For a connected reductive algebraic group, a Borel subgroup , and a maximal torus , the Weyl group indexes the double cosets:
The relative position of Borel subgroups is the Weyl group index of their simultaneous conjugacy orbit. After moving the first subgroup to a fixed , the second has the form and its index is the double coset . This identifies the diagonal action on pairs of flags with the Bruhat decomposition.
This is the Bruhat decomposition associated with a BN-pair. The Bruhat decomposition is the disjoint unionassociated with a BN-pair.
Articles by others on the same topic
Bruhat decomposition is a fundamental concept in the theory of Lie groups and algebraic groups, particularly in the study of algebraic varieties and symmetric spaces. It provides a way to decompose a group into pieces that can be analyzed more easily.