, Volume 20, Issue 1, pp 33-53

Bruhat-Chevalley Order in Reductive Monoids

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Abstract

Let M be a reductive monoid with unit group G. Let Λ denote the idempotent cross-section of the G × G-orbits on M. If W is the Weyl group of G and e, f ∈ Λ with ef, 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.