Analogue of the Bruhat decomposition for algebraic monoids

Trans. Amer. Math. Soc.

2001

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.

Section: Reductive Monoidsmentioning

confidence: 99%

“…Our focus in this paper is on the Bruhat decomposition for reductive monoids [17], where the Weyl group W is replaced by the Renner monoid R. The BruhatChevalley order on R, first studied by Renner [17], [18], remains quite mysterious. We studied this order in detail in an earlier paper [12] (with Pennell and Renner).…”

Section: Introductionmentioning

confidence: 99%

Shellability in reductive monoids

Trans. Amer. Math. Soc.

2001

“…Then each WeW is an interval in R and by [10] all maximal chains in R have the same length. R is an inverse semigroup.…”

Section: Proofmentioning

confidence: 99%

Bruhat-Chevalley Order in Reductive Monoids

Journal of Algebraic Combinatorics

2004

Abstract. 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 G 0 with Borel subgroup B 0 of G 0 , we show that the poset of B 0 × B 0 -orbits of X (with respect to Zariski closure inclusion) is Eulerian.

“…In particular the author [16] has proved the existence of a cross-section lattice Λ and has shown that the orbit monoids are of the form M (G, P, P − , L/K) for some pair of opposite parabolics P, P − , K L = P ∩ P − . Renner [24] has shown the existence of the finite inverse monoid…”

Section: Reductive Monoids and Monoids Of Lie Typementioning

confidence: 99%

Monoid Hecke algebras

Trans. Amer. Math. Soc.

1997

Abstract. This paper concerns the monoid Hecke algebras H introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group W . We use this to:1. find a description of the irreducible representations of H, 2. find an explicit isomorphism between H and the monoid algebra of the Renner monoid R, 3. extend the Kazhdan-Lusztig involution and basis to H, and 4. prove, for a W × W orbit of R, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials.