Shellability in reductive monoids

Abstract

Abstract. 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 -orbit in the Renner monoid R. Such an orbit is the retract of a 'universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.

Weak order on complete quadrics

“…The case of a symmetric space is investigated by Richardson and Springer in [15,16], and the combinatorics of the special case of the variety of smooth quadrics is investigated by Incitti [8]. The Bruhat-Chevalley order on algebraic monoids is investigated by Putcha and Renner in [13], [10], [11], [12]. Other recent work includes [5] and [1].…”

Big cells and LU factorization in reductive monoids

“…For the combinatorial properties of this order on W eW , see [5]. By [7], [8], the length (σ) is defined as…”

The rook monoid is lexicographically shellable

“…It is shown by Putcha in [13] that the subposets W eW ⊆ R (e ∈ Λ) are lexicographically shellable. Note that the cross section lattice Λ ⊆ R is an (upper) semimodular lattice, hence it is shellable.…”