Bruhat-Chevalley Order in Reductive Monoids

Abstract

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.

Orders of the Renner monoids

“…Also, let λ * (e) = f e λ(f ) and W * (e) = f e W (f ). The following results are due to Putcha and Renner [10]; they can also be found in [9,12] (for example, of [9, formulas …”

Nilpotent variety of a reductive monoid

“…The order has been determined in [5] and more explicitly in [11,12]. In particular for e, e ∈ , x, y ∈ W , e ≤ e ⇒ xey ≤ xe y (12) and for y ∈ D(e) −1 , y ∈ D(e ) −1 , ey ≤ e y ⇐⇒ y ≤ zy for some z ∈ W (e)…”

“…In particular for e, e ∈ , x, y ∈ W , e ≤ e ⇒ xey ≤ xe y (12) and for y ∈ D(e) −1 , y ∈ D(e ) −1 , ey ≤ e y ⇐⇒ y ≤ zy for some z ∈ W (e)…”

Conjugacy decomposition of reductive monoids

“…A successful strategy [10], [11] for studying the poset (R, ≤) has been to first study the W × W -orbits W eW , e ∈ and then to find some connecting maps between these orbits to describe the order on R. We wish to apply this strategy tõ R. We begin by noting thatR We proceed to take a closer look at the posetR(e) when W (e) = W * (e). Theorem 2.9.…”

“…Finally we expect the intervalsR(e), e ∈ , to be connected in a way analogous to the way that the intervals W eW , e ∈ , are connected in R, cf. [11]: 13. Let e, f ∈ , e < f .…”