Affine Weyl Group Of Type Research Articles

The relation [formula omitted] on some elements of the affine Weyl group [formula omitted

Let (W,S) be the affine Weyl group of type C˜n with S its Coxeter generator set. Let Λ‾2n+1 be the set of all partitions λ=(λ1,...,λr) of 2n+1 such that ∑j=12k+1λj is odd for any k∈N with 2k+1⩽r. For any J⊊S, let wJ be the longest element in the parabolic subgroup of W generated by J. We define a map ϕ‾:{wJ|J⊊S}⟶Λ‾2n+1 and study the preorder ⩽LR on the set {wJ|J⊊S} and its relation with the partial order ⩽ on the set {ϕ‾(wJ)|J⊊S}, where iterating star operations and primitive pairs play an important role.

Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

Let (W,S) be a Coxeter system and write PW(q) for its Poincaré series. Lusztig has shown that the quotient PW(q2)/PW(q) is equal to a certain power series LW(q), defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in W. The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization LWJ(s,q)∈Z[[s,q]] depending on a subset J⊂S. This new power series specializes to LW(q) when s=−1 and is given explicitly by a sum of rational functions over the involutions which are minimal length representatives of the double cosets of the parabolic subgroup WJ in W. When W is an affine Weyl group, we consider the renormalized power series TW(s,q)=LWJ(s,q)/LW(q) with J given by the generating set of the corresponding finite Weyl group. We show that when W is an affine Weyl group of type A, the power series TW(s,q) is actually a polynomial in s and q with nonnegative coefficients, which turns out to be a q-analogue recently studied by Cigler of the Chebyshev polynomials of the first kind, arising in a completely different context.

Gröbner-Shirshov bases of some Weyl groups

In this paper, we obtain Gr\{o}bner-Shirshov (non-commutative) bases for the $n$-extended affine Weyl group $\widetilde{W}$ of type $A_1$, elliptic Weyl groups of types $A_{1}^{(1,1)}$, $A_{1}^{(1,1)^{*}}$ and the $2$-extended affine Weyl group of type $A_{2}^{(1,1)}$ with a generator system of a $2$-toroidal sense. It gives a new algorithm for getting normal forms of elements of these groups and hence a new algorithm for solving the word problem in these groups.

Decomposition numbers of quantized walled Brauer algebras

In this paper, we establish an explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) \(q\)-Schur algebras over a field \(\kappa \). This enables us to use Ariki’s result (J Math Kyoto Univ 36(4):789–808, 1996) and Varagnolo–Vasserot’s result (Duke Math J 100(2):267–297, 1999) to compute such decomposition numbers via inverse Kazhdan–Lusztig polynomials associated with extended affine Weyl groups of type \(A\) if the ground field is \({\mathbb {C}}\).

Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy

Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang.

Excedances in classical and affine Weyl groups

Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We derive some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type B˜n, C˜n and D˜n and determine the generating functions of their distributions. This paper is inspired to one of E. Clark, R. Ehrenborg in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups.

Hyperplane arrangements and diagonal harmonics

In 2003, Haglund's {\sf bounce} statistic gave the first combinatorial interpretation of the $q,t$-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type $A$. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement - which we call the Ish arrangement. We prove that our statistics are equivalent to the {\sf area'} and {\sf bounce} statistics of Haglund and Loehr. In this setting, we observe that {\sf bounce} is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to extended Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to the elementary symmetric functions.

Pattern characterization of rationally smooth affine Schubert varieties of type A

Schubert varieties in finite dimensional flag manifolds G/P are a well-studied family of projective varieties indexed by elements of the corresponding Weyl group W. In particular, there are many tests for smoothness and rational smoothness of these varieties. One key result due to Lakshmibai and Sandhya is that in type A the smooth Schubert varieties are precisely those that are indexed by permutations that avoid the patterns 4231 and 3412. Recently, there has been a flurry of research related to the infinite dimensional analogs of flag manifolds corresponding with G being a Kac–Moody group and W being an affine Weyl group or parabolic quotient. In this paper we study the case when W is the affine Weyl group of type A or the affine permutations. We develop the notion of pattern avoidance for affine permutations. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.

Excedances in classical and affine Weyl groups

Based on the notion of colored and absolute excedances introduced by Bagno and Garber we give an analogue of the derangement polynomials. We obtain some basic properties of these polynomials. Moreover, we define an excedance statistic for the affine Weyl groups of type $\widetilde{B}_n, \widetilde {C}_n$ and $\widetilde {D}_n$ and determine the generating functions of their distributions. This paper is inspired by one of Clark and Ehrenborg (2011) in which the authors introduce the excedance statistic for the group of affine permutations and ask if this statistic can be defined for other affine groups. Basée sur la notion des excédances colorés et absolu introduits par Bagno and Garber, nous donnons un analogue des polynômes des dérangements. Nous obtenons quelques propriétés de base de ces polynômes. En outre, nous définissons une excédance statistique pour le groupes de Weyl affines de type $\widetilde{B}_n, \widetilde {C}_n$ et $\widetilde {D}_n$ et nous déterminons les fonctions génératrices de leurs distributions. Cet article est inspirè d'un article de Clark et Ehrenborg (2011) dans lequel les auteurs introduisent les excedances pour le groupe des permutations affine et demander si cette statistique peut être éèfinie pour les autres groupes affines.

