combining this with a slight extension of results of \textit{M. B. Can} et al. [J. Comb. Theory, Ser. A 137, 207--225 (2016; Zbl 1325.05183)], we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms. symmetric varieties; Schubert polynomials; wonderful compactification; equivariant cohomology; weak order; parabolic induction Compactifications; symmetric and spherical varieties, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Wonderful symmetric varieties and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In a long series of interesting and sometimes deep articles the authors have exploited the properties of the cohomology ring and Grothendieck ring of flag manifolds. The present article extends formulas for Schur functions, used to prove that the representation ring of the symmetric group is a Hopf algebra, to formulas for (what the authors call) Schubert and Grothendieck polynomials, that are generalizations of Schur functions. As a result of their formulas they obtain a beautiful formula concerning reduced representations of the symmetric group.
Unfortunately their work is now so far developed in terminology and notation that it is hard for non-experts to read it. In addition, their presentation is extremely condensed and computational. Perhaps time has come to collect their contributions in a leisurely written monograph? Why not a successor to Macdonald's book on symmetric polynomials [\textit{I. G. Macdonald}, ''Symmetric functions and Hall polynomials'' (1979; Zbl 0487.20007)]? Schubert polynomial; Grothendieck polynomial; cohomology ring of flag manifold; Grothendieck ring of flag manifolds; Hopf algebra; representations of the symmetric group Lascoux, Alain; Schützenberger, Marcel-Paul, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. I Math., 295, 11, 629-633, (1982) Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry, Representations of finite symmetric groups, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \({\mathfrak S}_ \sigma(x_ 1,x_ 2,\dots)\) indexed by permutations have been introduced and investigated by \textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)], \textit{M. Demazure} [Ann. Sci. École Norm. Sup., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009)], and by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]; see also their paper [Symmetry and flag manifolds, Lect. Notes in Math. 996, 118-144 (1983; Zbl 0542.14031)].
In this paper the theory of Schubert polynomials is recovered using the nilCoxeter algebra \({\mathfrak C}_ n\) with the identity element \(e\), given by its generators and defining relations as the \(K\)-algebra
\[
\begin{multlined} {\mathfrak C}_ n=\Bigl\langle u_ 1,\dots, u_{n-1}\mid u^ 2_ i= 0\;(i\in I_{n-1}),\;u_ i u_ j= u_ j u_ i\\ (\text{for }| i- j|\geq 2),\text{ and } u_ i u_{i+1} u_ i= u_{i+1} u_ i u_{i+1} (\text{for } i\in I_{n-2})\Bigr\rangle\end{multlined}
\]
over any commutative ring \(K\); here \(I_ n= \{1,2,\dots, n\}\). This algebra can be faithfully represented by the algebra of operators generated by \(\Phi_ i\) \((i\in I_{n-1})\),
\[
\Phi_ i(\sigma)= \begin{cases} \sigma\tau_ i &\text{if } \ell(\sigma\tau_ i)= \ell(\sigma)+1;\\ 0 & \text{otherwise}.\end{cases}
\]
Here, \(\sigma\) is any permutation in the symmetric group \({\mathcal S}_ n\) defined on \(I_ n\), \(\tau_ i\) \((i\in I_{n-1})\) is the `adjacent' transposition \((i,i+1)\), and \(\ell(\sigma)\) is the length of \(\sigma\in {\mathcal S}_ n\) defined as the minimal \(p\) such that \(\sigma= \tau_{a_ 1}\cdot\tau_{a_ 2}\cdot\dots\cdot \tau_{a_ p}\) for some \(a_ j\in I_{n-1}\). A sequence \(a= (a_ 1,\dots, a_ p)\), \(a_ j\in I_{n-1}\) is called a reduced decomposition of \(\sigma\) if \(p= \ell(\sigma)\). \(R(\sigma)\) denote the set of all reduced decompositions for \(\sigma\). For any reduced decomposition \(a= (a_ 1,\dots, a_ p)\) let us identify the monomial \(u_{a_ 1} u_{a_ 2}\cdots u_{a_ k}\) in \({\mathfrak C}_ n\) with \(\tau_{a_ 1} \cdot \tau_{a_ 2}\cdot\dots\cdot \tau_{a_ k}\) in \({\mathcal S}_ n\); the defining relations for \({\mathfrak C}_ n\) guarantee the correctness of such notation, and we see that \({\mathcal S}_ n\) gives a \(K\)-basis for \({\mathfrak C}_ n\). As usual, denote by \(\langle f,\sigma\rangle\) the coefficient of \(\sigma\in {\mathcal S}_ n\) in the \(K\)- expression for \(f\in {\mathfrak C}_ n\). Further, denote
\[
A_ i(x)= (e+ xu_{n-1})\cdot (e+ xu_{n-2})\cdot\dots\cdot (e+ xu_ i)
\]
for any \(i\in I_{n-1}\), \(\bar x= (x_ 1,\dots, x_{n-1})\), \({\mathfrak S}(\bar x)= A_ 1(x_ 1)\cdot A_ 2(x_ 2)\cdot \dots\cdot A_{n-1}(x_{n- 1})\) and let \({\mathfrak S}_ \sigma(\bar x)= \langle{\mathfrak S}(\bar x),\sigma\rangle\). Among the results of this paper is Theorem 2.2 saying that \({\mathfrak S}_ \sigma(\bar x)\) is a Schubert polynomial. The authors prove also (Lemma 2.3) that in the case of \(\text{char } K= 0\),
\[
{\mathfrak S}_ \sigma(1,\dots, 1)= {1\over p!} \sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} a_ 1\cdots a_ p.
\]
Also proved is the \(q\)-analogue of this last formula conjectured by \textit{I. Macdonald} [Notes on Schubert polynomials, LACIM, Université du Québec, Montréal (1991)]:
\[
{\mathfrak S}_ \sigma(1,q,\dots, q^{n-2})= {1\over [1]\cdot[2]\cdot\dots\cdot [p]}\sum_{(a_ 1,\dots, a_ p)\in R(\sigma)} [a_ 1]\cdot\dots\cdot [a_ p]q^{\sum_{\{i\mid a_ i\leq a_{i+1}\}}} i,
\]
where \([t]= 1+ q+\cdots+ q^{t-1}\). Schubert polynomials; nilCoxeter algebra; reduced decomposition S. Fomin and R. P. Stanley. ''Schubert polynomials and the NilCoxeter algebra''. Adv. Math. 103(1994), pp. 196--207.DOI. Symmetric functions and generalizations, Representations of finite symmetric groups, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics Schubert polynomials and the nilCoxeter algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An in-depth discussion on the analysis of the function \({E_a }({X;q,t})\) for the special case \(t=0\) is made in this paper, wherein the author shows that the specialized function `\({E_a}({X;q,0})\) stabilizes to \(\omega {P_\mu }({X;0,t})\)', where \({P_\mu }({X;0,t})\) denotes the Hall-Littlewood polynomials and \(\omega\) is the `involution on symmetric functions'. The author also `relates \({E_a}({X;q,0})\) to the (finite) type A Demazure characters \(E_a ({X; 0,0})\)'.
In his another paper [``Weak dual equivalence for polynomials'', Preprint, \url{arXiv:1702.04051}], the author has developed the theory of weak dual equivalence and introduced the `standard key tableaux to develop a theory of type A Demazure characters' which is invoked by him in this paper to give a combinatorial proof of the fact that on grouping `together the terms in the fundamental slide expansion of \({E_a}({X;q,0})\), the coefficients of \({E_a}({X;q,0})\), when expanded into Demazure characters, are polynomials in \(q\) with nonnegative integer coefficients.' This treatment here parallels the earlier treatment of `the use of dual equivalence' by the author in his work [Forum Math. Sigma 3, Article ID e12, 33 p. (2015; Zbl 1319.05135)] regarding the fundamental quasisymmetric expansion of \({H_\mu}({X;q,t})\) (the transformed Macdonald symmetric functions in type A).
The first significant result of the paper is:
Theorem 3.6. The specialized nonsymmetric Macdonald polynomial \({E_a}({X;q,0})\) is given by
\[
{E_a}({X;q,0}) = \sum_{T \in {\text{SKD}}(a)} {{q^{{\text{maj}}(T)}}{{\mathcal{F}}_{{\text{des}}(T)}}(X)},
\]
where \({{\text{SKD}}(a)}\) denotes the standard key tabloids of shape \(a\), \(\mathcal{F}_a\) denotes the fundamental slide polynomial (see [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]) defined on the finite set \(X\) of variables \(x_1, \ldots, x_n\) by the relation \({{\mathcal{F}}_a}(X) = \sum_{b \geqslant a; {\text{flat}}(b){\text{ refines flat}}(a)}{{X^b}}\) in which `\({\text{flat}}(a)\) is the composition obtained by removing zero parts from \(a\)', \({{\text{des}}(T)}\) denotes the weak descent composition of \(T\) for a standard filling \(T\) of a key diagram and \({{\text{maj}}(T)}\) represents `the sum of the legs of all cells \(c\) (of a key diagram) such that the entry in \(c\) is strictly greater than the entry immediately to its left.' Another important result is the following theorem:
Theorem 4.7. For a weak composition \(a\) such that \(\text{SKD}(a)\) has no virtual elements, the maps \(\left\{ \psi_i \right\}\) on \(\text{SKD}(a)\) give a weak dual equivalence for \((\text{SKD}(a),\text{des})\).
The Demazure character \({\kappa _a}(X)\) is given by the author in [loc. cit., arXiv:1702.04051] and in this paper he beautifully develops the relation between the functions \({E_a}({X;q,0})\) and Demazure characters in the following result:
Theorem 4.9. The specialized nonsymmetric Macdonald polynomial \({E_a}({X;q,0})\) given by \({E_a}({X;q,0}) = \sum_{T \in {\text{YKD}}(a)} {{q^{{\text{maj}}(T)}}{\kappa _{{\text{des}}(T)}}}. \) In particular, \({E_a}({X;q,0})\) is a positive graded sum of Demazure characters.
The following theorem is a landmark result of this paper:
Theorem 5.6. For a weak composition \(a\), we have
\[
\lim_{m \to \infty } {E_{{0^m} \times a}}({X;q,0}) = \omega {H_{{\text{sort}}(a)'}}({X;0,q}) = \omega {H_{{\text{sort}}(a)}}({X;q,0})
\]
By defining the nonsymmetric Kostka-Foulkes polynomial \({K_{a,b}}(q)\) by the relation \({E_b}({X;q,0}) = \sum_a {{K_{a,b}}(q){\kappa _{\text{a}}}(X)} \) the author redevelops the Theorem 5.6 in terms of Kostka-Foulkes polynomials as follows:
Corollary 5.7. Given a weak composition \(b\) with column lengths \(\mu\) such that \({{\text{SKT}}(b)}\) has no virtual Yamanouchi elements, we have
\[
{K_{\lambda ,\mu }}(t) = \sum_{{\text{sort}}({{\text{flat}}(a)}) = \lambda '} {{K_{a,b}}(t)}.
\]
The reviewer finds the paper an important and valuable contribution to the theory of nonsymmetric Macdonald polynomials and their interconnection with Demazure characters. Macdonald polynomials; Demazure characters; Kostka-Foulkes polynomials; Macdonald's symmetric functions; Hall-Littlewood symmetric functions; Jack symmetric functions; nonsymmetric Macdonald polynomials; Schubert calculus; plethystic substitution; Yamanouchi; weak descent composition; non-symmetric Kostka-Foulkes polynomial; Demazure atom Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Symmetric functions and generalizations, Classical problems, Schubert calculus Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in \textit{M. Chan} et al. [Trans. Am. Math. Soc. 370, No. 5, 3405--3439 (2018; Zbl 1380.05007)], and it generalizes a formula of \textit{C. Lenart} [Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] and a recent result of \textit{V. Reiner} et al. [J. Comb. Theory, Ser. A 158, 66--125 (2018; Zbl 1391.05269)] to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms of skew Grothendieck polynomials. Schur functions; Grothendieck polynomials; insertion algorithms; set-valued tableaux; Brill-Noether theory Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Combinatorial relations on skew Schur and skew stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Young tableaux are well studied in representation theory and combinatorics [see the classic books: \textit{G. James} and \textit{A. Kerber}, ``The representation theory of the symmetric group'' (1981; Zbl 0491.20010) or \textit{G. E. Andrews}, ``The theory of partition'' (1976; Zbl 0371.10001)]. The well-known author gives the definitions and main theorems in these fields. But his main emphasis is on the application of Young tableaux in (algebraic) geometry. Defining equations for Grassmannians and flag varieties are given. We find Schubert varieties in flag manifolds, Chern classes are introduced, and the geometry of flag varieties is used to construct the Schubert polynomials of Lascoux and Schützenberger. The basic facts about intersection theory on Grassmannians are given, proofs at some times are delegated to the standard books of \textit{R. Hartshorne} [``Algebraic geometry'' (3rd edition 1983; Zbl 0531.14001)] and \textit{I. R. Shafarevich} [``Basic algebraic geometry. I and II'' (2nd edition 1994; Zbl 0797.14001 and Zbl 0797.14002)] on algebraic geometry. The wealth of presented material makes this excusable.
There are numerous exercises with answers in each chapter and a lot of references to be found at the end of the book. Young tableau; Grassmannian; Schubert polynomials W. Fulton, \textit{Young tableaux: With Applications to Representation Theory and Geometry}, vol. 35 of \textit{London Mathematical Society Student Texts}. Cambridge University Press (1996). Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to group theory, Research exposition (monographs, survey articles) pertaining to combinatorics, Representations of finite symmetric groups, Representations of finite groups of Lie type, Representations of Lie and linear algebraic groups over real fields: analytic methods Young tableaux. With applications to representation theory and geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study multiplication of any Schubert polynomial \(\mathfrak{S}_w\) by a Schur polynomial \(s_{\lambda}\) (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions \(\lambda\), including hooks and the \(2\times 2\) box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of \(\lambda\) is a hook plus a box at the \((2,2)\) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov.{
}This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients. Schubert polynomials; symmetric functions; Fomin-Kirillov algebra Mészáros, Karola; Panova, Greta; Postnikov, Alexander, Schur times Schubert via the Fomin-Kirillov algebra, Electron. J. Combin., 21, 1, Paper 1.39, 22 pp., (2014) Symmetric functions and generalizations, Classical problems, Schubert calculus Schur times Schubert via the Fomin-Kirillov algebra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Recall: Given a map of vector bundles \(\varphi : E \to F\) of ranks \(e\), respectively \(f\), the degeneracy locus \(D_k(\varphi)\) of the points where \(\mathrm{rank}(\varphi )\leq k\) (assume \(k \leq \min (e,f)\)) has codimension at most \((e-k)(f-k)\). When this is the codimension, the homology class of \(D_k(\varphi)\) is expressed (Thom-Porteous formula) as a multi-Schur function in Chern classes of \(E\) and \(F\).
Moreover, there is a (Schur) complex which gives a linear resolution of a Cohen-Macaulay coherent sheaf with support \(D_k(\varphi)\).
The formula for the homology class of \(D_k(\varphi)\) was generalized by \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)] to degeneracy loci defined by an array of numbers corresponding to conditions on ranks of \(E_p \to F_q\) for a map between a flag \(E_\bullet\) of subbundles of \(E\) and a flag of quotients \(F_\bullet\) of \(F\).
In this paper, parallelizing the above situation, one constructs maximal Cohen-Macaulay modules supported on these generalized Schubert degeneracy loci. The complexes introduced here extend the Schubert functors of Kraśkiewcz and Pragacz. This short presentation cannot give the whole richness of the paper under review. complexes; degenearcy loci; Schubert varieties; cohomology classes; Schubert polynomials; polynomial functors Sam, SV, Schubert complexes and degeneracy loci, J. Algebra, 337, 103-125, (2011) Syzygies, resolutions, complexes and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Schubert complexes and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The symmetric Grothendieck polynomials representing Schubert classes in the K theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type \(A_n\) crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate our crystal structure to combinatorial models using excited Young diagrams, Gelfand-Tsetlin patterns via the 5-vertex model, and biwords via Hecke insertion to compute symmetric Grothendieck polynomials. \(A_n\) crystal structure; Schur polynomials; K theory of Grassmannians; Schubert classes Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Quantum groups (quantized enveloping algebras) and related deformations, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Crystal structures for symmetric Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex variety and let \(E_1\to \cdots \to E_{n-1} \to E_n\to F_n \to F_{n-1}\to \cdots \to F_1\) be a sequence of vector bundles and morphism over \(X\) such that \(\text{rank}(F_i) =\text{rank} (E_i)\) for \(1\leq i\leq n\). For any permutation \(w\in S_{n+1}\) let \(\Omega_w\) be the degeneracy locus \(\{x\in X| \text{rank}(E_q(x)\to F_p(x)) \leq r_w(p,q), \forall 1 \leq p,q\leq n\}\), where \(r_w(p,q)\) is the number of \(i\leq p\) such that \(w(i) \leq q\). \textit{W. Fulton} [Duke Math. J. 96, No.~3, 575--594 (1999; Zbl 0981.14022)] gave a formula for the cohomology class of \(\Omega_w\) in \(H^\ast(X,\mathbb Z)\) as a universal Schubert polynomial in the Chern classes of the vector bundles involved, when the maps are sufficiently general.
