Let G be a connected reductive group and B be a Borel subgroup. The Bruhat decomposition of G expresses G as a disjoint union of double cosets BwB, where w lies in the Weyl group W. We can ask when BwB is contained in the (Zariski) closure of Bw'B. Chevalley gave a combinatorial criterion to this in terms of w and w'. In this talk we will discuss this criterion and give some examples. If time permits we will give a rough sketch of a proof.

This talk will be based on Chevalley’s paper “Sur les Décompositions Cellulaires des Espaces G/B."