The Shi arrangement and the Ish arrangement

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ ceilings and $d$ degrees of freedom, etc. Moreover, all of these results hold in the greater generality of deleted Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

Hyperplane Arrangements and Diagonal Harmonics

In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

The Shi arrangement and the Ish arrangement

This paper is about two arrangements of hyperplanes. The first — the Shi arrangement — was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type A. The second — the Ish arrangement — was recently defined by the first author who used the two arrangements together to give a new interpretation of the q,t-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry'' between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with c "ceilings'' and d "degrees of freedom'', etc. Moreover, all of these results hold in the greater generality of "deleted'' Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labellings of Shi and Ish regions and a new set partition-valued statistic on these regions. Cet article traite de deux arrangements d'hyperplans. Le premier — arrangement Shi — a été introduit par Jian-Yi Shi pour décrire les cellules de Kazhdan-Lusztig du groupe de Weyl affine de type A. Le deuxième — arrangement Ish — a été récemment défini par le premier auteur pour donner une nouvelle interprétation des nombres q,t-Catalan de Garsia et Haiman. Ici nous définissons une mystérieuse "symétrie combinatoire" entre les deux arrangements et nous montrons que cette symétrie conserve un grand nombre d'informations. Par exemple, les arrangements Shi et Ish ont le même polynôme caractéristique, le même nombre de régions, de régions bornées, de régions dominantes, de régions avec c "plafonds'' et d "degrés de liberté'', etc. En outre, ces résultats se généralisent aux arrangements Shi et Ish "deleted'' correspondant à un sous-graphe arbitraire du graphe complet. Nos preuves reposent sur des étiquetages combinatoires des régions Shi et Ish, et sur une nouvelle statistique associée.

Kazhdan–Lusztig coefficients for an affine Weyl group of type [formula omitted

In this paper we compute the leading coefficients μ ( u , w ) of the Kazhdan–Lusztig polynomials P u , w for an affine Weyl group of type B ˜ 2 . By using the a-function of a Coxeter group defined by Lusztig, we compute most μ ( u , w ) explicitly. With part of these values μ ( u , w ) , we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type B ˜ 2 . We also show that the conjectural formula concerning some semilinear equations needs a modification.

Hypergeometric τ Functions of the q-Painlevé Systems of Type (A2+A1)(1)

We consider a $q$-Painlev\'e III equation and a $q$-Painlev\'e II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $\tau$ functions.

Some non-trivial Kazhdan-Lusztig coefficients of an affine Weyl group of type à n

In this paper we show that the leading coefficient µ(y,w) of some Kazhdan-Lusztig polynomials P y,w with y,w in an affine Weyl group of type à n is n + 2. This fact has some consequences on the dimension of first extension groups of finite groups of Lie type with irreducible coefficients.

Projective Reduction of the Discrete Painleve System of Type (A2 + A1)(1)

We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type (A2 + A1)(1). We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the τ functions.

Kazhdan-Lusztig Cells in Affine Weyl Groups of Rank 2

In this paper we determine the partition into Kazhdan-Lusztig cells of the affine Weyl groups of type $\tB_{2}$ and $\tG_{2}$ for any choice of parameters. Using these partitions we show that the semicontinuity conjecture of Bonnaf\'e holds for these groups.

On the Affine Schur Algebra of Type A

Let n, r ∈ ℕ. The affine Schur algebra (of type A) over a field K is defined to be the endomorphism algebra of certain tensor space over the extended affine Weyl group of type A r−1. By the affine Schur–Weyl duality it is isomorphic to the image of the representation map of the action on the tensor space when K is the field of complex numbers. We show that can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra (defined in Section 3) action on the tensor space, and it equals to the “dual” of a certain formal coalgebra related to this semigroup. By these approaches, we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur–Weyl duality.

Generalized Induction of Kazhdan-Lusztig cells

Following Lusztig, we consider a Coxeter group W together with a weight function. Geck showed that the Kazhdan-Lusztig cells of W are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of W which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of certain parabolic subgroups of W are cells in the whole group, and we decompose the affine Weyl group of type G into left and two-sided cells for a whole class of weight functions.

Extended affine Weyl groups of type A 1

It is known that elliptic Weyl groups, extended affine Weyl groups of nullity 2, have a finite presentation called the generalized Coexter presentation. Similar to the finite and affine case this presentation is obtained by assigning a Dynkin diagram to the root system. Then there is a prescription to read the generators and relations from the diagram. Recently a similar presentation is given for simply laced extended affine Weyl groups of nullity 3 and rank>1. Employing a new method, we complete this work by giving a similar presentation for nullity 3 extended affine Weyl groups of type A 1.