The main result of the present article is:
For \(w\in S_{n+1}\) we have
\[
[\mathcal O_{\Omega_w}] =\sum (-1)^{\l(u_1\cdots u_{2n-1}w)} G_{u_1}(E_2-E_1)\cdots G_{u_n}(F_n-E_n)\cdots G_{u_{2n-1}}(F_1-F_2)
\]
in \(K(X)\), where the sum is over all factorizations \(w=u_1\cdots u_{2n-1}\) in the degenerate Hecke algebra such that \(u_i \in S_{\min(i,2n-1)+1}\) for each \(i\), and \(G_u(E-E')\) is the stable Grothendieck polynomial of the permutation \(u\).
This result generalizes a previous result of the authors [Duke Math. J. 122, No.~1, 125--143 (2004; Zbl 1072.14067)].
\textit{A. Buch} [Duke Math. J. 115, No.~1, 75--103 (2002; Zbl 1052.14056)] proved the quiver formula
\[
[\mathcal O_{\Omega_w}] =\sum_\lambda c_{w,\lambda}^{(n)} G_{\lambda^1}(E_2-E_1)\cdots G_{\lambda^n}(F_n-E_n)\cdots G_{\lambda^{2n-1}} (F_1-F_2)
\]
where the sum is over finitely many sequences of partitions \(\lambda=(\lambda^1,\dots, \lambda^{2n-1})\) and where the \(c_{w,\lambda}^{(n)}\) are quiver coefficients and \(G_\alpha =G_{w_\alpha}\) is the stable Grothendieck polynomial for the Grassmannian permutation \(w_\alpha\) corresponding to \(\alpha\). The main result of the present article, together with a result of \textit{A. Lascoux} [Transition on Grothendieck polynomials. Physics and Combinatorics, 2000 (Nagoya), World Sci. Publishing, River Edge, HJ, 2001, 164--179)], proves that these coefficients have alternating signs. In fact, define integers \(a_{w,\beta}\) such that \(G_w=\sum a_{w,\beta}G_\beta\) where the sum is over all permutations \(\beta\). Then the main result is equivalent to the explicit combinatorial formula for quiver coefficients:
\[
c_{w,\lambda}^{(n)} =(-1)^{| \lambda| -\l(w)} \sum_{u_1\cdots u_{2n-1}=w} | a_{u_1,\lambda^1} a_{u_2,\lambda^2}\cdots a_{u_{2n-1},\lambda^{2n-1}}|.
\]
The proof of the main result is based on a special case of this formula proved by \textit{A. Buch} [loc. cit.] together with a Cauchy identity given by \textit{A. N. Kirillov} [J. Math. Sci., New York 121, No.~3, 2360--2370 (2004); translation from Zap. Nauchn. Semin. POMI 283, 123--139 (2001; Zbl 1063.05134)].
As a consequence of their result the authors obtain new formulas for the double Grothendieck polynomials of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)]. degeneracy loci; Schubert polynomials; quiver variety; quiver coefficient; words \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \bauthor\binitsA. \bsnmKresch, \bauthor\binitsH. \bsnmTamvakis and \bauthor\binitsA. \bsnmYong, \batitleGrothendieck polynomials and quiver formulas, \bjtitleAmer. J. Math. \bvolume127 (\byear2005), no. \bissue3, page 551-\blpage567. \endbarticle \OrigBibText ----, Grothendieck polynomials and quiver formulas , Amer. J. Math. 127 (2005), no. 3, 551-567. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Combinatorial aspects of representation theory, \(K\)-theory of schemes Grothendieck polynomials and quiver formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We present combinatorial rules (one theorem and two conjectures) concerning three bases of \(\mathbb Z[x_{1},x_{2},\dots ]\). First, we prove a ``splitting'' rule for the basis of Key polynomials [\textit{M. Demazure}, Bull. Sci. Math., II. Ser. 98(1974), 163--172 (1975; Zbl 0365.17005)], thereby establishing a new positivity theorem about them. Second, we introduce an extension of Kohnert's [Bayreuther Mathematische Schriften 38, 1--97 (1991; Zbl 0755.05095)] ``moves'' to conjecture the first combinatorial rule for a certain deformation [\textit{A. Lascoux}, in: Physics and combinatorics. Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, August 21--26, 2000. Singapore: World Scientific. 164--179 (2001; Zbl 1052.14059)] of the Key polynomials. Third, we use the same extension to conjecture a new rule for the Grothendieck polynomials [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 295, 629--633 (1982; Zbl 0542.14030)]. Kohnert's moves; Grothendieck polynomials C. Ross and A. Yong. ''Combinatorial Rules for Three Bases of Polynomials''. Sém. Lothar. Combin. 74 (2015), Art. B74a.URL. Combinatorial aspects of representation theory, Symmetric functions and generalizations, Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds Combinatorial rules for three bases of polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series [the authors, SIAM J. Discrete Math. 31, No. 3, 1953--1989 (2017; Zbl 1370.05007); J. Comb. Theory, Ser. A 154, 350--405 (2018; Zbl 1373.05026)]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications. skew standard tableaux; product formulas; hook length; lozenge tilings; Schubert polynomials Combinatorial aspects of representation theory, Permutations, words, matrices, Classical problems, Schubert calculus Hook formulas for skew shapes. III: Multivariate and product formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The coefficients of the Kazhdan-Lusztig polynomials \(P_{v,w}(q)\) are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for \(h\)-polynomials \(H_{v,w}(q)\) of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for \(H_{v,w}(q)\). We introduce drift configurations to formulate a new and compatible combinatorial rule for \(P_{v,w}(q)\). From our rules we deduce, for these cases, the coefficient-wise inequality \(P_{v,w}(q) \preceq H_{v,w}(q)\). Kazhdan-Lusztig polynomials; Hilbert series; Schubert varieties Li L., Yong A.: Kazhdan-Lusztig polynomials and drift configurations. Algebra Number Theory 5(5), 595--626 (2011) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Hecke algebras and their representations Kazhdan-Lusztig polynomials and drift configurations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The notion of set-valued Young tableaux was introduced by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by flagged set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram \(D(w)\) of a permutation \(w\), a set-valued Rothe tableau of shape \(D(w)\) is a filling of finite nonempty subsets of positive integers into the squares of \(D(w)\) such that the rows are weakly decreasing and the columns are strictly increasing. We show that the double Grothendieck polynomials of 1432-avoiding permutations can be generated by flagged set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our formula specializes to the tableau formula for double Grothendieck polynomials due to \textit{T. Matsumura} [J. Algebr. Comb. 49, No. 3, 209--228 (2019; Zbl 1416.05297)]. Employing the properties of tableau complexes given by Knutson et al. [loc. cit.], we obtain two alternative tableau formulas for the double Grothendieck polynomials of 1432-avoiding permutations. Grothendieck polynomial; Schubert polynomial; 1432-avoiding permutation; set-valued Rothe tableau; tableau complex Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of commutative algebra, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Set-valued Rothe tableaux and Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\rho(q)=a_1q+\cdots+a_dq^d\) be a polynomial of degree \(d\), with non-negative integral coefficients and without constant term. Let \(a=\rho(1)\) and \(N=2d+a\). We exhibit a quasi-homogeneous hypersurface \(V_\rho\subset \mathbb{C}^{N+1}\) such that the \(m\)-th intersection cohomology Betti number of \(V_\rho\) is \(a_i\) for \(m=2i\), and 0 otherwise. Explicitly, let \(x_1,y_1,\dots, x_d\), \(y_d,z_0,z_1,\dots, z_a\) be indeterminates and, for \(s=1,\dots,d\), let \(\pi_s\) denote the product of the \(z_i\) for \(1\leq i\leq a_1+\cdots+a_s\). Then \(V_\rho\) is defined by the polynomial \(F_\rho=x_1y_1+\pi_1x_2y_2+\cdots+\pi_{d-1}x_dy_d+\pi_dz_0\). This is a consequence of earlier work of the author about Schubert varieties. Intersection cohomology; hypersurfaces; Schubert varieties; Kazhdan-Lusztig polynomials Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Grassmannians, Schubert varieties, flag manifolds Construction of affine hypersurfaces with prescribed intersection cohomology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple algebraic group containing opposite Borel subgroups \(B^{+}\) and \(B^{-}\). \textit{V. Deodhar} [Commun. Algebra 13, 1379--1388 (1985; Zbl 0579.14046)] gave a stratification of the flag variety \(G/B\) of \(G\), refining the stratification into both \(B^{+}\) and \(B^{-}\) orbits. Each stratum is isomorphic to a product of copies of the complex numbers and the complex numbers without zero. In this article, the authors give an analogue of the Deodhar stratification for the double flag variety (the product of \(G/B\) with itself). They also discuss connections to positivity, Poisson geometry and cluster algebras. algebraic group; Lie group; Deodhar Stratification; generalized minor; chamber minor; double Schubert cell; double Bruhat cell; total positivity; cluster algebra; flag variety; Poisson structure; Bruhat decomposition Ben Webster and Milen Yakimov, A Deodhar-type stratification on the double flag variety, Transform. Groups 12 (2007), no. 4, 769-785. Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over the reals, the complexes, the quaternions A Deodhar-type stratification on the double flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use a theorem of \textit{S. Tolman} and \textit{J. Weitsman} [in: Northern California symplectic geometry seminar. AMS Transl. Ser. 2, 196(45), 251--258 (1999; Zbl 0955.57023)] to find explicit formulæ~ for the rational cohomology rings of the symplectic reduction of flag varieties in \(\mathbb{C}^n\), or generic coadjoint orbits of \(\text{SU}(n)\), by (maximal) torus actions. We also calculate the cohomology ring of the moduli space of \(n\) points in \(\mathbb{P}^k_\mathbb{C}\), which is isomorphic to the Grassmannian of \(k\) planes in \(\mathbb{C}^n\), by realizing it as a degenerate coadjoint orbit. weight varieties; symplectic reduction; Schubert polynomials Goldin R.F., The cohomology ring of weight varieties and polygon spaces, Adv. in Math., 2001, 160, 175--204 Grassmannians, Schubert varieties, flag manifolds, Momentum maps; symplectic reduction, Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology The cohomology ring of weight varieties and polygon spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Kostant polynomials play a crucial role in the Schubert calculus on \(G/B\) for a semisimple Lie group \(G\) and its Borel subgroup \(B\). These polynomials which are characterized by vanishing properties on the orbits of a regular point under the action of the Weyl group have nonzero values on the corresponding certain elements of higher Bruhat order. The author succeeded in giving explicit forms of these values. It should be emphasized that his description is very minute. Kostant polynomials; Schubert calculus; semisimple Lie group S. C. Billey, ''Kostant Polynomials and the Cohomology Ring for G/B,'' Duke Math. J. 96(1), 205--224 (1999). Semisimple Lie groups and their representations, Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds Kostant polynomials and the cohomology ring for \(G/B\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author develop a general formalism for equivariant Schubert calculus of Grassmannians consisting of a basis theorem, Pieri formula and Giambelli formula in the previous works. In this paper he present an extract of of the theory containing the essential features of the ring. In particular he emphasize the importance of Goresky-Kottwitz- MavPherson (GKM) conditions.
The formalism in this paper are influenced by the combinatorial formalism given by Knutson and Tau for equivariant cohomology of Grassmannians and of use of factorial Schur polynomials on equivariant quantum cohomology of Grassmannians. factorial Schur polynomials; equivariant quantum cohomology; GKM conditions; equivariant Schubert calculus Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A relation between symmetric polynomials and the algebra of classes, motivated by equivariant Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a Schubert variety in a flag variety and denote by \(\text{IH}_X\) the intersection homology sheaf associated to X. The authors present an algorithm to calculate the characteristic class CC(\(\mathbb C\)) of the constant sheaf \({\mathbb C}_X\) and the characteristic class CC(\(\text{IH}_X\)) of the intersection homology sheaf \(\text{IH}_X\). Their main interest are the Hermitian symmetric spaces \(H_G\) of a classical Lie group \(G\). In particular they can show that the characteristic classes CC(\(\text{IH}_X\)) for \(X\subset H_G\) are irreducible if and only if the root system of \(G\) is simply laced. Kazhdan Lusztig polynomials; Schubert varieties; Hermitian symmetric spaces; intersection homology sheaf; characteristic classes; root system Boe, B.; Fu, J., Characteristic cycles in Hermitian symmetric spaces, Can. J. Math., 49, 417-467, (1997) Grassmannians, Schubert varieties, flag manifolds, General properties and structure of real Lie groups, Integral geometry, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Characteristic classes and numbers in differential topology Characteristic cycles in Hermitian symmetric spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The context of the research is in the theory of double Schubert polynomials (of Billey-Haiman for Lie type C and Ikeda-Mihalcea-Naruse for Lie type B, D) as representative polynomials of equivariant Schubert classes of the equivariant cohomology ring of symplectic (Lie type C) and (odd, even) orthogonal flag manifolds (Lie type B, D). The main results contain:
\begin{itemize}
\item The generators for the kernel of the natural map from the ring of Schubert polynomials to the equivariant cohomology ring of flag manifolds type C (Theorem 1) and type D (Theorem 3).
\item The relations between the double Schubert polynomials of Billey-Haiman and the double theta polynomials of Tamvakis-Wilson (Theorem 2) in working with type C. The relationship between the double Schubert polynomials of Ikeda-Mihalcea-Naruse and the double eta polynomials of Tamvakis (Theorem 4) in working with type D.
\item The results in working with type B are nearly identical to type C through an identity between polynomials in subsection 5.1.
\end{itemize}
These main results are a similar, extended continuation of the program laid out in the work of Lascoux and Schützenberger in [\textit{A. Lascoux} and \textit{P. Pragacz}, Mich. Math. J. 48, 417--441 (2000; Zbl 1003.05106); \textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031); Lect. Notes Math. None, 118--144 (1983; Zbl 0542.14031)] for flag manifolds type A. The method used to obtain the main results included:
\begin{itemize}
\item For generators of the kernel of natural maps (Theorems 1, 3): It used the idea from [\textit{H. Tamvakis}, Math. Ann. 314, No. 4, 641--665 (1999; Zbl 0955.14037), Lemma 1] with the transition equations of [\textit{S. Billey}, Discrete Math. 193, No. 1--3, 69--84 (1998; Zbl 1061.05510); \textit{T. Ikeda} et al., Adv. Math. 226, No. 1, 840--886 (2011; Zbl 1291.05222)] to write the Schubert polynomials in this kernel as an explicit linear combination of these generators.
\item For the relations between the double Schubert polynomials with double theta, eta polynomials (Theorems 2, 4): It is based on the equality of the multi-Schur Pfaffian and its orthogonal analog with certain double Schubert polynomials (Propositions 4, 12).
\end{itemize} Schubert polynomials; theta and eta polynomials; Weyl group invariants; flag manifolds; equivariant cohomology Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Actions of groups on commutative rings; invariant theory, Classical problems, Schubert calculus Schubert polynomials, theta and eta polynomials, and Weyl group invariants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are refined by the key polynomials of Lascoux-Schützenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms. Schubert polynomials; Lascoux-Schützenberger key polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Slide multiplicity free key polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives \(\hat{\mathfrak{S}}^{\mathrm{FPF}}_z\) akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions \(\hat{F}^{\mathrm{FPF}}_z\), which are stable limits of the polynomials \(\hat{\mathfrak{S}}^{\mathrm{FPF}}_z\), are Schur \(P\)-positive. To do so, we construct an analogue of the Lascoux-Schützenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for \(\hat{\mathfrak{S}}^{\mathrm{FPF}}_z\) when \(z\) is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur \(P\)-functions, and show that the decomposition of \(\hat{F}^{\mathrm{FPF}}_z\) into Schur \(P\)-functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety. Schubert polynomials; Schur \(P\)-positivity Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Fixed-point-free involutions and Schur \(P\)-positivity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected reductive linear algebraic group over \(\mathbb C\) with an involution \(\theta\). Denote by \(K\) the subgroup of fixed points. In certain cases, the \(K\)-orbits in the flag variety \(G/B\) are indexed by the set of twisted identities \(\iota=\{\theta(w-1)w\mid w\in W\}\) in the Weyl group \(W\). Assume we are in such a case. A good example is when \(K=\text{Sp}_{2n}(\mathbb C)\), \(G=\text{GL}_{2n}(\mathbb C)\). Under the assumption, a criterion is established for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a ``Bruhat graph'' whose vertices form a subset of \(\iota\). Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on \(\iota\) is rank symmetric. -- For the proofs one shows that some Kazhdan-Lusztig-Vogan polynomials still have the required properties under the above assumption.
