Separable elements in Weyl groups

Minuscule Elements of Weyl Groups

Fully Commutative Elements in the Weyl and Affine Weyl Groups

Let W be a Weyl or affine Weyl group and let Wc be the set of fully commutative elements in W . We associate each w ∈ Wc to a digraph G(w). By using G(w), we give a graph-theoretic description for Lusztig’s a-function on Wc and describe explicitly all the distinguished involutions of W . The results verify two conjectures in our case: one was proposed by myself in [15, Conjecture 8.10] and the ...

Involutions in Weyl Groups

Let G be a split real group with Weyl group W . Let E be an irreducible representation of W . Let V be the stable Lie algebra version of the coherent continuation representation of W . The main result of this paper is a formula for the multiplicity of E in V . The formula involves the position of E in Lusztig’s set ∐ M(G). The paper treats all quasi-split groups G as well.

Reduced decompositions in Weyl groups

Let R be a root system with fixed basis ~' and let W be its Weyl group. For every element w ~ W, there exists a natural correspondence between reduced decompositions of w and some linear orders of the set of inversions of w. We study combinatorial properties of these orders. Furthermore, in the case of a symmetric group, for any permutation w we construct a linear action of the symmetric group ...

A Root System Criterion for Fully Commutative and Short Braid-Avoiding Elements in Affine Weyl Groups

We provide simple characterizations of short-braid avoiding and fully commutative elements in an affine Weyl group W , generalizing results of Fan and Stembridge for finite Weyl groups. Our results rely on the combinatorics of the compatible subsets of the root system of W .