Some Kazhdan–Lusztig Coefficients of Affine Weyl Group of Type B~ 2

2021 ◽  
Vol 28 (04) ◽  
pp. 541-554
Author(s):  
Ge Feng ◽  
Liping Wang
Keyword(s):  
Weyl Group ◽  
Bruhat Order ◽  
Length Function ◽  
Type B ◽  
Type Formula ◽  
Affine Weyl Group ◽  
Polynomial Formula

Let [Formula: see text] be the affine Weyl group of type [Formula: see text], on which we consider the length function [Formula: see text] from [Formula: see text] to [Formula: see text] and the Bruhat order [Formula: see text]. For [Formula: see text] in [Formula: see text], let [Formula: see text] be the coefficient of [Formula: see text] in Kazhdan–Lusztig polynomial [Formula: see text]. We determine some [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is the lowest two-sided cell of [Formula: see text] and [Formula: see text] is the higher one. Furthermore, we get some consequences using left or right strings and some properties of leading coefficients.

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_{n-1}^{(1)}$ WITH HIGHER DEGREE LAX OPERATORS

2007 ◽  
Vol 18 (07) ◽  
pp. 839-868 ◽  
Author(s):  
HAJIME NAGOYA
Keyword(s):  
Weyl Group ◽  
Heisenberg Algebra ◽  
Painlevé Equations ◽  
Differential Systems ◽  
Classical Systems ◽  
Painleve Equations ◽  
Type Formula ◽  
Affine Weyl Group ◽  
Lax Operator ◽  
Group Symmetries

Quantum Painlevé systems of type [Formula: see text] [13] are the quantizations of the second, fourth and fifth Painlevé equations and their generalizations [1, 15, 26]. These quantized systems have the Lax representations as in the classical systems. As a polynomial in an element of a Heisenberg algebra of [Formula: see text], the degrees of those Lax operators are 2 or 3. In this paper, we shall deal with the Lax operator whose degree is greater than or equal to 2. Using this Lax operator, we systematically construct the differential systems with the affine Weyl group symmetries of type [Formula: see text] and the commuting Hamiltonians.

QUANTUM PAINLEVÉ SYSTEMS OF TYPE $A_l^{(1)}$

2004 ◽  
Vol 15 (10) ◽  
pp. 1007-1031 ◽  
Author(s):  
HAJIME NAGOYA
Keyword(s):  
Weyl Group ◽  
Discrete Systems ◽  
Painlevé Equation ◽  
Differential Systems ◽  
Type Formula ◽  
Affine Weyl Group ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Group Symmetries

We propose quantum Painlevé systems of type [Formula: see text]. These systems, for l=1 and l≥2, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type [Formula: see text] studied by Noumi and Yamada [13], respectively. These quantizations enjoy the affine Weyl group symmetries of type [Formula: see text] as well as the Lax representations. The quantized systems of type [Formula: see text] and type [Formula: see text](l=2n) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type [Formula: see text] and [Formula: see text], respectively.

scholarly journals Maximal Newton polygons via the quantum Bruhat graph

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Elizabeth T. Beazley
Keyword(s):  
Weyl Group ◽  
Quantum Cohomology ◽  
Bruhat Order ◽  
Flag Variety ◽  
Necessary Condition ◽  
Newton Polygons ◽  
Weighted Directed Graph ◽  
Affine Weyl Group ◽  
Bruhat Graph ◽  
Quantum Bruhat Graph

International audience This paper discusses a surprising relationship between the quantum cohomology of the variety of complete flags and the partially ordered set of Newton polygons associated to an element in the affine Weyl group. One primary key to establishing this connection is the fact that paths in the quantum Bruhat graph, which is a weighted directed graph with vertices indexed by elements in the finite Weyl group, encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. In addition, using some geometry associated to the poset of Newton polygons, one obtains independent proofs for several combinatorial statements about paths in the quantum Bruhat graph and its symmetries, which were originally proved by Postnikov using the tilted Bruhat order. An important geometric application of this work is an inequality which provides a necessary condition for non-emptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. Cet article étudie une relation surprenante entre la cohomologie quantique de la variété de drapeaux complets et l'ensemble partiellement ordonné de polygones de Newton associé à un élément du groupe de Weyl affine. L’élément clé pour établir cette connexion est le fait que les chemins dans le graphe de Bruhat quantique, qui est un graphe orienté pondéré dont les sommets sont indexés par des éléments du groupe de Weyl fini, encodent des chaînes saturées dans l'ordre de Bruhat fort sur le groupe de Weyl affine. Cette correspondance est aussi fondamentale dans les travaux de Lam et Shimonozo qui établissent l'isomorphisme de Peterson entre la cohomologie quantique de la variété de drapeaux finie et l'homologie de la Grassmannienne affine. De plus, en utilisant la géométrie associée à l'ensemble partiellement ordonné des polygones de Newton, on obtient des preuves indépendantes pour plusieurs assertions combinatoires sur les chemins dans le graphe de Bruhat quantiques et les symétries de ce graphe, qui ont été originellement démontrées par Postnikov en utilisant l'ordre de Bruhat incliné. Une application géométrique importante de ce travail est une inégalité qui donne une condition nécessaire pour que certaines variétés de Deligne-Lusztig affines dans la variété de drapeaux affine soient non-vides.

The affine Weyl group and modular invariant partition functions

1988 ◽  
Vol 205 (2-3) ◽  
pp. 281-284 ◽  
Author(s):  
D. Altschüler ◽  
J. Lacki ◽  
Ph. Zaugg
Keyword(s):  
Weyl Group ◽  
Partition Functions ◽  
Modular Invariant ◽  
Affine Weyl Group

scholarly journals Casselman’s Basis of Iwahori Vectors and the Bruhat Order

2011 ◽  
Vol 63 (6) ◽  
pp. 1238-1253 ◽  
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Weyl Group ◽  
Bruhat Order ◽  
Group Element ◽  
The Other ◽  
Intertwining Operators ◽  
Transition Matrices ◽  
Natural Basis ◽  
Spherical Representation ◽  
Langlands Parameters ◽  
Combinatorial Condition

AbstractW. Casselman defined a basis fu of Iwahori fixed vectors of a spherical representation of a split semisimple p-adic group G over a nonarchimedean local field F by the condition that it be dual to the intertwining operators, indexed by elements u of the Weyl group W. On the other hand, there is a natural basis , and one seeks to find the transition matrices between the two bases. Thus, let and . Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then , where z are the Langlands parameters for the representation and α runs through the set S(u, v) of positive coroots (the dual root systemof G) such that with rα the reflection corresponding to α. The condition is conjecturally always satisfied if G is simply-laced and the Kazhdan–Lusztig polynomial Pw0v,w0u = 1 with w0 the long Weyl group element. There is a similar formula for conjecturally satisfied if Pu,v = 1. This leads to various combinatorial conjectures.

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