In the special case \(K=\text{Sp}_{2n}(\mathbb C)\), \(G=\text{SL}_{2n}(\mathbb C)\), the author strengthens the criterion by showing that only the degree of a single vertex, the ``bottom one'', needs to be examined. This generalises a result of Deodhar for type \(A\) Schubert varieties. connected reductive linear algebraic groups; flag varieties; Schubert varieties; rational smoothness; symmetric orbit closures; Bruhat graphs; Kazhdan-Lusztig polynomials Dyer, M.J.: Hecke algebras and reflections in Coxeter groups. Ph.D. thesis, University of Sydney (1987) Linear algebraic groups over the reals, the complexes, the quaternions, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Cohomology theory for linear algebraic groups, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Criteria for rational smoothness of some symmetric orbit closures. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper under review, the authors give an explicit formula for the degree of the Grothendieck polynomial \(\mathfrak{G}_w\) of a Grassmannian permutation \(w\) in the symmetric group \(\mathfrak{S}_n\). Their method uses a formula of [\textit{C. Lenart}, Ann. Comb. 4, No. 1, 67--82 (2000; Zbl 0958.05128)] that expresses \(\mathfrak{G}_w\) in terms of Schur polynomials. The authors then use their degree formula to give an explicit formula for the Castelnuovo-Mumford regularity of (the homogeneous coordinate ring of) the Grassmannian matrix Schubert variety \(X_w\) associated to \(w\). The authors also give a counterexample to a conjectured formula, as well as a corrected formula, for the regularity of (the affine coordinate rings of) standard open patches of certain Grassmannian Schubert varieties appearing in [\textit{M. Kummini} et al., Pac. J. Math. 279, No. 1--2, 299--328 (2015; Zbl 1342.14103)]. They further give a conjectured formula for the regularity of standard open patches of arbitrary Grassmannian Schubert varieties. Castelnuovo-Mumford regularity; Grothendieck polynomial; Schubert variety Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Combinatorial aspects of commutative algebra Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce several new (and recall one old) tableau models for Schubert polynomials. Applications include a bijective proof of Kohnert's rule. Rothe tableaux; balanced tableaux; Schubert polynomials Assaf, S., Combinatorial models for Schubert polynomials, (2017), preprint Combinatorial aspects of representation theory, Classical problems, Schubert calculus Tableau models for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The representation theory of affine Lie algebras can be encoded by semi-infinite flag varieties. These can also be interpreted as spaces of rational maps, and so play a major role in the computation of the quantum \(K\)-theory of flag varieties. \textit{A. Braverman} and \textit{M. Finkelberg} [J. Am. Math. Soc. 27, No. 4, 1147--1168 (2014; Zbl 1367.17011); Math. Ann. 359, No. 1--2, 45--59 (2014; Zbl 1367.17010); J. Lond. Math. Soc., II. Ser. 96, No. 2, 309--325 (2017; Zbl 1384.17009)] used this last interpretation to give fundamental properties, such as normality, rationality of the singularities, an analogue of the Borel-Weil theorem, the computation of quantum \(J\) functions (in the line of Givental-Lee) and its connection with \(q\)-Whittaker functions.
This article has two main objectives. The first is to extend a cohomology formula for line bundles given by Braverman and Finkelberg to include a special class of twisted sheaves. The other main objective is to generalize the other results to all Schubert varieties giving more applications to representation theory. This last extension provides a natural realization of specializations of non-symmetric Macdonald polynomials together with some difference equations characterizing them. This generalizes the link of non-symmetric Macdonald polynomials to representation theory of algebras currently defined by \textit{A. Braverman} and \textit{M. Finkelberg} [Zbl 1367.17010], \textit{C. Lenart} et al. [Int. Math. Res. Not. 2015, No. 7, 1848--1901 (2015; Zbl 1394.05143); ``A uniform model for Kirillov-Reshetikhin crystals. II: Alcove model, path model, and \(P=X\)'', Int. Math. Res. Not. 2017, No. 14, 4259--4319 (2017; \url{doi:10.1093/imrn/rnw129}); Transform. Groups 22, No. 4, 1041--1079 (2017; Zbl 1428.05325)], \textit{I. Cherednik} and \textit{D. Orr} [Math. Z. 279, No. 3--4, 879--938 (2015; Zbl 1372.20009)], \textit{S. Naito} et al. [Trans. Am. Math. Soc. 370, No. 4, 2739--2783 (2018; Zbl 1432.17015)], and \textit{E. Feigin} and \textit{I. Makedonskyi} [Sel. Math., New Ser. 23, No. 4, 2863--2897 (2017; Zbl 1407.17028)].
Let \(G\) be a simply connected simple algebraic group and let \(W\) be its Weyl group with set of simple reflections \(\{s_i\}.\) Let \(\Lambda\) be the weight lattice, \(\Lambda_+\) the set of dominant weights, let \(Q^\vee\) be the co-root lattice of \(G\). Consider the space \(\mathcal Q\) of rational maps from \(\mathbb P^1\) to \(G/B\) with subspace \(\mathcal Q(w)\) defined as the closure of the set of rational maps whose value at \(0\) is in a Schubert variety corresponding to \(w\in W\). This space has a natural line bundle \(\mathcal O(\lambda)\) for each \(\lambda\in\Lambda\).
Associated to \(G\), the current algebra is defined as \(\mathfrak g[z]=\text{Lie}G\otimes_{\mathbb C}\mathbb C[z]\), and \(G\) comes with the associated Iwahori subalgebra \(\mathcal J\). For each \(\lambda\in\Lambda_+\), \(\mathfrak g[z]\) has a representation \(W(\lambda)\) called a global Weyl module. \textit{M. Kashiwara} [Publ. Res. Inst. Math. Sci. 41, No. 1, 223--250 (2005; Zbl 1147.17306)] defined its Demazure submodule \(W(\lambda)_w\) for \(w\in W\) as the cyclic \(\mathcal J\)-module generated by a vector with weight \(w\lambda\in\Lambda\). As these last modules are graded, this defines their character \(\text{ch}W(\lambda)_w\in\mathbb C(\!(q)\!)[\Lambda].\)
The first main theorem is stated more or less verbatim, as it is stated in the category of ind-schemes:
Theorem A. For each \(\lambda\in\Lambda\) and \(w\in W\); {\parindent=0.8cm \begin{itemize}\item[1.] The ind-scheme \(\mathcal Q(w)\) is normal, and projectively normal; \item[2.] There is an isomorphism of \(\mathcal J\)-modules \(H^i(\mathcal Q(w),\mathcal O_{\mathcal Q(w)}(\lambda))^\ast=\begin{cases} W(\lambda)_w,\;(i=0,\lambda\in\Lambda_+)\\\{0\}\text{ otherwise }\end{cases};\) \item[3.] For each \(i\in\mathsf{I}\) so that \(s_iw>w\), \(\text{ch}W(\lambda)_{s_i w}=D_i(\text{ch} W(\lambda)_w)\), where \(D_i\) is a Demazure operator acting on \(\mathbb C (\!(q)\!)[\Lambda]\); \item[4.] There exists a Demazure operator \(D_{t\beta}\) for each \(\beta\in\mathcal Q^\vee\) such that \(\langle\beta,w\alpha\rangle>0\) for every positive root \(\alpha\). This operator is mutually commutative, and \(D_{t\beta}(\text{ch}W(\lambda)_w)=q^{-\langle\beta,w\lambda\rangle}\cdot\text{ch}W(\lambda)_w\).
\end{itemize}} The author remarks that (2) to (4) above are regarded as semi-infinite analogues of the Demazure character formula due to Demazure-Joseph-Kumar in the ordinary setting.
We state the second main result (more or less) verbatim:
Theorem B. For each \(w\in W\) and \(\lambda\in\Lambda_+\), the module \(W(\lambda)_w\) admits a free action of a certain polynomial ring and its specialization to \(\mathbb C\) gives the Feigin-Makedonskyi module \(W_{w\lambda}\). In particular, \(\Gamma(\text{FI}_G^\frac{\infty}{2})(w),\mathcal O_{\text{FI}_G^\frac{\infty}{2}(w)}(\lambda))^\ast\cong W_{w\lambda},\) where \(\text{FI}_G^\frac{\infty}{2}(w)\) is a variant of \(Q(w)\).
The third theorem is a result of the comparison of Cerednik-Orr's recursive formula for non-symmetric Macdonald polynomials specialized to \(t=\infty\) with the authors construction, verbatim:
Theorem C. For each \(\lambda\in\Lambda_+\) and \(w\in W\), there exists an \((\mathbf{I}\rtimes\mathbb G_m)\)-equivariant sheaf \(\mathcal E_w(\lambda)\) such that \(\text{ch}H^0(Q(w),\mathcal E_w(\lambda))^\ast=(\prod_{i\in I}\prod_{k=1}^{\langle\alpha_i^\vee,\lambda_w\rangle}\frac{1}{1-q^k})\cdot E^\dagger_{-w\lambda}(q^{-1},\infty),\) where \(\lambda_w\) is a dominant weight determined by \(\lambda\) and \(w\), and \(E^\dagger_{-w\lambda}(q,t)\) is the bar-conjugate of a non-symmetric Macdonald polynomial, in addition, \(H(Q(w),\mathcal E_w(\lambda))=\{0\}\) for \(i>0\).
To prove the results, an analogue of the Kodaira vanishing theorem is presented. This leads to the fourth main result of the article, which gives relations between different specializations of non-symmetric Macdonald polynomials.
The article starts off with two sections on the preliminary material regarding current algebra representations and semi-infinte flag varieties. This involves many very complicated definitions, and includes a lot of high-end results which the author proves because of lack of references. Then the necessary observation that the semi-infinite flag variety is projectively normal is proved, so that the proof of Theorem A can be given by algebraic manipulations.
The last sections contain the proofs of the remaining main results. These results are simple observations from the author's point of view, but it would take a lot of labour to understand them fully, at least for this reviewer. We understand that the theory of ind-schemes, limiting to derived schemes and a touch of deformation theory are important ingredients, and they are put together in a concise and detailed manner to prove these very important results. semi-infinite flag varieties; Schubert varieties; affine Lie algebras; quantum \(K\)-theory; Borel-Weil theorem; current algebra; non-symmetric Macdonald polynomials; Weyl module Kato, S., Demazure character formula for semi-infinite flag manifolds Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Frobenius induction, Burnside and representation rings, Representation theory of associative rings and algebras, Character groups and dual objects Demazure character formula for semi-infinite flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give Kazhdan-Lusztig polynomials \(P_\mu(\nu)\) for all vexillary permutations \(\mu\) (i.e. \(\mu\) does not contain any substring \(jikh\), \(i<j<h<k\)), and all \(\nu\). We use the embedding of the symmetric group into its enveloping lattice, and the characterization of a permutation by the set of bi-Grassmannian ones which are below it with respect to the Ehresmann-Bruhat order. Schubert varieties; Bruhat order; Kazhdan-Lusztig polynomials; vexillary permutations Alain Lascoux, Polynômes de Kazhdan-Lusztig pour les variétés de Schubert vexillaires, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 667 -- 670 (French, with English and French summaries). Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Kazhdan-Lusztig polynomials for vexillary Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(w\in S_n\) be a permutation. Consider a polynomial \(P_w(q)=\sum_{u\leq w} q^{l(u)}\), where the sum is over all permutations \(u\in S_n\) below \(w\) in the strong Bruhat order. The polynomial \(P_w(q)\) is the Poincaré polynomial of the Schubert variety \(X_w=BwB/B\) in the flag variety \(\text{SL}(n,\mathbb{C})/B\).
Define the inversion hyperplane arrangement \(\mathcal{A}_w\) as the collection of the hyperplanes \(x_i-x_j=0\) in \(\mathbb{R}^n\), for all inversions \(1\leq i< j\leq n\), \(w(i)>w(j)\). Let \(R_w(q)=\sum_r q^{d(r_0,r)}\) be the generating function that counts regions \(r\) of the arrangement \(\mathcal{A}_w\) according to the distance \(d(r_0,r)\) from the fixed initial region \(r_0\) with \((1,2,\dots,n)\in r_0\).
The main result of the paper is the claim that \(P_w(q)=R_w(q)\) if and only if the Schubert variety \(X_w\) is smooth. In this case, an explicit factorization of the polynomial \(P_w(q)\) as a product of \(q\)-numbers \([e_1+1]_q\dots [e_n+1]_q\) is obtained. The numbers \(e_1,\dots,e_n\) are called exponents. They can be computed using the left-to-right maxima (aka records) of the permutation \(w\). Here the inversion graph \(G_w\), whose edges correspond to inversions in \(w\), is a chordal graph. The numbers \(e_1,\dots,e_n\) are the roots of the chromatic polynomial \(\chi_{G_w}(t)\) of the inversion graph. The polynomial \(\chi_{G_w}(t)\) is also the characteristic polynomial of the inversion hyperplane arrangement \(\mathcal{A}_w\). Chordal graphs and perfect elimination ordering are the main technical tools of the paper under review.
In the final section, a generalization of the construction to other root systems is proposed. For an element \(w\) of the Weyl group, the arrangement \(\mathcal{A}_w\) is defined as the collection of hyperplanes \(\alpha(x)=0\) for all roots \(\alpha\) such that \(\alpha>0\) and \(w(\alpha)<0\). The authors conjecture that coincidence of polynomials \(P_w(q)\) and \(R_w(q)\) corresponds to rational smoothness of the Schubert variety \(X_w\) in the generalized flag variety \(G/B\). flag varieties; Schubert varieties; hyperplane arrangements; Bruhat order; Poincaré polynomials Oh, S; Postnikov, A; Yoo, H, Bruhat order, smooth Schubert varieties, and hyperplane arrangements, J. Comb. Theory Ser. A, 115, 1156-1166, (2008) Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Bruhat order, smooth Schubert varieties, and hyperplane arrangements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Each integer partition \(\lambda\) has an associated Schur symmetric function \(s_{\lambda}\), and these Schur functions form a basis for the \(\mathbb Q\)-algebra of symmetric functions, which is also freely generated by the power sum symmetric functions \(p_r\). The Murnaghan-Nakayama rule is the expansion in the Schur basis of the product by a power sum,
\[
p_r \cdot s_{\lambda} = \sum_{\mu} (-1)^{\operatorname{ht}(\frac{\mu}{\lambda})+ 1}s_{\mu},
\]
the sum over all partitions \(\mu\) such that \(\frac{\mu}{\lambda}\) is a rim hook of size \(r\) and \(\operatorname{ht}(\frac{\mu}{\lambda}))\) is the height (number of rows) of \(\frac{\mu}{\lambda}\).
Products \(p_{\lambda} = p_{\lambda_1}p_{\lambda_k}\) of power sums form another basis for symmetric functions, and the change of basis matrix between these two is the character table for the symmetric group. In this way, the Murnaghan-Nakayama rule gives a formula for the characters of the symmetric group.
In this paper, the authors establish a version of the Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming from the Chern character. Like the classical rule, both rules are multiplicity-free signed sums. Murnaghan-Nakayama rule; Schubert calculus; Schubert polynomials; quantum cohomology Symmetric functions and generalizations, Classical problems, Schubert calculus Two Murnaghan-Nakayama rules in Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(q_1, \dots, q_n\) be some variables and consider the ring \(K := \mathbb Z[q_1,\dots,q_n]/( \prod_{i=1}^n q_i)\). We show that there exists a \(K\)-bilinear product \(\star\) on \(H^*(F_n;\mathbb Z)\otimes K\) which is uniquely determined by some quantum cohomology like properties (most importantly, a degree two relation involving the generators and an analogue of the flatness of the Dubrovin connection). Then we prove that \(\star\) satisfies the Frobenius property with respect to the Poincaré pairing of \(H^*(F_n;\mathbb Z)\); this leads immediately to the orthogonality of the corresponding Schubert type polynomials. We also note that if we pick \(k\in\{1,\dots,n\}\) and we formally replace \(q_k\) by 0, the ring \((H^*(F_n;\mathbb Z)\otimes K,\star)\) becomes isomorphic to the usual small quantum cohomology ring of \(F_n\), by an isomorphism which is described precisely. flag manifolds; cohomology; quantum cohomology; periodic Toda lattice; Schubert polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Homology and cohomology of homogeneous spaces of Lie groups A quantum type deformation of the cohomology ring of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a complex algebraic variety with a \(\Gamma\)-action, where \(\Gamma\) is a finite group. Denote by \(H^i(X)\) the complex de Rham cohomology of \(X\) and by \(H^i_c(X)\) the complex cohomology with compact supports, both regarded as \(\Gamma\)-modules. It is known that each cohomology space \(H^i_c(X)\) has a \(\Gamma\)-invariant `weight filtration' whose graded quotients \(Gr_mH^i_c(X)\) give rise to a `weight \(m\)' equivariant Euler characteristic
\[
E^\Gamma_{m,c}(X):=\sum_j(-1)^j Gr_m H^j_c(X)\in R(\Gamma).
\]
Here \(R(\Gamma)\) denotes the Grothendieck ring of \(\Gamma\).
The following polynomials are defined (for \(g\in\Gamma\)):
\[
P^\Gamma_X(t):=\sum_j H^j(X)t^j,\quad Q^\Gamma_{X,c}(t):=\sum_m E^\Gamma_{m,c}(X)t^m,
\]
\[
P^\Gamma_X(g,t):=\sum_j\text{trace}(g,H^j(X))t^j,\quad Q^\Gamma_{X,c}(t):=\sum_m\text{trace}(g,E^\Gamma_{m,c}(X))t^m.
\]
If \(\Gamma=1\) the superscript is dropped and the modules are replaced by their dimensions.
The purpose of the paper is, to compute these polynomials in the case where \(X=G_{rs}\), the set of regular semisimple elements of the complex connected reductive algebraic group \(G\), or \(X={\mathfrak G}_{rs}\) the corresponding set in the Lie algebra \(\mathfrak G\) of \(G\).
