Shellability in reductive monoids

Putcha, Mohan

doi:10.1090/S0002-9947-01-02806-9

Shellability in reductive monoids
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Trans. Amer. Math. Soc. 354 (2002), 413-426 Request permission

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 $B\times B$-orbits (with respect to Zariski closure inclusion) within a $G\times G$-orbit of $M$. This is the same as studying a $W\times W$-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.

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