For any finite Coxeter system $(W,S)$ we construct a certain noncommutative algebra, so-called {\it bracket algebra}, together with a familiy of commuting elements, so-called {\it Dunkl elements.} Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the group $W.$ We prove this conjecture for classical Coxeter groups and $I_2(m)$. We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and $G_2,$ the algebra generated by Dunkl elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define {\it quantum Bruhat representation} of the corresponding bracket algebra. We study in more detail relations and structure of $B_n$-, $D_n$- and $G_2$-bracket algebras, and as an application, discover {\it Pieri type formula} in the $B_n$-bracket algebra. As a corollary, we obtain Pieri type formula for multiplication of arbitrary $B_n$-Schubert classes by some special ones. Our Pieri type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type $A.$ We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements which describes ``noncommutative differential geometry on a finite Coxeter group'' in a sense of S. Majid.
In this paper, we study the decomposition of Bruhat intervals in a Coxeter group with respect to cosets of a parabolic subgroup. Our main result is that the intersection of a lower Bruhat interval with a parabolic coset contains a unique maximal element. As an application, we give a decomposition formula for the Poincaré polynomial of a Coxeter group element. We also show that the fibers of standard parabolic projection maps on Schubert varieties are themselves Schubert varieties.
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …
We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the Coxeter group ${\mathcal B…
A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.
We define an analog of David Little's algorithm for reduced words in type B, and investigate its main properties. In particular, we show that our algorithm preserves the recording tableau of Kraśkiewicz insertion, and that it provides a bijective realization of the Type B transition equations in Schubert calculus. Many other aspects of type A theory carry over to this new setting. Our primary tool is a shifted version of the dual equivalence graphs defined by Assaf and further developed by Roberts. We provide an axiomatic characterization of shifted dual equivalence graphs, and use them to pro…
We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haiman's enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of "nearly-maximally labelled" staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors…
We give an algebra-combinatorial constructions of (noncommutative) generating functions of double Schubert and double $β$-Grothendieck polynomials corresponding to the full flag varieties associated to the Lie groups of classical types $A,B, C$ and $D$. Our approach is based on the decomposition of certain `` transfer matrices `` corresponding to the exponential solution to the quantum Yang--Baxter equations associated with either NiCoxeter or IdCoxeter algebras of classical types. The "triple"~$β$-Grothendieck polynomials ${\mathfrak{G}}_{w}^{W}(X,Y,Z)$ we have introduced, satisfy, among othe…
We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval $[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular…
Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σand an element in Πdetermine a {\em subword complex}, as introduced in our paper on Gröbner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to s…
Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $α\in R$, $v,w\in W$, we denote by $\partial_α$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew divided difference operators acting on the coinvariant algebra of $W$. Generalizing the work of Liu, we prove that $\partial_{w/v}$ can be written as a polynomial with nonnegative coefficients in $\partial_α$ where $α\in R^+$. In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on $W$ after Bazlov. We draw …
We present a general conjecture on the divisibility of a certain expression in terms of Kostka numbers and their close variants. This conjecture is closely related to a variant of the period-index problem of noncommutative algebra, with partial implications in both directions. We present a description of the connection between these two problems via Schubert calculus as motivation and evidence for the conjecture before turning to a proof of the conjecture in a family of cases.
We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the special linear and symplectic groups, respectively. For the odd orthogonal groups, a Hopf-algebra isomorphism is given, identifying (co)homology Schubert classes with symmetric functions. For the even orthogonal groups, we conjecture an approximate model of (co)homology via symmetric functions. In the process, we develop type B and type D non-commutative k-Schur…
The Kazhdan-Lusztig polynomials for finite Weyl groups arise in the geometry of Schubert varieties and representation theory. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no completely combinatorial interpretation for them is known in general. Deodhar (1990) has given a framework for computing the Kazhdan-Lusztig polynomials, which generally involves recursion. We define embedded factor pattern avoidance for general Coxeter groups and use it to characterize when Deodhar's algorithm yields a simple combinatorial formula for the Kazhdan-Lu…
Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. 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 …
Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementa…
A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob--Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to type B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of type B and D and quasisymmetric functions and noncommutative symmetric functions of type B and D. Other applications are also provided.
Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W.$ We describe all the reduced express…
Let $G$ be a simple algebraic group of adjoint type over $\mathbb{C},$ whose root system is of type $F_{4}.$ Let $T$ be a maximal torus of $G$ and $B$ be a Borel subgroup of $G$ containing $T.$ Let $w$ be an element of Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w.$ In this article, we study the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ i…
We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu rela…
