Extending the Bruhat order and the length function from the Weyl group to the Weyl monoid

Bruhat-Chevalley Order in Reductive Monoids

Let M be a reductive monoid with unit group G. Let denote the idempotent cross-section of the G × G-orbits on M . If W is the Weyl group of G and e, f ∈ with e ≤ f , we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M . For the canonical compactification X of a semisimple group G0 with Bo...

Shellability in Reductive Monoids

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let M be a reductive monoid with unit group G, Borel subgroup B and Weyl group W . We study the partially ordered set of B×Borbits (with respect to Zariski closure inclusion) within a G × G-orbit of M . This is the same as studying a W ×W -or...

A Weighted Enumeration of Maximal Chains in the Bruhat Order

Given a finite Weyl group W with root system , assign the weight α ∈ to each covering pair in the Bruhat order related by the reflection corresponding to α. Extending this multiplicatively to chains, we prove that the sum of the weights of all maximal chains in the Bruhat order has an explicit product formula, and prove a similar result for a weighted sum over maximal chains in the Bruhat order...

R-Polynomials of Finite Monoids of Lie Type

This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, M . It is shown by Putcha in [12] that the Kazhdan-Lusztig involution ([6]) can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using the ...

A Necessary Condition for Weyl-Heisenberg Frames