This problem can be reduced to analogous \(W\)-equivariant problems concerning respectively maximal tori or toral algebras \(T_{rs}\) and \({\mathcal T}_{rs}\), and \(W=N_G(T)/T\) the corresponding Weyl group.
In Section 4, explicit formulae for the \(P_X(t)\) and \(Q_{X,c}(t)\), \(X\in\{G_{rs},{\mathfrak G}_{rs}\}\) are given in terms of certain polynomials \(P_{T_{rs}}(w,t)\) (in the group case) and \(P_{M_W}(w,t)\) (in the Lie algebra case), which depend on elements \(w\in W\). The polynomials \(P_{M_W}(w,t)\) are explicitly known (see the references in the paper for exceptional groups and groups of type \(A\) and \(B\) as well as the paper [\textit{P. Fleischmann} and \textit{I. Janiszczak}, Manuscr. Math. 72, No. 4, 375-403 (1991; Zbl 0790.52006)] for type \(D\)).
On applying his results to the variety of regular semisimple elements in the Lie algebra, \({\mathfrak G}L_n(\mathbb{C})\), the author obtains stability results for the Betti numbers of this variety.
In the last section the previous results are applied, with the help of \(\ell\)-adic cohomology, to regular semisimple varieties of algebraic groups over finite fields and their Lie algebras. Under certain technical assumptions, explicit formulae are derived for the number of \(\mathbb{F}_q\)-rational points in the regular semisimple variety of the Lie algebras of the general linear group or Lie algebras of type \(B_n\), \(C_n\). Notice that the corresponding formula in Proposition (8.9) for \(B_n\), \(C_n\) contains a slight misprint. It should read
\[
q^{n^2}\prod^n_{k=1} (q^{2k}-1)\sum_{\substack{\lambda=(\lambda^+,\lambda^-)\\ \lambda^+=(i^{m_i}),\;\lambda^-=(j^{n_j})}} \prod_i {q^+_i(-q^{-1})\choose m_i}(q^i-1)^{-m_i}\times\prod_j {q^-_j(-q^{-1})\choose n_j}(q^j+1)^{-n_j}.
\]
[For a different approach to these formulae see also: \textit{P. Fleischmann} and \textit{I. Janiszczak}, J. Algebra 155, No. 2, 482-528 (1993; Zbl 0809.20007); \textit{P. Fleischmann}, Finite Fields Appl. 4, No. 2, 113-139 (1998)]. Poincaré polynomials; complex algebraic varieties; complex de Rham cohomology; Euler characteristic; Grothendieck rings; regular semisimple elements; complex connected reductive algebraic groups; Lie algebras; maximal tori; toral algebras; \(\ell\)-adic cohomology; numbers of rational points G. I. Lehrer, The cohomology of the regular semisimple variety, J. Algebra 199(2) (1998), 666\Ndash689. \small\texttt DOI: 10.1006/jabr.1997.7195. Cohomology theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Linear algebraic groups over finite fields, Cohomology of Lie (super)algebras, Classical real and complex (co)homology in algebraic geometry The cohomology of the regular semisimple variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate \(K\)-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynomials. We combinatorially prove this expansion is determined by Yamanouchi set-valued tableaux. A by-product of our results is a dual approach proof for Buch's \(K\)-theoretic Littlewood-Richardson rule for the product of stable Grothendieck polynomials. Grothendieck polynomials; Littlewood-Richardson rule; tableaux Li, H., Morse, J., Shields, P.: Structure constants for \(K\)-theory of Grassmannians revisited (2016) (preprint). arXiv:1601.04509 Combinatorial aspects of representation theory, Classical problems, Schubert calculus Structure constants for \(K\)-theory of Grassmannians, revisited | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Via the methods of analytic inspiration, based on the use of the formulas of integral representation attached to the proper holomorphic mappings of an affine domain \(U\subset\mathbb C^n\) to \(\mathbb C^n\), the author rediscovers the generalized formulae for the Jacobi multidimensional residues, obtained by C.~A.~Berenstein, A.~Vidras and A.~Yger, and extends these results to the singular setting (at infinity), or generally to the transcendental setting, restricting oneself to the strictly affine setting without recursing to the compactification of the domain in which one works. multidimensional residues; Grothendieck residue; divisors; algebraic cycles; differential forms; currents; Jacobi residue formulae; analytic inspiration; integral representations; Bochner-Martinelli formula; toroidal residues; Laurent polynomials Residues for several complex variables, Integral representations; canonical kernels (Szegő, Bergman, etc.), Integration on analytic sets and spaces, currents, Divisors, linear systems, invertible sheaves, Proper holomorphic mappings, finiteness theorems The Jacobi formulae and analytic methods | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Rat_ d\) be the space of degree d rational maps modulo post composition with Moebius transformations, and \(\rho(d):=d^{- 1}\binom{2d-2}{d-1}\) the Catalan number. - The author proves that the number of classes in \(Rat_ d\) with a given branch set is generically equal to the Catalan number \(\rho(d)\).
Let \(Poly_ d\) denote the space of polynomials whose degree is \(\leq d\), and \(G_ 2(Poly_ d)\) the Grassmann manifold of two dimensional subspaces of \(Poly_ d\). There is an embedding of \(Rat_ d\) into \(G_ 2(Poly_ d):\) \(Rat_ d\ni [R]\to X_ R\in G_ 2(Poly_ d),\) where \(R=P/Q\) and \(X_ R\) is generated by P and Q. There is a complex analytic map \(\Phi_ d: G_ 2(Poly_ d)\to {\mathbb{P}}^{2d-2}\) which connects a subspace generated by P and Q to the polynomial \(PQ'-P'Q\) whose roots are the critical points of P/Q. The author computes the degree of \(\Phi_ d\) by transforming the problem to one of enumerative geometry and by using the Schubert calculus. Grassmann manifold of two dimensional subspaces of space of polynomials; rational maps; Catalan number; enumerative geometry; Schubert calculus \beginbarticle \bauthor\binitsL. \bsnmGoldberg, \batitleCatalan numbers and branched coverings by the Riemann sphere, \bjtitleAdv. Math. \bvolume85 (\byear1991), no. \bissue2, page 129-\blpage144. \endbarticle \OrigBibText L. Goldberg, Catalan numbers and branched coverings by the Riemann sphere, Adv. Math. 85 (1991), no. 2, 129-144. \endOrigBibText \bptokstructpyb \endbibitem Coverings of curves, fundamental group, Grassmannians, Schubert varieties, flag manifolds, Polynomials and rational functions of one complex variable, Riemann surfaces; Weierstrass points; gap sequences, Enumerative problems (combinatorial problems) in algebraic geometry Catalan numbers and branched coverings by the Riemann sphere | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Our purpose is to describe some remarks on Schur polynomials, which play an important role in the theory of Chow ring. In {\S} 2, we describe the relationship between the Schur polynomials and the Schubert cycles of a Grassmann variety. In {\S} 3, we prove the duality of Schur polynomials using the theory of the Chow ring of Grassmann variety. - In {\S} 4, we present the theorem of W. Fulton and R. Lazarsfeld for numerically positive polynomials for ample vector bundles. - In {\S} 5 and {\S} 6, we introduce the Gysin's projection formula for flag bundles and give a formal proof of this formula. Schur polynomials; Schubert cycles; Chow ring of Grassmann variety; flag bundles Grassmannians, Schubert varieties, flag manifolds, Parametrization (Chow and Hilbert schemes) Remarks on Schur polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a chain of vector bundle maps \(E_0\to E_1\to\cdots\to E_n\) over a base space \(B\) and an integer array \(r=(r_{i,j})_{i\leq j}\) one can associate a degeneracy locus \(\Omega_r\) in \(B\): the set of points over which the rank of \(E_i\to E_j\) is at most \(r_{i,j}\). The cohomology class represented by this class in \(B\) can be expressed in terms of the Chern classes of the vector bundles. This expression has remarkable combinatorial properties, studied by various authors, e.g. it has a certain positivity property. One can also consider the \(K\)-theoretic analogue of this expression. Buch wrote this quiver \(K\)-polynomial in terms of stable Grothendieck polynomials, and conjectured that the coefficients have alternating signs. In the paper under review the author proves another form of the quiver \(K\)-polynomial, the ``stable double component formula'', a sum of expressions labeled by ``lace diagrams''. As a corollary he proves the conjecture of Buch on the signs of the coefficients. quiver polynomials; cohomology class; Chern classes; Grothendieck polynomials; conjecture of Buch Ezra Miller, Alternating formulas for \(K\) -theoretic quiver polynomials, Duke Math. J. 128 (2005), 1-17. Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Alternating formulas for \(K\)-theoretic quiver polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert varieties appear in a natural way in many parts of mathematics and its applications. They are on the crossroad of representation theory, combinatorics and algebraic, analytic and differential geometry. The structure of the singularities of the Schubert varieties is extremely complicated, but still manageable. This makes Schubert varieties extremely interesting objects and explain why the work on these varieties has contributed so much to mathematics. There is a vast literature on their properties, in particular on their singularities and their cohomology. The literature list of the present book contains nearly 160 items, most of them on singularities, and most of them are written after 1984. The book started out as a survey article. This is reflected in the text which consists mostly of explanation of concepts and statements of results. There are very few proofs, and the background material is limited to a minimum. In spite of this the book is more than 200 pages long. This illustrates well the size of the task that the authors have undertaken. That they have succeeded in writing a readable text is partly due to the fact that they both are experts in the area. About one fifth of the articles in the reference list are written by the authors. The book is closer to a survey of recent work on the singularities of Schubert varieties, than a textbook. What distinguishes it from a survey is that it contains a wealth of examples and calculations. This makes the book extremely valuable to those that want to learn the area.
The contents of the book is: Chapter 2: This chapter contains background material on Schubert varieties, Bruhat-Chevalley orders and parabolic groups, weights, algebraic groups, Weyl groups, and representations of semi-simple algebraic groups. It would take several courses, and books, to cover properly this material alone. -- Chapter 3: To compensate for the rapid introduction of concepts in chapter 2 this chapter contains concrete computations of the Bruhat-Chevalley order in the classical groups, the grassmannians, the flag manifolds and their Schubert varieties. -- Chapter 4: This chapter is devoted to the study of the tangent space to a Schubert variety. -- Chapter 5: In this chapter results by P. Polo on the singular locus of a Schubert variety is recalled, and Polo's root system description of the tangent space is described. This is used to obtain a very detailed treatment of the classical groups. -- Chapter 6: Here Kazhdan-Lusztig polynomials and their use for characterizing rational singularities is explained. -- Chapter 7: In this chapter Shrawan Kumars criteria for smoothness, and generic smoothness are discussed. The relation between nil-Hecke rings and characters on the tangent cone is explained. This chapter also contains some proofs. -- Chapter 8: In this chapter combinatorial algorithms for testing smoothness and rational smoothness of Schubert varieties for the classical groups are given. The last four chapters are devoted to the singularities of special varieties. -- Chapter 9: In this chapter the minuscule and cominuscule cases are discussed. The major part is devoted to the small resolutions of A. Zelevinsky and of P. Sankaran and P. Vanchinatan. Also the description of their tangent space made by M. Brion and Polo is given. -- Chapter 10: This chapter is devoted to the rank \(2\) case. -- Chapter 11: In this chapter some results on the relation between smoothness and factorization of the Poincaré polynomial are described. -- Chapter 12: In this chapter determinantal varieties are treated.
The book contains a wealth of results, and a large number of very useful and illustrating examples. It treats many central topics. The authors seem to concentrate on the work treating singularities of Schubert varieties only. There is a large body of work on the cohomology of manifold where the singularities of the Schubert varieties enter in an essential way, and where many of the techniques for treating singularities originated. It would have been very interesting to have a short description of this work and to get an explanation of the relations to the work on the singularities of Schubert varieties. Also it would have made the reference list more complete.
The book is well written, and the background material clearly presented. It is valuable to both experts and beginners. Beginners will find a way to get an idea about the area before indulging into the large and technically complicated background material. Together with the experts they will find useful detailed examples, together with lists of minimal bad patterns and the singular loci of the special groups. The experts will find a wealth of interesting material, much of it new. Schubert varieties; singular loci; roots; Weyl groups; tangent cone; parabolic groups; Bruhat-Chevalley orders; grassmannian; Kazhdan-Lusztig polynomials; classical groups; nil-Hecke ring; minuscule groups; Poincaré polynomial; determinantal varieties; Dynkin diagrams Billey, Sara; Lakshmibai, V., Singular loci of Schubert varieties, Progress in Mathematics, vol. 182, (2000), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA, MR 1782635 Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Determinantal varieties Singular loci of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of ``admissible sequences'' by \textit{R. W. Richardson} and \textit{T. A. Springer} [Geom. Dedicata 35, No. 1--3, 389--436 (1990; Zbl 0704.20039); ibid. 49, No. 2, 231--238 (1994; Zbl 0826.20045)]. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of ``involution Schubert polynomials'' can be viewed as a Billey-Jockusch-Stanley type formula for cohomology class representatives of \(\mathrm{O}_n\)- and \(\mathrm{Sp}_{2 n}\)-orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type \(B\). permutations; involutions; reduced words; Schubert polynomials; Stanley symmetric functions; Bruhat order; spherical varieties; Coxeter groups Z. Hamaker, E. Marberg, and B. Pawlowski. ''Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures''. 2015. arXiv:1508.01823. Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Reflection and Coxeter groups (group-theoretic aspects) Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type \(A\). We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to express \(k\)-Schur functions in terms of power sum symmetric functions. We also provide the definition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of the affine flag variety. affine flag variety; affine Fomin-Kirillov algebra; affine nilCoxeter algebra; affine Schubert polynomials; \(k\)-Schur function; Murnaghan-Nakayama rule Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Combinatorial description of the cohomology of the affine flag variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The context of the research is as follows. Let \(G\) be a semisimple complex algebraic group, \(K\) be a closed subgroup of \(G\). Let \(P\) be a parabolic subgroup of \(G\) and \(B_K\) is a Borel subgroup of \(K\). Then \(G/P \times K/B_K\) is a double flag variety. In this paper, the author considers three cases:
\begin{itemize}
\item[(I)] \(G=SL(2n)\), \(K=(GL(n) \times GL(n)) \cap G\), \(G/P\) is the Grassmannian \(Gr(n,2n)\),
\item[(II)] \(G=Sp(2n)\), \(K=GL(n)\), \(G/P\) is the Lagrangian Grassmannian \(LG(2n)\),
\item[(III)] \(G=SO(2n)\), \(K=GL(n)\), \(G/P\) is the orthogonal Grassmannian \(OG(2n)\).
\end{itemize}
The main result of the paper is:
\begin{itemize}
\item[1.] Descriptions of \(K\)-orbits on \(G/P \times K/B_K\) intersecting certain subset of \(G/P\) in terms of rank condition (Proposition 13 for (I), Proposition 34 for (II), (II))
\item[2.] Statements that the cohomology classes of the \(K\)-orbits are represented by the back-stable double Schubert polynomials (Theorem for (I)), back-stable involution Schubert polynomials (Theorem 41 for (II) and Theorem 43 for (III)).
\end{itemize}
Then, the author obtains some corollaries:
\begin{itemize}
\item[a.] Geometric interpretations of the involution Stanley symmetric functions \(2^{cyc(y)}F_y^{O}\), \(F_z^{Sp}\) (Theorem 44).
\item[b.] A new proof that \(2^{cyc(y)}F_y^{O}\) is Schur \(Q\) positive and \(F_z^{Sp}\) is Schur \(P\) positive (Corollary 45), which are proved in [\textit{Z. Hamaker} et al., Int. Math. Res. Not. 2019, No. 17, 5389--5440 (2019; Zbl 1459.05338); J. Comb. 11, No. 1, 65--110 (2020; Zbl 1427.05226)].
\item[c.]A generalization of Pfaffian formulas of involution Schubert polynomials and Stanley symmetric functions for I-Grassmannian, fpf-I-Grassmannian involution in [\textit{Z. Hamaker} et al., Int. Math. Res. Not. 2019, No. 17, 5389--5440 (2019; Zbl 1459.05338); J. Comb. 11, No. 1, 65--110 (2020; Zbl 1427.05226)] to vexillary involution (Theorems 39,49,54,57).
\end{itemize}
The key point used to obtain the main results is the interpretation graph Schubert varieties of the as certain degeneracy when \(w,z\) are vexillary.
The structure of the paper is as follows. Section 2 prepares background about cohomology rings of flag varieties, degeneracy loci, Schubert varieties, and vexillary permutations. Sections 3 shows the main result 1. 2. for (I). Sections 4 prepares the background for involution graph Schubert varieties. Section 5 shows the main results 1. 2. for (II), (III), and then corollaries a. b. Section 6 shows corollaries c. The last section gives tableau formulas for back-stable involution Schubert polynomials in terms of shifted tableaux in vexillary settings. flag manifolds; Schubert varieties; Grassmannian; Lagrangian Grassmannian; orthogonal Grassmannian; Shubert polynomials; Stanley symmetric functions Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial aspects of commutative algebra Universal graph Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{S. K. Donaldson} [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)] has introduced polynomials on the second cohomology of any simply connected closed real four-manifold with odd \(b^ +_ 2\). Their importance results from their diffeomorphism invariance and they are used to distinguish different differentiable structures with the same underlying topology. But they are not easy to compute.
