Weyl Group Element Research Articles

Extended Affine Weyl Groups

In this paper we study the Weyl groups of reduced extended affine root systems, the root systems of extended affine Lie algebras. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements (in terms of some naturally arisen transformations) which is crucial in the further study of extended affine Weyl groups. We use this to give a presentation, called a presentation by conjugation, for an important subclass of extended affine Weyl groups. Using a new notion, called the index which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems.

Minuscule Elements of Weyl Groups, the Numbers Game, andd-Complete Posets

Certain posets associated to a restricted version of the numbers game of Mozes are shown to be distributive lattices. The posets of join irreducibles of these distributive lattices are characterized by a collection of local structural properties, which form the definition ofd-complete poset. In representation theoretic language, the top “minuscule portions” of weight diagrams for integrable representations of simply laced Kac–Moody algebras are shown to be distributive lattices. These lattices form a certain family of intervals of weak Bruhat orders. These Bruhat lattices are useful in studying reduced decompositions of λ-minuscule elements of Weyl groups and their associated Schubert varieties. Thed-complete posets have recently been proven to possess both the hook length and the jeu de taquin properties.

Fundamental weights, permutation weights and Weyl character formula

For a finite Lie algebra of rank N, the Weyl orbits of strictly dominant weights contain number of weights, where is the dimension of its Weyl group . For any , there is a very peculiar subset for which we always have For any dominant weight , the elements of are called permutation weights. It is shown that there is a one-to-one correspondence between the elements of and where is the Weyl vector of . The concept of the signature factor which enters the Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant , calculation of the character for the irreducible representation will then be provided by multiplicity rules governing the generalized Schur functions. The main idea is again to express everything in terms of the so-called fundamental weights with which we obtain a quite relevant specialization in applications of the Weyl character formula. To provide simplifications in practical calculations, a reduction formula governing the classical Schur functions is also given. As the most suitable example, , which requires a sum over Weyl group elements, is studied explicitly. This will be instructive also for an explicit application of multiplicity rules. As a result, it will be seen that the Weyl or Weyl-Kac character formulae find explicit applications no matter how large the rank of the underlying algebra.

On the spaltenstein-steinberg map for classical lie algebras

We compute the fibers of the canonical map from Weyl group elements to orbital varieties for a Lie algebra of type B,C, or D. We also compute one irreducible component of the associated variety of any simple highest weight module in any classical type, thereby sharpening a result of Joseph and partially proving a conjecture of Tanisaki.

Crystals for Demazure Modules of Classical Affine Lie Algebras

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of typesA(1)n,B(1)n,C(1)n,D(1)n,A(2)2n−1,A(2)2n, andD(2)n+1. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show that if the highest weight islΛ0, the Demazure crystal has a remarkably simple structure.

Demazure modules and vertex models: The sl̂(2) case

We characterize, in the case of sl∧(2),the crystal base of the Demazure module Ew(Λ) in terms of extended Young diagrams or paths for any dominant integral weight Λ and Weyl group element w. Its character is evaluated via two expressions, “bosonic” and “fermionic.”

Irreducible Components of Fixed Point Subvarieties of Flag Varieties

Suppose G is a connected reductive algebraic group, P is a parabolic subgroup of G, L is a Levi factor of P, and e is a regular nilpotent element in Lie L. We assume that the characteristic of the underlying field is good for G. Choose a maximal torus, T, and a Borel subgroup, B, of G, so that T⊆B∩L, B ⊆ P and e ∈ Lie B. Let β be the variety of Borel subgroups of G and let 𝔙e be the subset of 𝔙 consisting of Borel subgroups whose Lie algebras contain e. Finally, let W be the Weyl group of G with respect to T. For ω ∈ W let 𝔒ω be the B-orbit in 𝔙 containing ωB. We consider the intersections 𝔒ω ∩ 𝔙e. The main result is that if dim 𝔒ω ∩ 𝔙e = dim 𝔙e, then 𝔒ω ∩ 𝔙e is an affine space. Thus, the irreducible components of 𝔙e are indexed by Weyl group elements. It is also shown that if G is of type A, then this set of Weyl group elements is a right cell in W.

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedure for computing analogues of Young’s seminormal representations of the symmetric groups . The method is to generalize the Jucys – Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori – Hecke algebras . The combinatorics of these elements allow one to compute irreducible representations explicitly and often very easily . In this paper we do these computations for Weyl groups and Iwahori – Hecke algebras of types A n , B n , D n , G 2 . Although these computations are within reach for types F 4 , E 6 , and E 7 , we shall , in view of the length of the current paper , postpone this to another work . In reading this paper , I would suggest that the reader begin with § 3 , the symmetric group case , and go back and pick up the generalities from §§ 1 and 2 as they are needed . This will make the motivation for the material in the earlier sections much more clear and the further examples in the later sections very easy . The realization that the Jucys – Murphy elements for Weyl groups and Iwahori – Hecke algebras come from the very natural central elements in (2 . 1) and Proposition 2 . 4 is one of the main points of this paper . There is a simple concrete connection (Proposition 2 . 8) between Jucys – Murphy type elements in Iwahori – Hecke algebras and Jucys – Murphy elements in group algebras of Weyl groups . I know that the analogues of the Jucys – Murphy elements in Weyl groups of types B and D will be new to some of the experts and known to others . These Jucys – Murphy elements for types B and D are not new ; similar elements appear in the paper of Cherednik [ 7 ] , but I was not able to recognize them there until they were pointed out to me by M . Nazarov . I extend my thanks to him for this . Some people were asking me for Jucys – Murphy elements in type G 2 as late as June 1995 . In July 1995 I was told that it was not known how to quantize the elements of Cherednik , that is , to find analogues of them in the Iwahori – Hecke algebras of types B and D . Of course , this had been done already in 1974 , by Hoefsmit . I have chosen to state my results in terms of the general mechanism of path algebras which I have defined in § 1 . This is a technique which I learned from H . Wenzl during our work on the paper [ 30 ] . It is a well-known method in several fields (with many dif ferent terminologies) . I shall mention here only a few of the