Weyl Group - Bruhat Decomposition

Bruhat Decomposition

If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and a maximal torus T = T₀ is chosen to lie in B, then we obtain the Bruhat decomposition

which gives rise to the decomposition of the flag variety G/B into Schubert cells (see Grassmannian).

The structure of the Hasse diagram of the group is related geometrically to the cohomology of the manifold (rather, of the real and complex forms of the group), which is constrained by Poincaré duality. Thus algebraic properties of the Weyl group correspond to general topological properties of manifolds. For instance, Poincaré duality gives a pairing between cells in dimension k and in dimension n - k (where n is the dimension of a manifold): the bottom (0) dimensional cell corresponds to the identity element of the Weyl group, and the dual top-dimensional cell corresponds to the longest element of a Coxeter group.