Over a smooth projective surface these polynomials are closely related to moduli spaces of holomorphic rank-two vector bundles. Recently \textit{K. G. O'Grady} [Invent. Math. 107, No. 2, 351-395 (1992)] introduced algebro- geometric analogues of Donaldson's polynomials. More recently \textit{J. Morgan} [``Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues'', Preprint (1992)] has shown that these two kinds of polynomials coincide (under certain circumstances).
In the paper under review the author defines similar to O'Grady symmetric polynomials on the second cohomology in the case of a ruled surface \(X\) over a curve \(C\). He uses the explicit description of the moduli space of rank two bundles on ruled surfaces of his earlier paper [``Moduli spaces of stable rank-2 bundles on ruled surfaces''. I. (Preprint); see also J. Reine Angew. Math. 433, 201-219 (1992; Zbl 0753.14031)] and a universal rank-two sheaf parametrized by that. The polynomial is defined with the aid of the first Pontryagin class of the projective bundle associated to this universal sheaf. From the computation of Chern classes of a vector bundle over the Picard torus of \(C\) the author can explicitly compute his polynomials defined before. \(\mu\)-stable; Grothendieck-Riemann-Roch; Donaldson polynomials; moduli space of rank two bundles on ruled surfaces Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Characteristic classes and numbers in differential topology, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Fine and coarse moduli spaces Symmetric polynomials constructed from moduli of stable sheaves with odd \(c_ 1\) on ruled surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the author proves a tableau formula for the eta polynomials of \textit{A. S. Buch} et al. [``A Giambelli formula for even orthogonal Grassmannians'', Preprint, \url{arXiv:1109.6669}] and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group \(\widetilde{W}_n\) of type \(D_n\).
The formula for the eta polynomial \(H_\lambda(x,y)\) is written as a sum of monomials over certain fillings of the Young diagram of \(\lambda\) called typed \(k^\prime\)-bitableaux. The proof is the result of various reduction formulas which are then combined to obtain a main tableau formula for \(H_\lambda(x , y)\). In the last section, the author relates this theory to the type D Schubert polynomials and Stanley symmetric functions in [\textit{S. Billey} and \textit{M. Haiman}, J. Am. Math. Soc. 8, No. 2, 443--482 (1995; Zbl 0832.05098); \textit{T. K. Lam}, B and D analogues of stable Schubert polynomials and related insertion algorithms. Cambridge, MA: Massachusetts Institute of Technology (PhD Thesis) (1995)], and study the skew elements of \(\widetilde{W}_{n+1}\). Stanley symmetric functions; type D Schubert polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations A tableau formula for eta polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this well written and many faceted article several analogs of properties of Schur polynomials are provided for Schubert polynomials.
The author gives three different constructions of Bott-Samelson varieties for a general reductive group with a Borel subgroup, and gives a characterization of the Bott-Samelson variety in terms of certain incidence conditions.
In the case when the group is \(Gl(n)\) the results are given as combinatorial interpretations, and it is shown that in this case the coordinate ring of the Bott-Samelson variety consists of generalized Schur modules. This allows the author to compute the generalized Schur polynomials that are the characters of these modules. Using the same arguments as in an earlier paper [\textit{P. Magyar}, Adv. Math. 134, No. 2, 328-366 (1998; Zbl 0911.14024)] the Borel-Weil theorem is proved, and a version of Demazures character formula is worked out in order to obtain a new expression for generalized Schur polynomials.
The theory is used to compute the Schubert polynomials associated to permutations, the theorem of Kraskiewicz and Pragacz is proved, and three new explicit formulas for Schubert polynomials are given.
The essential ingredient in the combinatorial approach are chamber families used by \textit{A. Berenstein, A. Fomin} and \textit{A. Zelevinsky} [Parametrizations of canonical bases and totally positive matrices, Adv.Math. 122, 49-149 (1996)], associated to reduced decompositions of elements in the Weyl group via its wiring diagram. This is translated into the language of generalized Young diagrams. Schubert polynomials; Weyl character formula; Demazure character formula; Borel-Weil theorem; Bott-Samelson varieties; generalized Schur functions; generalized Schur modules; wiring diagram; chamber set; reductive group; Young diagrams Magyar P., Schubert polynomials and Bott-Samelson varieties, Comment. Math. Helv., 1998, 73(4), 603--636 Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Representations of quivers and partially ordered sets, Classical problems, Schubert calculus Schubert polynomials and Bott-Samelson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A flagged Weyl module is a representation of the group of invertible upper triangular \(n\times n\) matrices, associated with a diagram \(D\) with \(n\) columns. Schubert polynomials and key polynomials are special cases of the dual chararacters of these modules. The principal specialization of Schubert polynomials has long been of interest. The authors prove a lower bound on the principal specialization of the dual characters of flagged Weyl modules; when specialized to Schubert polynomials, this gives a new proof of a conjecture of Stanley originally proved in [\textit{A. E. Weigandt}, Algebr. Comb. 1, No. 4, 415--423 (2018; Zbl 1397.05205)]. They also characterize the diagrams which give equality. In the case of equality, all nonzero coefficients of the character are 1. There is a conjectural characterizaton of all cases in which the nonzero coefficients are all 1, which is known for Schubert and key polynomials. There is also a conjecture for when the known upper bound gives equality. flagged Weyl module; Schubert polynomials; key polynomials; Schur functions; principal specialization Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Principal specialization of dual characters of flagged Weyl modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials How to multiply two Schubert polynomials is a notorious open problem in Schubert calculus. The author addresses the special case where a Schubert polynomial is multiplied by a Schur polynomial. His result is a (not very efficient but still beautiful) description of the expansion coefficients in this product as the number of pairs of an RC-graph as introduced by \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, 123-143 (1996; Zbl 0852.05078)] and a Young tableau, which have to be related in a certain way. The proof of this result is entirely based on the insertion algorithm for RC-graphs due to \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)]. Schubert polynomials; Schur functions; Littlewood-Richardson rule; Monk's rule; Pieri's rule; RC-graphs Kogan, M.: RC-graphs and a generalized Littlewood--Richardson rule. Int. Math. Res. Not. 2001(15), 765--782 (2001) Algebraic combinatorics, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus RC-graphs and a generalized Littlewood-Richardson rule | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)] published a characterization of singular Schubert varieties in flag manifolds using the notion of pattern avoidance. This was the first time pattern avoidance was used to characterize geometrical properties of Schubert varieties. Their results are very closely related to work of Haiman, Ryan and Wolper, but Lakshmibai-Sandhya [loc. cit.] were the first to use that language exactly. Pattern avoidance in permutations was used historically by Knuth, Pratt, Tarjan, and others in the 1960's and 1970's to characterize sorting algorithms in computer science. Lascoux and Schützenberger also used pattern avoidance to characterize vexillary permutations in the 1980's.
Now, there are many geometrical properties of Schubert varieties that use pattern avoidance as a method for characterization including Gorenstein, factorial, local complete intersections, and properties of Kazhdan-Lusztig polynomials. These are what we call consequences of the Lakshmibai-Sandhya [loc. cit.] theorem. We survey the many beautiful results, generalizations, and remaining open problems in this area. We highlight the advantages of using pattern avoidance characterizations in terms of linear time algorithms and the ease of access to the literature via Tenner's database of permutation pattern avoidance. Schubert varieties; flag manifolds; pattern avoidance; Kazhdan-Lustig polynomials; Coxeter groups; Bruhat order H. Abe and S. C. Billey. ''Consequences of the Lakshmibai-Sandhya Theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry''. To appear in Adv. Studies of Pure Math. 2014. arXiv:1403.4345. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Reflection and Coxeter groups (group-theoretic aspects) Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all \(a\)-coefficients of the DAHA superpolynomials upon the substitution \(q \mapsto qt\) satisfy the Riemann Hypothesis for sufficiently small \(q\) for uncolored algebraic knots, presumably for \(q \leq 1/2\) as \(a = 0\). This can be partially extended to algebraic links at least for \(a = 0\). Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stöhr zeta-functions are discussed. double affine Hecke algebras; Jones polynomials; HOMFLY-PT polynomials; plane curve singularities; compactified Jacobians; Hilbert scheme; Khovanov-Rozansky homology; iterated torus links; Macdonald polynomial; Hasse-Weil zeta-function; Riemann hypothesis Knots and links in the 3-sphere, Plane and space curves, Hecke algebras and their representations, Braid groups; Artin groups, Root systems, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Singular homology and cohomology theory Riemann hypothesis for DAHA superpolynomials and plane curve singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Pipe dreams represent permutations pictorially as a series of crossing pipes. Recent applications of pipe dreams include the calculation of Schubert polynomials, fillings of moon polyominoes, and in the combinatorics of antidiagonal simplicial complexes. These applications associate pipe dreams to words of elementary symmetric transpositions via a canonical mapping. However, this canonical mapping is by no means the only way of mapping pipe dreams to permutation words. We define sensical mappings from pipe dreams to words and prove sensical mappings are in bijection with standard shifted tableaux of triangular shape. We characterize the set of pipe dreams associated to a given word (under any sensical map) using step ladder moves. These moves induce a partial order on the set of pipe dreams mapping to a given word, yielding a distributive lattice. pipe dreams; RC-graphs; permutations; Schubert polynomials; posets Permutations, words, matrices, Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds On the relationship between pipe dreams and permutation words | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. This proof extends to a principal specialization due to \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)]. Such a proof has been sought for over 20 years. Our bijective tools also allow us to solve a problem posed by \textit{S. Fomin} and \textit{A. N. Kirillov} [J. Algebr. Comb. 6, No. 4, 311--319 (1997; Zbl 0882.05010)] using work of Wachs, Lenart, Serrano and Stump. These results extend earlier work by the third author [``A Markov growth process for Macdonald's distribution on reduced words'', Preprint, \url{arXiv:1409.7714}] on a Markov process for reduced words of the longest permutation. Young tableaux; Ferrers shape; reduced words; identity; Stanley's formula; Macdonald's formula; Schubert polynomials; enumeration of plane partitions; permutations; shapes Combinatorial aspects of partitions of integers, Exact enumeration problems, generating functions, Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Markov processes A bijective proof of Macdonald's reduced word formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, \(K\)-theoretic deformations of Schur polynomials. Grothendieck polynomials; total positivity; symmetric functions; combinatorial \(K\)-theory Symmetric functions and generalizations, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Positive specializations of symmetric Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0592.00025.]
This is a survey article summarizing recent results that the Italian school has achieved in the matter of characterizing some special algebraic varieties as partial line spaces (PLS). These last are those pairs (S,\({\mathcal L})\) where S is a non-empty set and \({\mathcal L}\) is a proper family of subsets of S such that: (1) any \(\ell \in {\mathcal L}\) contains at least two points, i.e. \(| \ell | \geq 2\). - (2) \({\mathcal L}\) is a covering of S. - (3) \(\ell,\ell '\in {\mathcal L},\ell \neq \ell '\) implies \(| \ell \cap \ell '| \leq 1.\)
That general definition allows room for further axioms, leading this approach to characterizations of Grassmann varieties, pseudoproduct spaces and C. Segre's varieties, Veronese varieties and Schubert varieties. Also, since a graph is a PLS this allows to provide graph theoretical characterizations as well. A large list of references which contain the original development of the topic is provided in the present survey too. collinear; flag space; double point; bibliography; survey; algebraic varieties; partial line spaces; Grassmann varieties; pseudoproduct spaces; Segre's varieties; Veronese varieties; Schubert varieties Tallini G.: Partial linear spaces and algebraic varieties. Symp. Math. 28, 203--217 (1986) Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Research exposition (monographs, survey articles) pertaining to geometry, Grassmannians, Schubert varieties, flag manifolds Partial line spaces and algebraic varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author computes the number of points, in the Galois field \(GF(q)\) with \(q\) elements, of the Grassmann variety of \(d\)-planes in projective \(n\)-space, and all its Schubert subvarieties, all defined over \(GF(q)\). He observes in the case of Grassmann manifolds that the formula he obtains is the same as the one given by Sylvester for Gauss polynomials. number of points of Grassmann variety; Gauss polynomials; Sylvester's formula; Galois geometries; Galois field; Schubert subvarieties Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Enumerative problems (combinatorial problems) in algebraic geometry, Finite ground fields in algebraic geometry A geometric interpretation of an equality by Sylvester | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider Buch's rule for \(K\)-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong. Grassmannian; Richardson varieties; Grothendieck polynomials; Schur multiplicity free Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A combinatorial approach to multiplicity-free Richardson subvarieties of the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use classical Schubert calculus to give a direct geometric proof of the quantum version of Monk's formula [see \textit{S. Fomin, S. Gelfand} and \textit{A. Postnikov}, J. Am. Math. Soc. 10, No. 3, 565-596 (1997; Zbl 0912.14018)]. quantum Schubert polynomials Anders Skovsted Buch, A direct proof of the quantum version of Monk's formula, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2037 -- 2042. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds A direct proof of the quantum version of Monk's formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a local ring R, let \({\mathcal C}_ R\) be the category of R-modules of finite length and finite projective dimension, and denote by \(K_ 0({\mathcal C}_ R)\) its Grothendieck group. The main result of this paper is that if Spec(R) is a rational double point of a surface over an algebraically closed field of characteristic zero, then \(K_ 0({\mathcal C}_ R)={\mathbb{Z}}\). The second result is that if \(f: X\to Y\) is a resolution of singularities of a normal quasiprojective surface X with only quotient singularities, then the induced map \(f^*: K_ 0(X)\to K_ 0(Y)\) of Grothendieck groups (of vector bundles) induces an isomorphism \(F_ 0K_ 0(X)\to F_ 0K_ 0(Y)\) of Chow groups, where \(F_ 0K_ 0(X)\subset K_ 0(X)\) is the subgroup generated by the classes of smooth points. The author makes some remarks about the situation in characteristic \(p,\) and computes \(K_ 0({\mathcal C}_ R)\) up to p-torsion for a particular class of examples. modules of finite length; Grothendieck group; rational double point of a surface; resolution of singularities; Chow groups Srinivas, V., Modules of finite length and Chow groups of surfaces with rational double points, Ill. J. math., 31, 36-61, (1987) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Commutative rings and modules of finite generation or presentation; number of generators, Grothendieck groups, \(K\)-theory and commutative rings, Singularities in algebraic geometry Modules of finite length and Chow groups of surfaces with rational double points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this note, we give Gysin formulas for partial flag bundles for the classical groups. We then give Gysin formulas for Schubert varieties in Grassmann bundles, including isotropic ones. All these formulas are proved in a rather uniform way by using the step-by-step construction of flag bundles and the Gysin formula for a projective bundle. In this way we obtain a comprehensive list of new general formulas. The content of this paper was presented by Piotr Pragacz at the International Festival in Schubert Calculus in Guangzhou, November 6--10, 2017. push forward; Gysin maps; Segre polynomials; classical flag bundles; Kempf-Laksov bundles; Schubert bundles Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry Flag bundles, Segre polynomials, and push-forwards | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules. equivariant D-modules; Kostka polynomials; Poisson-de Rham homology; W-algebras; Springer fibers; nilpotent cone; Harish-Chandra homomorphism; Grothendieck-Springer resolution Bellamy, G., Schedler, T.: Kostka polynomials from nilpotent cones and Springer fiber cohomology. arXiv:1509.02520 (2015) Poisson algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grassmannians, Schubert varieties, flag manifolds Filtrations on Springer fiber cohomology and Kostka polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Involution Schubert polynomials represent cohomology classes of \(K\)-orbit closures in the complete flag variety, where \(K\) is the orthogonal or symplectic group. We show they also represent \(\mathsf{T}\) -equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables \(x_i+x_j\) , and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting. Schubert polynomials; pipe dreams; involutions; symmetric groups; spherical varieties Grassmannians, Schubert varieties, flag manifolds, Compactifications; symmetric and spherical varieties, Symmetric functions and generalizations Involution pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider a lattice of \(n\) sites arranged around a ring, with the \(n\) sites occupied by particles of weights \(\{1,2,\dots,n\}\); the possible arrangements of particles in sites thus corresponds to the \(n\)! permutations in \(S_n\). The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on the set of permutations, in which two adjacent particles of weights \(i<j\) swap places at rate \(x_i-y_{n+1-j}\) if the particle of weight \(j\) is to the right of the particle of weight \(i\). (Otherwise nothing happens.) In the case that \(y_i=0\) for all \(i\), the stationary distribution was conjecturally linked to Schubert polynomials by \textit{T. Lam} and \textit{L. Williams} [Exp. Math. 21, No. 2, 189--192 (2012; Zbl 1243.05240)], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues by \textit{A. Ayyer} and \textit{S. Linusson} [Adv. Appl. Math. 57, 21--43 (2014; Zbl 1329.60326)] and \textit{C. Arita} and \textit{K. Mallick} [J. Phys. A, Math. Theor. 46, No. 8, Article ID 085002, 11 p. (2013; Zbl 1278.82039)]. In the case of general \(y_i\), \textit{L. Cantini} [``Inhomogenous Multispecies TASEP on a ring with spectral parameters'', Preprint, \url{arXiv:1602.07921}] showed that \(n\) of the \(n\)! states have probabilities proportional to double Schubert polynomials. In this paper we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns 2413, 4132, 4213 and 3214. We show that there are \(\frac{(2+\sqrt{2})^{n-1} + (2-\sqrt{2})^{n-1}}{2}\) evil-avoiding permutations in \(S_n\), and for each evil-avoiding permutation \(w\), we give an explicit formula for the steady state probability \(\psi_w\) as a product of double Schubert polynomials. We also show that the Schubert polynomials that arise in these formulas are flagged Schur functions, and give a bijection in this case between multiline queues and semistandard Young tableaux. Schubert polynomials; TASEP; multiline queues Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Schubert polynomials and the inhomogeneous TASEP on a ring | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0731.00008.]
