The Bruhat Decomposition

Abstract

The Bruhat decomposition was discovered quite late in the history of Lie groups, which is surprising in view of its fundamental importance. It was preceded by Ehresmann’s discovery of a closely related cell decomposition for flag manifolds. The Bruhat decomposition was axiomatized by Tits in the notion of a Group with (B, N) pair or Tits’ system. This is a generalization of the notion of a Coxeter group, and indeed every (B, N) gives rise to a Coxeter group. We have remarked after Theorem 25.1 that Coxeter groups always act on simplicial complexes whose geometry is closely connected with their properties. As it turns out a group with (B N) pair also acts on a simplicial complex, the Tits’ building. We will not have space to discuss this important concept but see Tits [163] and Abramenko and Brown [1].