In this paper the Schur \(S\)-polynomials and the Schur \(Q\)-polynomials are studied. These polynomials are then applied to elimination theory, and Schubert calculus for Grassmannians of isotropic subspaces. Schur polynomials; elimination theory; Schubert calculus for Grassmannians Pragacz, Piotr, Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials.Topics in invariant theory, Paris, 1989/1990, Lecture Notes in Math. 1478, 130-191, (1991), Springer, Berlin Grassmannians, Schubert varieties, flag manifolds Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In their paper [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011)], \textit{W. Kraśkiewicz} and \textit{P. Pragacz} defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product of Kraśkiewicz-Pragacz modules always has KP filtration, i.e. a filtration whose each successive quotients are isomorphic to KP modules. In this paper we explicitly construct such filtrations for certain special cases of these tensor product modules, namely \(S_w \otimes S^d(K^i)\) and \(S_w \otimes \bigwedge^d(K^i)\), corresponding to Pieri and dual Pieri rules for Schubert polynomials. Schubert polynomials; Schubert functors; Kraśkiewicz-Pragacz modules; Schubert calculus Classical problems, Schubert calculus, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Kraśkiewicz-Pragacz modules and Pieri and dual Pieri rules for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This text grew out of an advanced course that the author taught at the Fourier Institute (Grenoble, France) during the 1995-96 academic year. The first part (chapter I) is of purely algebraic-combinatorial nature and provides a modern, very thorough introduction to the classical topic of symmetric functions, with a special emphasis on the combinatorics of Schur polynomials. The various properties of the family of Schur polynomials, which have proved to be a powerful tool in the representation theory of groups, are described by means of the combinatorial operations on Young diagrams (or Young tableaux) and the insertion algorithm of D. Knuth, and the corresponding various (classical and more recent) identities such as Pieri's formulae, Giambelli's formula, the rule of Littlewood-Richardson, and others are derived in the course of the discussion. This introductory chapter also contains a section on Kostka-Foulkes polynomials, including the proof of the Foulkes conjecture by A. Lascoux and M.-P. Schützenberger (1978), and some comments on the action of the symmetric group on the set of (semi-standard) Young diagrams.
The second part (chapter II) is devoted to the study of the so-called Schubert polynomials, which were introduced by A. Lascoux and M.-P. Schützenberger about twenty years ago. These polynomials, defined in terms of divided differences, are closely related to the Bruhat order on symmetric groups as well as to certain Hecke algebras of these groups, to the celebrated Yang-Baxter equation, and to certain Schur functions. All this plentiful, fascinating and fairly recent material is thoroughly covered by chapter II, together with numerous combinatorial applications. The third part (chapter III) is, in contrast, of purely geometric nature. Its main topic is the algebro-geometric, enumerative study of Schubert varieties inside Grassmannians and flag manifolds. The classical fact that the homology classes of Schubert varieties can be represented by Schur polynomials, and also by the more recently introduced Schubert polynomials, makes it possible to interpret most of the combinatorial results of the previous chapters in the language of algebraic geometry. Moreover, since Schubert varieties are also universal models for certain degeneracy loci of vector bundle morphisms, Schur or Schubert polynomials can be used to compute the homology classes of those degeneracy loci in terms of the characteristic classes of the involved vector bundles. This method is described and illustrated in the course of chapter III, whereat the basic concepts and facts from algebraic geometry (e.g.: Grassmannians, Schubert varieties, Chern classes of vector bundles, flag varieties, degeneracy loci of vector bundle maps) are briefly explained for the purpose of self-containedness of the text.
In addition, the author has included a concise appendix to the text, in which the fundamental notions from algebraic topology (i.e., singular homology and cohomology groups, fundamental classes and Poincaré duality, and some concepts from algebro-geometric intersection theory) are neatly compiled. As for the geometric part of this excellent text, the reader should be referred to the recent, more detailed, and really brilliant lecture notes ``Schubert varieties and degeneracy loci'' by \textit{W. Fulton} and \textit{P. Pragacz} [Lect. Notes Math. 1689 (1998)]. The text under review comes with numerous further-leading exercises and remarks, and with a rich bibliography, which makes the study of it very profitable. symmetric functions; Schur polynomials; Young diagrams; Schubert polynomials; Schubert varieties; degeneracy loci of vector bundle morphisms; Grassmannians; Chern classes; flag varieties Manivel, Laurent, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, 2-85629-066-3, Cours Spécialisés [Specialized Courses] 3, vi+179 pp., (1998), Société Mathématique de France, Paris Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Research exposition (monographs, survey articles) pertaining to combinatorics, Symmetric functions and generalizations, Representations of finite symmetric groups, Combinatorial aspects of representation theory Symmetric functions, Schubert polynomials and degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors introduce a new set of combinatorially defined nonsymmetric functions whose symmetrizations are Molev's dual Schur functions. \textit{A. I. Molev} [Electron. J. Comb. 16, No. 1, Research Paper R13, 44 p. (2009; Zbl 1182.05128)] described some properties of dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions and a multiplication rule for the dual Schur functions. Schur functions are an old subject and much is known about them. They are studied in relation to many different subjects from a number of different points of view. In this work they follow the Lascoux-Schützenberger approach [\textit{A. Lascoux} and \textit{M.-P. Schützenberger}, C. R. Acad. Sci., Paris, Sér. I 294, 447--450 (1982; Zbl 0495.14031)], viewing Schur functions as (symmetric) special cases of Schubert polynomials. From this point of view, it is natural to ask how one can define a larger set of nonsymmetric functions, which will include Molev's dual Schur functions as their symmetric counterparts. This theme is the main focus of their work. On the algebraic geometry side, they obtain a duality formula for the Schubert classes in Grassmannians in terms of rational Schubert (key) polynomials. Also they point out that a dominant rational Schubert polynomial can be described as a configuration of lines as in the work of \textit{S. Fomin} and \textit{A. N. Kirillov} [Discrete Math. 153, No. 1--3, 123--143 (1996; Zbl 0852.05078)]. dual Schur functions; Schubert polynomials; configuration of lines Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Rational Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations \(w\) with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\) for some \(h\). Kazhdan-Lusztig polynomials; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Hecke algebras and their representations Permutations with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X_w\) be the Schubert subvariety of the complete flag variety associated to a permutation \(w\) in the symmetric group \(S_n\). It is important to know which \(X_w\) are singular and where \(X_w\) is singular. The authors survey many recent results concerning this problems. They introduce a new combinatorial notion, a generalization of pattern avoidance, which they call interval pattern avoidance, and use this to explore the singularities of Schubert varieties and their local invariants.
In the last sections of the paper a computation approach to the problems based on the Macaulay 2 is discussed. The associated commutative algebra is that of Kazhdan-Lusztig ideals (a class of ideals generalizing classical determinantal ideals). The authors wrote the Macaulay 2 code Schubsingular as an exploratory complement to this paper. It is available at the authors' websites. Schubert varieties; singularities; Kazhdan-Lusztig polynomials; determinantal ideals A. Woo, A. Yong, Governing singularities of Schubert varieties, J. Algebra 320 (2008), no. 2, 495--520. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Governing singularities of Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Involution Schubert polynomials represent cohomology classes of \(K\)-orbits in the complete flag variety, where \(K\) is the orthogonal or symplectic group. We show that they also represent \(T\)-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables \(x_i + x_j\), and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. In Knutson and Miller's approach to matrix Schubert varieties [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)], pipe dream formulas reflect Gröbner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting. Schubert polynomials; pipe dreams; spherical orbits Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Involution pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials are explicit representatives for Schubert classes in the cohomology ring of a flag variety. Those of type \(A_n\) were introduced by \textit{A. Lascoux} and \textit{M. P. Schürzenberger} [Polynomes de Schubert, C. R. Acad. Sci. Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031)]. \textit{S. Billey} and \textit{M. Haiman} [Schubert polynomials for the classical groups, J. Am. Math. Soc. 8, No. 2, 443-482 (1995; Zbl 0832.05098)] extended the theory of \(A_n\)-Schubert polynomials for the groups of type \(B_n, C_n\) and \(D_n\) and their flag varieties using combinatorial methods. The starting point for the theory of Schubert polynomials is the observation of \textit{I. N. Bernstein, I. M. Gelfand} and \textit{S. I. Gelfand }[Schubert cells and cohomology of the spaces \(G/P\), Russ. Math. Surveys 28, No. 3, 1-26 (1973; Zbl 0286.57025)] that all Schubert classes can be computed by applying a sequence of divided difference operators to the cohomology class of highest codimension (the Schubert class of a point). For the type \(A_n\) Lascoux and Schürzenberger found a particular polynomial to represent the top cohomology class which yields Schubert polynomials that represent the Schubert classes simultaneously for all \(n\), the top polynomial. Billey and Haiman [loc. cit.] described the top polynomials of type \(B_n, C_n\) and \(D_n\). In the paper under review the author follows closely the original algebraic approach of Lascoux and Schürzenberger in type \(A_n\). He is able to present simple formulas for the top polynomials of type \(C_n\) and \(D_n\). He uses creation operators for \(Q\)-Schur and \(P\)-Schur functions which also allows him in types \(B_n, C_n\) and \(D_n\) to give: (1) formulas for the easy computation with all divided differences, (2) recursive structures, and (3) simplified derivations of basic properties. Schubert polynomial; Schur function; top polynomials; classical groups; Schubert variety Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory Schubert polynomials of types A-D | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review deals with type \(B\) analogues of the Schubert polynomials of Lascoux and Schützenberger. Let \(W=W_n\) be the Coxeter group of type \(A_n\) (i.e. the symmetric group \(S_{n+1}\)) with the natural action of \(W\) on \({\mathbb C}[x_1,\ldots,x_{n+1}]\). Let \(I_W\) be the ideal generated by the \(W\)-symmetric polynomials without constant terms. The Schubert polynomials \(X_w\) are laballed with the elements \(w\in W\) and are a distingushed linear basis of \({\mathbb C}[x_1,\ldots,x_{n+1}]/I_W\).
The authors choose five properties in the ``ordinary'' \(A\) case: (0) \(X_w\) is homogeneous of degree the length \(l(w)\) of \(w\) with respect to the natural set of generators \(s_i\) of \(W\). (1) \(X_w\) can be recursively defined by divided difference operators by \(\partial_i(X_{ws_i})=X_w\) if \(l(ws_i)=l(w)+1\). (2) For any \(u,v\in W\) and for a sufficiently large \(m\), in the multiplication of \(X_u\) and \(X_v\) one can get rid of the unpleasant ``mod \(I_W\)'', i.e. \(X_uX_v=\sum_{w\in W_m}c_{uv}^wX_w\). (3) The polynomials \(X_w\) are with nonnegative integer coefficients. (4) The Schubert polynomials are stable with respect to the natural embedding \(W_n\subset W_m\), \(n<m\), and the corresponding projection
\[
{\mathbb C}[x_1,\ldots,x_{m+1}]\longrightarrow {\mathbb C}[x_1,\ldots,x_{n+1}]
\]
sending the extra variables to 0.
The authors show that it is impossible to transfer all these properties to the case of the hyperoctahedral group \(B_n\), i.e. the group of symmetries of the \(n\)-dimensional cube. They require the properties (0) and (4) and construct \(B_n\)-Schubert polynomials satisfying also the conditions (1)-(2) and (2)-(3). The constructions are based on the exponential solution of the Yang-Baxer equation in the nil-Coxeter algebra . The authors show that the two kinds of Schubert polynomials are related by a certain ``change of variables''. Finally, they construct a family of polynomials \(X_w\) and conjecture that they satisfy the conditions (0), (1), (3) and (4). Recently the conjecture was solved affirmatively by Tao Kai Lam. Other interesting results involving, e.g., Stanley symmetric functions of type \(B\), are also obtained in this paper. Schubert polynomials; Yang-Baxer equation; symmetric functions; Coxeter groups Fomin, S.; Kirillov, A. N., Combinatorial \(B_n\)-analogues of Schubert polynomials, Trans. Am. Math. Soc., 348, 3591-3620, (1996) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Combinatorial \(B_ n\)-analogues of Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams. Schubert polynomials; bumpless pipe dreams Symmetric functions and generalizations, Classical problems, Schubert calculus Bijective proofs of Monk's rule for Schubert and double Schubert polynomials with bumpless pipe dreams | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A fundamental open problem in combinatorial representation theory and algebraic geometry is to give closed formulas for calculations in the cohomology ring of the full flag variety \(\mathcal F:=SL_n/B\) in terms of the basis of Schubert classes. The paper under review answers a part of this question by expressing the classes of certain subvarieties of the flag variety, called \textit{regular semisimple Hessenberg varieties}, in terms of the Schubert classes. Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics.
The authors give a ``Giambelli formula'' expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, they show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. They also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and they give closed combinatorial formulas for the coefficients in many cases. They introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use their results to determine when such schemes are reduced. Schubert polynomials; Hessenberg varieties Anderson, D; Tymoczko, J, \textit{Schubert polynomials and classes of Hessenberg varieties}, J. Algebra, 323, 2605-2623, (2010) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Schubert polynomials and classes of Hessenberg varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simply-connected simple algebraic group over \(\mathbb{C}\) and let \(K\subset C\) be a maximal compact subgroup with maximal torus \(T\subset K\). Let \({\mathcal G}r_G\) be the affine Grassmannian associated to \(G\). Then, \({\mathcal G}r_G\) is homotopy equivalent to the based continuous loop group \(\Omega_e(K)\). Let \(W\) be the (finite) Weyl group of \(G\) and let \(W_{\text{aff}}\) be the corresponding affine Weyl group. For any \(w\in W_{\text{aff}}/W\), let \(\Omega_w\subset{\mathcal G}r_G\) be the Bruhat cell. We denote by \(\sigma_w\in H_*({\mathcal G}r_G,\mathbb{Z})\) (resp., \(\sigma^w\in H^*({\mathcal G}r_G,\mathbb{Z})\)) the corresponding Schubert homology (resp., cohomology) class. Since \(\Omega_e(K)\) is a topological group, \(H_*({\mathcal G}r_G,\mathbb{Z})\) and \(H^*({\mathcal G}r_G,\mathbb{Z})\) are dual Hopf algebras.
Let \(\mathbb{A}_{\text{aff}}\) denote the Kostant-Kumar nil-Hecke ring associated to \(W_{\text{aff}}\). There is a canonical embedding of the \(T\)-equivariant cohomology \(S:= H^T(pt)\) of a point into \(\mathbb{A}_{\text{aff}}\). Then, as shown by Peterson, the centralizer \(Z_{\mathbb{A}_{\text{aff}}}(S)\) of \(S\) in \(\mathbb{A}_{\text{aff}}\) is isomorphic with the \(T\)-equivariant homology \(H_T({\mathcal G}r_G,\mathbb{Z})\). The \(\mathbb{Z}\)-algebra \(\mathbb{A}_{\text{aff}}\otimes_S\mathbb{Z}\) admits a subalgebra \(\mathbb{B}'\) called the affine Fomin-Stanley subalgebra. Lam shows that \(\mathbb{B}'\) is isomorphic with \(Z_{\mathbb{A}_{\text{aff}}}(S)\otimes_S\mathbb{Z}\); in particular, by Peterson's result, \(\mathbb{B}'\) is isomorphic with \(H_*({\mathcal G}r_G,\mathbb{Z})\) as a Hopf algebra. As a corollary of this, one obtains that \(\mathbb{B}'\) is a commutative algebra.
When \(G= SL_n(\mathbb{C})\), Bott identified \(H^*({\mathcal G}r_G,\mathbb{Z})\) (resp., \(H^*({\mathcal G}r_G,\mathbb{Z})\)) with a certain subring (resp., a certain quotient ring) of the ring \(\Lambda_{\mathbb{Z}}\) of symmetric functions over \(\mathbb{Z}\) in infinitely many variables. The principal result of this paper of Lam identifies the bases \(\sigma_w\) and \(\sigma^w\) as elements of \(\Lambda_{\mathbb{Z}}\). These are, respectively, the \(k\)-Schur functions (introduced by Lapointe-Lascoux-Morse) and the affine Schur functions (introduced by Lapointe-Morse and also Lam). In the homology this was conjectured by M. Shimozono and in the cohomology this was conjectured by J. Morse.
The above identifications allow one to conclude that the product structure constants for the \(k\)-Schur functions as well as the affine Schur functions are nonnegative integers by virtue of the corresponding nonnegativity results for the product of the homology Schubert classes \(\{\sigma_w\}\) (due to Peterson) and the product of the cohomology Schubert classes \(\{\sigma^w\}\) (due to Kumar-Nori). Schubert polynomials; symmetric functions; Schubert calculus; affine Grassmannian; based continuous loop group; affine Weyl group; Kostant Kumar; centralizer; Fomin Stanley subalgebra; Hopf algebra T. Lam. ''Schubert polynomials for the affine Grassmannian''. J. Amer. Math. Soc. 21 (2008), pp. 259--281.DOI. Symmetric functions and generalizations, Classical problems, Schubert calculus Schubert polynomials for the affine Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe a bijective proof of Macdonald's reduced word identity using pipe dreams and Little's bumping algorithm. The proof extends to a principal specialization of the identity due to \textit{S. Fomin} and \textit{R. P. Stanley} [Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)]. Our bijective tools also allow us to address a problem posed by \textit{S. Fomin} and \textit{A. N. Kirillov} [J. Algebr. Comb. 6, No. 4, 311--319 (1997; Zbl 0882.05010)], using work of \textit{M. L. Wachs} [J. Comb. Theory, Ser. A 40, 276--289 (1985; Zbl 0579.05001)], \textit{C. Lenart} [J. Algebr. Comb. 20, No. 3, 263--299 (2004; Zbl 1056.05146)] and \textit{L. Serrano} and \textit{C. Stump} [Electron. J. Comb. 19, No. 1, Research Paper P16, 18 p. (2012; Zbl 1244.05239)]. reduced words; Schubert polynomials; RC graphs; pipe dreams; bijective proof; algorithmic bijection Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus, Permutations, words, matrices A bijective proof of Macdonald's reduced word formula | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The remarkable role of Schur functions and Schubert polynomials is known in the group representation theory; [\textit{I. N. Bernstein}, \textit{I. M. Gelfand}, \textit{S. I. Gelfand}, Usp. Mat. Nauk 28, No.3, 3-26 (1973; Zbl 0286.57025)] and [\textit{W. Kraskiewicz}, \textit{P. Pragacz}, C. R. Acad. Sci., Paris, Sér. I 304, 209-211 (1987; Zbl 0642.13011)]. This important paper is a concise and readable summary (the proofs are not given) of a ``noncommutative lifting'' of these notions and the corresponding basic facts. In this lifting the notions `tableau' and `nilplaxtic relation' are central.
Let us fix some (ordered) alphabet \(X=\{x_ 1<x_ 2<...\}\). In the free monoid \(X^*\) a word \(w=x_{i_ 1}x_{i_ 2}...x_{i_ k}\) with \(x_{i_ 1}>x_{i_ 2}>...>x_{i_ k}\) is called column; its length \(| w|\) is k and its contents C(w) is the set \(\{x_{i_ 1},...,x_{i_ k}\}\). If for two given columns v and w there exists a nonincreasing injection C(v)\(\to C(w)\), then it is said that w majorizes v (denoted by \(w\gg v)\). A finite product \(t=v_ 1v_ 2..\). of columns such that \(v_ 1\gg v_ 2\gg..\). is called a tableau; its shape is the partition \((| v_ 1|,| v_ 2|,...)\) of \(| t|\). A tableau t is called standard if its shape is a permutation on the set \(\{\) 1,2,...,\(| t| \}\); t is called key if for all k, \(v_{k+1}\) is a subword of \(v_ k\). On \(X^*\) there exist two important equivalences: the plaxtic (\(\equiv)\) and nilplaxtic (\(\cong)\) relation. The first is generated by elementary relations (PL1) \(a_ ka_ ia_ j\equiv a_ ia_ ka_ j\), \(a_ ja_ ia_ k\equiv a_ ja_ ka_ i\) and (PL2) \(a_ ja_ ia_ j\equiv a_ ja_ ja_ i\), \(a_ ja_ ia_ i\equiv a_ ia_ ja_ i\), (PL1) and (PL2) both in the case \(i<j<k\). The second is given also by (PL1) and (PL2) in all cases excluding that of i, j with \(j=i+1\)- then (PL2) is replaced by (Nil PL): \(a_ ia_{i+1}a_ i\cong a_{i+1}a_ ia_{i+1}\), \(a_ ia_ i=0\). The Schensted construction is primarily concerned with the plaxtic relation, it extends to the nilplaxtic case (Th. 1). The nilplaxtic relations contain Coxeter relations for transpositions. So the set of reduced decompositions of any permutation \(\mu\) is a disjoint union of nilplaxtic classes. Any such class (not containing 0) contains exactly one tableau t and its elements are in 1-1 correspondence with standard tableaux of the same shape as t. Using these remarkable facts the paper describes the way how general (noncommutative) Schubert polynomials \(X_{\mu}\) can be introduced as elements of \({\mathbb{Z}}<X>\) and how they can be found as certain sums of tableaux connected with the reduced decompositions of \(\mu\). Schur functions; Schubert polynomials; nilplaxtic relation; alphabet; monoid; word; Schensted construction; Coxeter relations; transpositions; nilplaxtic classes; standard tableaux; sums of tableaux; reduced decompositions Lascoux, A.; Schützenberger, M. P., Tableaux and noncommutative Schubert polynomials, Funct. Anal. Appl., 23, 223-225, (1990) Representations of finite symmetric groups, Free semigroups, generators and relations, word problems, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Combinatorial identities, bijective combinatorics, Symmetric functions and generalizations Noncommutative Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the \(K\)-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2. dual stable Grothendieck polynomials; symmetric functions; Schur functions; plane partitions; Young tableaux Galashin P, Grinberg D and Liu G 2016 Refined dual stable Grothendieck polynomials and generalized Bender--Knuth involutions \textit{Electron. J. Comb.}23 3--14 Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G = G(m, N)\) denote the Grassmannian of \(m\)-dimensional subspaces of \(\mathbb C_N\). To each integer partition \(\lambda = (\lambda_1, \ldots, \lambda_m)\) whose Young diagram is contained in an \(m \times (N- m)\) rectangle, they associate a Schubert class \(\sigma_{\lambda}\) in the cohomology ring of \(G\). The special Schubert classes \(\sigma_1, \ldots, \sigma_{N-m}\) are the Chern classes of the universal quotient bundle \(\mathcal Q\) over \(G(m, N)\); they generate the graded cohomology ring \(H^\ast(G,\mathbb Z)\). The classical Giambelli formula is an explicit expression for \(\sigma_{\lambda}\) as a polynomial in the special classes.
In this paper the authors prove: Let \(X\) be a symplectic or odd orthogonal Grassmannian. They prove a Giambelli formula which expresses an arbitrary Schubert class in \(H^\ast(X,\mathbb Z)\) as a polynomial in certain special Schubert classes. This polynomial will be called a theta polynomial, is defined using raising operators, and we study its image in the ring of Billey-Haiman Schubert polynomials. Giambelli formula; isotropic Grassmannians; raising operators; theta polynomials; Schubert polynomials Buch, A.; Kresch, A.; Tamvakis, H., \textit{A Giambelli formula for isotropic Grassmannians}, Selecta Math. (N.S.), 23, 869-914, (2017) Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds A Giambelli formula for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42--99 (1987; Zbl 0616.05005)] generalized the Robinson-Schensted-Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman-Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well. Schubert polynomials; Demazure characters; key polynomials; RSK; Edelman-Greene insertion; reduced words Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Symmetric functions and generalizations, Permutations, words, matrices, Exact enumeration problems, generating functions, Combinatorial identities, bijective combinatorics A generalization of Edelman-Greene insertion for Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove that twisted versions of Schubert polynomials defined by \(\widetilde{\mathfrak{S}}_{w_0} = x_1^{n-1}x_2^{n-2} \cdots x_{n-1}\) and \(\widetilde{\mathfrak{S}}_{ws_i} = (s_i+\partial_i)\widetilde{\mathfrak{S}}_w\) are monomial positive and give a combinatorial formula for their coefficients. In doing so, we reprove and extend a previous result about positivity of skew divided difference operators and show how it implies the Pieri rule for Schubert polynomials. We also give positive formulas for double versions of the \(\widetilde{\mathfrak{S}}_w\) as well as their localizations. combinatorial formula; positivity of skew divided difference operators; Pieri rule for Schubert polynomials Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Twisted Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main object of study is the space \(F_n\) of complete flags in \({\mathbb{C}}^n\) with the aim of describing intersections of pairs of Schubert cells. The Schubert cell decomposition \(D_f\) of \(F_n\) relative to the flag \(f\) consists of cells formed by all flags having a given set of dimensions of intersections with subspaces of \(f\). Associated to a pair of flags are their refined double strata. The first theorem in this very detailed study is that each refined double stratum is biholomorphically equivalent to the product of a complex torus by a complex linear space. Schubert cell; complete flags; Hodge structure; refined double stratum Boris Shapiro, Michael Shapiro, and Alek Vainshtein, On combinatorics and topology of pairwise intersections of Schubert cells in \?\?_{\?}/\Cal B, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 397 -- 437. Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Combinatorial aspects of matroids and geometric lattices, Homogeneous spaces and generalizations On combinatorics and topology of pairwise intersections of Schubert cells in \(SL_ n/B\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of \textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] can be expressed by a Jacobi-Trudi-type determinant formula generalizing the work of \textit{T. Hudson} and \textit{T. Matsumura} [Eur. J. Comb. 70, 190--201 (2018; Zbl 1408.14030)]. We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide. Grothendieck polynomials; flagged set-valued tableaux; vexillary permutations; Jacobi-Trudi formula Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Flagged Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The fundamental slide basis of polynomials was recently introduced by the authors [J. Comb. Theory, Ser. A 156, 85--118 (2018; Zbl 1381.05084)]. We survey positivity properties of this basis, and applications to the important Schubert and key bases of polynomials. fundamental slide polynomials; Schubert polynomials; key polynomials; fundamental quasisymmetric functions; quasi-Schur functions; Kohnert tableaux Combinatorial aspects of representation theory, Symmetric functions and generalizations, Classical problems, Schubert calculus Slide polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We make a broad conjecture about the \(k\)-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the \(k\)-Schur expansion of (1) Hall-Littlewood polynomials, proving the \(q = 0\) case of the strengthened Macdonald positivity conjecture from [\textit{L. Lapointe} et al., Duke Math. J. 116, No. 1, 103--146 (2003; Zbl 1020.05069)]; (2) the product of a Schur function and a \(k\)-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) \(k\)-split polynomials, solving a substantial special case of a problem of \textit{B. Broer} [Prog. Math. 123, 1--19 (1994; Zbl 0855.22015)] and \textit{M. Shimozono} and \textit{J. Weyman} [Eur. J. Comb. 21, No. 2, 257--288 (2000; Zbl 0956.05100)] on parabolic Hall-Littlewood polynomials. In addition, we prove the conjecture that the \(k\)-Schur functions defined via \(k\)-split polynomials [\textit{L. Lapointe} and \textit{J. Morse}, J. Comb. Theory, Ser. A 101, No. 2, 191--224 (2003; Zbl 1018.05101)] agree with those defined in terms of strong tableaux [\textit{T. Lam} et al., Affine insertion and Pieri rules for the affine Grassmannian. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1208.14002)]. Macdonald polynomials; Gromov-Witten invariants; Schubert structure constants; parabolic Hall-Littlewood polynomials; strong tableaux Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations \(k\)-Schur expansions of Catalan functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Suppose \(K\subseteq\mathrm{GL}(n,\mathbb C)\) is a closed subgroup which acts on the complete flag variety with finitely many orbits. When \(K\) is a Borel subgroup, these orbits are Schubert cells, whose study leads to Schubert polynomials and many connections to type A Coxeter combinatorics. When \(K\) is \(\mathrm{O}(n,\mathbb C)\) or \(\mathrm{Sp}(n,\mathbb C)\), the orbits are indexed by some involutions in the symmetric group. \textit{B. Wyser} and \textit{A. Yong} [Transform. Groups 22, No. 1, 267--290 (2017; Zbl 1400.14130)] described polynomials representing the cohomology classes of the orbit closures, and we investigate parallels for these ``involution Schubert polynomials'' of classical combinatorics surrounding type A Schubert polynomials. We show that their stable versions are Schur-P-positive, and obtain as a byproduct a new Littlewood-Richardson rule for Schur P-functions.
A key tool is an analogue of weak Bruhat order on involutions introduced by \textit{R. W. Richardson} and \textit{T. A. Springer} [Geom. Dedicata 35, No. 1--3, 389--436 (1990; Zbl 0704.20039); complements ibid. 49, No. 2, 231--238 (1994; Zbl 0826.20045)]. This order can be defined for any Coxeter group \(W\), and its labelled maximal chains correspond to reduced words for distinguished elements of \(W\) which we call atoms. In type A we classify all atoms, generalizing work of \textit{M. B. Can}, \textit{M. Joyce} and \textit{B. Wyser} [``Wonderful symmetric varieties and Schubert polynomials'', Ars Math. Contemp. (to appear)], and give a connection to the Chinese monoid of \textit{J. Cassaigne} et al. [Int. J. Algebra Comput. 11, No. 3, 301--334 (2001; Zbl 1024.20046)].We give a different description of some atoms in general finite \(W\) in terms of strong Bruhat order. Schubert polynomials; Coxeter combinatorics; spherical orbits Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects), Linear algebraic groups over arbitrary fields Involution Schubert-Coxeter combinatorics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One of the main concerns of this multifaceted paper is that of dealing with the quantum cohomology of flag varieties parametrizing inclusions of subspaces of a given vector space, as well as to supply an innovative description of the product structure of the affine Grassmannian \(\mathrm{Gr}\) associated to the group \(\mathrm{SL}_n({\mathbb C})\). Just not to loose the less experienced reader, we recall that the \(\mathrm{Gr}\) we alluded above is in fact the quotient of the group of all \(n\times n\) unimodular square matrices with entries in the ring of Laurent series modulo the action of the subgroup of unimodular square matrices with entries in the ring of formal power series with complex coefficients. The new sharp description that the authors offer for the cohomology \(H^*(\mathrm{Gr})\) and homology \(H_*(\mathrm{Gr})\), whose details are richly displayed in Section 5, definitely shed light of the intimate relationship of the subject, the so-called affine Schubert calculus, with the rich, celebrated and still mysterious theory of Macdonald polynomials. The latter is about some of the richest objects in mathematics. As the authors themselves declare in a enthusiastically inspiring introduction, Macdonald polynomials do not occur just in combinatorics, but also in the theory of double affine Hecke algebras, quantum relativistic systems, diagonal harmonics and Hilbert schemes of points in the plane.
The origin of the noble story told by the authors in the paper, has to do with a basic and a fundamental question. Nearly every professional mathematician is aware that symmetric polynomials admit a basis of Schur polynomials, parametrized by partitions of non-negative integers. Although we shall not recall here their definition, Macdonald polynomials may be seen as symmetric polynomials depending on two extra parameters, say \(t\) and \(q\), and it is then natural to wonder about the transition matrix relating them with the more familiar Schur polynomials. The still open conjecture is that the entries of the transition matrix, a generalization of the so called Kostka-Foulkes polynomial, are polynomials with non-negative integer coefficients. In other words, Macdonald polynomials relate positively to Schur polynomials, a conjecture that inspired many more researches whose output has been the dramatic emerging of the relationship with the affine Schubert calculus. The authors so come to cope with the problem of a more flexible description of the homology and cohomology of \(\mathrm{Gr}\) by introducing a clever new combinatorial tool, which is of crucial importance in all the paper, called Affine Bruhat Countertableaux (ABC). Their generating functions form a basis of \(H^*(\mathrm{Gr})\) and everything leads to a refinement of the Kotska-Foulkes polynomials. The authors deal also with the problem of providing a closer description of the constants structure of the quantum cohomology of flag varieties, where Gromov-Witten invariants related with the art of counting rational curves in homogeneous varieties. There are many more beautiful and interesting features, in this paper, that deserve to be discussed, but this at the price of giving a more detailed account of the fine technical combinatorial tools masterly employed by the authors. This cannot be evidently done in a review, but we can end it by quickly describing the organization. Let us start from the abstract: it already contains the juice of the article and say the reader what it can be found inside. The introduction is simply as beautiful as exciting and is enhanced by the second section where there is a useful interesting account of the related literature. The preliminaries are collected in Section 3: here the reader can be made aware with selected tools from the theory of symmetric functions, explain the basic vocabulary related with Ferrer diagrams, horizontal strips, addable corners, extremal cells and so on. Section 4 enters into the deep core of the paper, being devoted to the affine Pieri's rule, described in terms of sophisticated but versatile combinatorics. The explicit representative of Schubert classes is provided in this section, where the ABC order is also introduced. More relations with Macdonald polynomials are collected in Section 6. The rich reference list is still preceded by an appendix where the output of a Sage routine is displayed to check a conjecture on the equality of two kinds of symmetric functions and by section 7, eventually devoted to the quantum cohomology of the flags. Macdonald polynomials; Hall-Littlewood polynomials; affine Schubert calculus; quantum Schubert calculus; type-A affine Weyl group; affine Grassmannian; Gromov-Witten invariants; Bruhat order; weak \(k\)-Pieri rule; \(k\)-tableaux; affine Bruhat counter-tableaux Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Classical problems, Schubert calculus, Enumerative combinatorics Quantum and affine Schubert calculus and Macdonald polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The question of when two skew Young diagrams produce the same skew Schur function has been well studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the \(K\)-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons. symmetric functions; Grothendieck polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus Coincidences among skew stable and dual stable Grothendieck polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper is devoted to the study of the double affine Hecke algebra (DAHA) of type \(A_1\) and its degenerations. DAHA has two parameters \(q\) and \(t\), and has a natural polynomial representation. In a series of earlier papers, the first author constructed the non-symmetric generalizations of Macdonald polynomials defined as eigenvectors of a certain commutative subalgebra inside DAHA.
In present paper, the authors write explicit formulas for nonsymmetric Macdonald polynomials of type \(A_1\) and study their degenerations in the limit \(t\to 0\). In this limit, DAHA is transformed to nil-DAHA, which can be written explicitly by generators and relations, and nonsymmetric Macdonald polynomials are transformed to a certain nonsymmetric analogue of \(q\)-Whittaker functions. A possible connection to the work of \textit{A. Givental} and \textit{Y.-P. Lee} [Invent. Math. 151, No. 1, 193-219 (2003; Zbl 1051.14063)] on quantum \(K\)-theory of flag varieties is also discussed. double affine Hecke algebras; nil-DAHAs; nonsymmetric Macdonald polynomials; Whittaker functions I. Cherednik and D. Orr. ''One-dimensional nil-DAHA and Whittaker functions I''. Trans form. Groups 17 (2012), pp. 953--987.DOI. Hecke algebras and their representations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds One-dimensional nil-DAHA and Whittaker functions. I. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. S. Buch} et al. [Math. Ann. 340, No. 2, 359--382 (2008; Zbl 1157.14036)] defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials \(G_\pi\) indexed by permutations in the basis of stable Grothendieck polynomials \(G_\lambda\) indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of ``orthogonal'' and ``symplectic'' shifted analogues of \(G_\pi\) in Ikeda and Naruse's basis of \(K\)-theoretic Schur \(P\)-functions. symmetric groups; Grothendieck polynomials; Hecke insertion; Schur \(P\)-functions; flag varieties Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds A symplectic refinement of shifted Hecke insertion | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``The double discrete logarithm has attracted interest as a one-way function in cryptography, in particular in group signature schemes and publicly verifiable secret sharing schemes. We obtain lower bounds on the degrees of polynomials interpolating the double discrete logarithm in multiplicative subgroups of a finite field and in the group of points on an elliptic curve over a finite field, respectively. These results support the assumption of hardness of the double discrete logarithm if the parameters are properly chosen. Similar results for other cryptographic one-way functions including the discrete logarithm, the Diffie-Hellman mapping and related functions as well as functions related to the integer factoring problem have already been known to the literature. The investigations on the double discrete logarithm in this paper are motivated by these results on other cryptographic functions.''
In more detail it is proved: Theorem 3: Let \(p > 3\) be a prime, \(E\) an elliptic curve over \(\mathbb F_p\) and \(P\) a point on \(E\) of prime order \(t\). For \(1\leq k\leq t-1\) let the first coordinate of \(kP\) be denoted by \(x_k\). Let \(h\in\mathbb Z^*_t\) be an element of order \(m\geq 2\) and let \(S\) be a subset of \(\{0, 1,\dots, \min\{m, p\}-1\}\) of cardinality \(m-s\). Let \(F(X)\in\mathbb F_p[X]\) satisfy \(F(x_{h^n})=n\), \(n\in S\), then we have
\[
\deg(F)\geq \frac 1{4\cdot 2^{2(t-1)/m}} (\min\{m, p\}-2s).
\]
For non-supersingular elliptic curves over finite fields of characteristic 2 the bound \(\deg(F)\geq \frac{m-4s}{2h^2}\) is proved (Theorem 4). double discrete logarithm; interpolation polynomials; finite fields; elliptic curves Meletiou, G.C., Winterhof, A.: Interpolation of the double discrete logarithm. In: von~zur Gathen, J. Imaña, J.L., Koç, Ç.K. (eds.) Arithmetic of Finite Fields, 2nd International Workshop, WAIFI 2008, Siena, Italy, July 6-9, 2008. Lecture Notes in Computer Science, Vol. 5130, pp. 1-10. Springer (2008). Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry Interpolation of the double discrete logarithm | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study a twisted action of the symmetric group on the cohomology ring \(R\) of the variety of complete flags in a complex vector space \(V\) of dimension \(n\). It is known that \(R\) can be identified with the polynomial ring \(\mathbb{C}[x_1, \dots, x_n]\) in \(n\) variables modulo the ideal \(I^+\) of symmetric polynomials without constant term. The usual basis of \(R\) is given by Schubert polynomials (expressing the Schubert cycles in terms of the codimension one Schubert cycles represented by \(x_1, \dots, x_n\)). The usual Demazure operators \(\partial_i\) act on \(R\) and the authors study the action of the symmetric group on \(R\) given by the operators \(s_i = \sigma_i + \partial_i\), where \(\sigma_i\) denotes a simple transposition (reflection). The algebra generated by \(s_i\) and \(x_i\) is isomorphic to the degenerate affine Hecke algebra \(\mathcal H\) considered by Cherednik. The authors construct certain elements in \(\mathcal H\) (Yang-Baxter operators) and use these to define a bilinear form on \(R\). A distinguished basis with respect to this form is extracted (termed affine Schubert polynomials). This is then applied to computing Schubert expansions of Chern classes. cohomology ring; action of the symmetric group; degenerate affine Hecke algebra; variety of complete flags; Schubert polynomials; Schubert cycles Lascoux A., Leclerc B., Thibon J.-Y.: Twisted action of the symmetric group on the cohomology of the flag manifold. Banach Center Publications, Vol. 36, 1996, pp. 111--124 Grassmannians, Schubert varieties, flag manifolds, Classical real and complex (co)homology in algebraic geometry, Group actions on varieties or schemes (quotients) Twisted action of the symmetric group on the cohomology of a flag manifold | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author provides an alternative proof of the ``component formula'' of Knutson, Miller and Shimozono, replacing the ``Gröbner degeneration part'' of their original argument by a combinatorial result (Theorem 1 in the paper). Moreover, he outlines a geometric project suggested by the striking correspondence between the ``splitting'' formula for the ordinary Schubert polynomials of Lascoux and Schützenberger in terms of quiver coefficients (stated as Theorem 2 in the paper), and an analogous formula (Theorem 3 in the paper) for the BCD-Schubert polynomials of Billey and Haiman in terms of positive combinatorial coefficients. degeneracy loci; quiver polynomials; generalized Littlewood--Richardson coefficients; Schubert polynomials Yong, A.: On combinatorics of quiver component formulas. J. Algebr. Comb. 21, 351--371 (2005) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus On combinatorics of quiver component formulas | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors present the theory of Schur and Schubert polynomials, revisited from the point of view of generalized Thom polynomials. The Schur and Schubert polynomials are realized as first obstructions of certain fiber bundles. After presenting some results about the Thom polynomials for group actions, the authors obtain the calculation of these polynomials via the method of restriction equations. Then they obtain a new definition for the Schur and Schubert polynomials by applying the general method to compute them as Thom polynomials. They also redefine the double Schubert polynomials and the Kempf-Laksov-Schur polynomials Schur polynomials; Schubert polynomials; Thom polynomials; method of restriction equations Fehér, L.; Rimányi, R., Schur and Schubert polynomials as thom polynomials--cohomology of moduli spaces, Cent. eur. J. math., 1, 4, 418-434, (2003) Singularities of differentiable mappings in differential topology, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Schur and Schubert polynomials as Thom polynomials -- cohomology of moduli spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected complex reductive algebraic group, \(B\) a Borel subgroup of \(G\) and \(T\) a maximal torus in \(B\). By the Bruhat decomposition, the generalized flag variety \(G/B\) is the disjoint union of the subsets \(BwB/B\) where \(w\) runs over all elements of the Weyl group \(W\) of \(G\). The subset \(BwB/B\) is called a Schubert cell, and its closure \(X_w\) is called a Schubert variety. The action of \(B\) on \(G/B\) leaves stable Schubert varieties, and in general there are multiple intermediate parabolic subgroups \(B\subset P\subset G\) that leave a given Schubert variety stable.
The article under review focuses on the question whether, given such a Schubert variety \(X_w\) and parabolic subgroup \(P\), \(X_w\) is spherical under the action of a Levi subgroup \(H\) of \(P\). The latter means that a Borel subgroup \emph{of \(H\)} acts with an open orbit in \(X_w\). An obvious example is given by the full generalized flag variety \(G/B\) which, by the Bruhat decomposition again, is a spherical Schubert variety under the action of \(H=P=G\). The article studies, more precisely, a combinatorial criterion which conjecturally characterizes when such a property holds. This conjecture is proved for rank two simple groups in the paper, as well as some other special cases, and has since been proved for type A groups [\textit{Y. Gao} et al., ``Classification of Levi-spherical Schubert varieties'', Preprint, \url{arXiv:2104.10101}]. Without stating the technical criterion in detail here, it should be noted that the criterion depends only on the Weyl group. In particular, the definition of the combinatorial counterpart of the Schubert variety being spherical is extended in the paper to finite Coxeter systems.
A significant part of the article focuses on the type A case, that is, when \(G=GL_n\) and the Weyl group is \(\mathfrak{S}_n\). In this setting, the combinatorial counterpart of sphericality is given a conjectural reformulation in terms of a pattern avoidance property for permutations, which has since been proved by \textit{C. Gaetz} [``Spherical Schubert varieties and pattern avoidance'', Preprint, \url{arXiv:2104.03264}]. The sphericality property is further studied from a polynomial ring point of view by introducing the notion of split-symmetric polynomials (polynomials in \(n\)-variables which are symmetric in subsets of variables, according to a partition of \(n\)). This point of view allows to prove the aforementioned partial results towards the conjecture, and initial steps towards an algorithmic characterization of sphericality. This approach is also used in a companion paper [\textit{R. Hodges} and \textit{A. Yong}, ``Multiplicity-free key polynomials'', Preprint, \url{arXiv:2007.09229}]. Schubert varieties; spherical varieties; key polynomials; split symmetry Compactifications; symmetric and spherical varieties, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Group actions on varieties or schemes (quotients) Coxeter combinatorics and spherical Schubert geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this survey, we describe and relate various occurrences of quivers of type \(A\) (both finite and affine) and their canonical bases in combinatorics, in algebraic geometry and in representation theory. The ubiquity of these quivers makes them especially important to study: they are pervasive in very classical
topics (such as the theory of symmetric functions) as well as in some of the most recent and exciting areas of representation theory (such as representation theory of quantum affine algebras, affine Hecke algebras or Cherednik algebras).
There is a vast literature on the subject and we have been forced to make a choice in the selection of the results presented here. We believe that one of the main reasons for the omnipresence of quivers of type \(A\) is the fact that they are related (in more than one way) to classical and fundamental objects in geometric representation theory of type \(A\) such as (partial) flag varieties and nilpotent orbits. This is the point we tried to emphasize in this survey. For that reason, many interesting and important results are only alluded to or sketched here and we apologize to all those whose work we did not mention by lack of space and competence. Also, we do not give complete proofs of all results presented here and refer to the original papers whenever possible. Rather, we have tried to convey the fundamental ideas in a nontechnical way as much as possible. quivers of type \(A\); representation spaces; Ringel-Hall algebras; canonical bases; categories of representations; symmetric functions; quantum groups; flag varieties; nilpotent orbits; Schubert varieties; Kazhdan-Lusztig polynomials; Schur-Weyl duality; affine Hecke algebras Schiffmann, O., Quivers of type \textit{A}, flag varieties and representation theory, Fields inst. commun., vol. 40, 453-479, (2004), Amer. Math. Soc. Providence, RI Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations, Symmetric functions and generalizations Quivers of type \(A\), flag varieties and representation theory. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper gives a sufficient criterion for Schubert intersection numbers to vanish which can be executed in polynomial time. Schubert intersection numbers arise from Schubert varieties \(X_{w}\), which are certain subvarieties of the flag variety \(X\) indexed by permutations \(w \in S_{n}\). The Poincaré duals \(\sigma_{w} = [X_{w}]\) of Schubert varieties form a basis of the cohomology ring \(H^\ast(X)\) of the flag variety. A Schubert problem is a \(k\)-tuple \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\) of permutations in \(S_{n}\) such that the sum of the lengths is \(\binom{n}{2}\). The Schubert intersection number \(C_{w^{(1)}, w^{(2)}, \dots, w^{(k)}}\) is equivalently either
\begin{itemize}
\item the multiplicity of \(\sigma_{w_{0}}\) in \(\prod_{i = 1}^k \sigma_{w^{(i)}}\), or
\item the number of points in \(\bigcap_{i = 1}^k g_{i}X_{w^{(i)}}\), where each \(g_{i}\) lies in some dense open subset of \(GL_{n}\).
\end{itemize}
Finding a combinatorial counting rule for Schubert intersection numbers is a famous open problem. Many algorithms exist for computing them. As stated previously, the aim of the paper is to give an algorithm to decide whether or not they vanish in a given case.
The algorithm which the paper introduces proceeds roughly as follows. Given \(w \in S_{n}\), one can define the Rothe diagram \(D(w)\) as a certain subset of the boxes of the \(n \times n\) grid. Given a Schubert problem \((w^{(1)}, w^{(2)}, \dots, w^{(k)})\), one concatenates the Rothe diagrams \(D(w^{(i)})\) to obtain a diagram \(D\). One then considers the fillings of \(D\) which obey certain rules. If there are no such fillings, then the Schubert intersection number vanishes. There is then a polynomial-time algorithm to determine whether there are indeed no such fillings.
The arguments of the paper use generalised permutahedra, which are obtained from the standard permutahedron by degeneration. The particular generalised permutahedra that the authors are interested in are known as ``Schubitopes''. These are constructed from fillings of rectangular grids. In the case of the Rothe diagram \(D(w)\) mentioned above, the Schubitope obtained is the Newton polytope of the Schubert polynomial \(\mathfrak{S}_{w}\) of \(w\) by [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]. The crux is that the emptiness of the set of fillings from the previous paragraph is equivalent to the presence of a certain point in a Schubitope by \textit{A. Adve} et al. [Sémin. Lothar. Comb. 82B, Paper No. 52, 12 p. (2019; Zbl 1436.05115)]. This can then be decided in polynomial time by standard linear programming methods, given the algorithm mentioned above.
The paper finishes by comparing the test for vanishing of Schubert intersection numbers proven in the paper to three other tests from the literature. In each instance, there are some cases in which the test from the paper can decide and which the test from the literature cannot, and also some cases where the opposite is true. We mention finally that the paper contains a couple of variations on its main result. Schubitopes; Newton polytopes; Schubert polynomials; Schubert intersection numbers Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Linear programming Generalized permutahedra and Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using \textit{A. Kohnert}'s algorithm [Bayreuther Math. Schr. 38, 1--97 (1991; Zbl 0755.05095)], we associate a polynomial to any cell diagram in the positive quadrant, simultaneously generalizing Schubert polynomials and \(\mathrm{GL}_n\) Demazure characters. We survey properties of these Kohnert polynomials and their stable limits, which are quasisymmetric functions. As a first application, we introduce and study two new bases of Kohnert polynomials, one of which stabilizes to the skew-Schur functions and is conjecturally Schubert-positive, the other stabilizes to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; monomial slide polynomials Combinatorial aspects of representation theory, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Skew polynomials and extended Schur functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a Coxeter group generated by a reflection set \(S\) satisfying relations \((st)^{m_{st}}=e\) such that \(m_{st}=1\) if and only if \(s=t\). The Poincaré series \(P_w(q)=\sum_{x\leq w}q^{\ell(x)}\) of an element \(w\in W\) is a polynomial of degree \(l(w)\). An element \(w\) is said to be palindromic (or rationally smooth) if the coefficients of \(P_w(q)\) are the same whether read from top degree to bottom degree, or in reverse.
A triangle group is a Coxeter group with \(|S|=3\). A Coxeter group contains the triangle \((a,b,c)\) if there is a subset \(\{r,s,t\}\subseteq S\) such that \((a,b,c)=(m_{rs},m_{rt},m_{st})\). If \(S\) contains no such subset then we say \(W\) avoids the triangle \((a,b,c)\). Let \(\mathrm{HQ}:=\{(2,b,c)\mid b,c\geq 3\) and \(b<\infty\}\).
The main result that the authors obtain is the following: Let \(W\) be a Coxeter group which avoids all triangle groups in HQ. Then every 4-palindromic \(w\in W\) is palindromic. Furthermore, if \(W\) avoids all triangle groups \((3,3,c)\) where \(3<c<\infty\), then every 2-palindromic \(w\in W\) is palindromic. Coxeter groups; Poincaré polynomials; palindromic polynomials; palindromic elements; Schubert varieties; rational smoothness; triangle groups; pattern avoidance Richmond, E; Slofstra, W, Rationally smooth elements of Coxeter groups and triangle group avoidance, J. Algebra. Comb., 39, 659-681, (2014) Reflection and Coxeter groups (group-theoretic aspects), Generators, relations, and presentations of groups, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Rationally smooth elements of Coxeter groups and triangle group avoidance. | 0 |
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