Geometry of Coxeter Groups

affine Weyl groups b₁ basis Borel Bruhat decomposition Bruhat order Bruhat posets called Chapter classical cn+k cntk coefficients cohomology coinvariant coinvariant algebra combinatorial completes the proof complex reflection groups compute Corollary corresponding coset Coxeter graph Coxeter group Coxeter system d₂ define denote dimension example finite Coxeter groups finite group finitely-generated flag manifold fundamental degrees G₁ G₁₂ G₂ geometric realization Grassmannian group G Hence homogeneous hyperplane induction intersection invariant theory irreducible isomorphism lattice Lemma length function Lie algebras linear Math matrix miniscule weight Möbius function node notation parabolic subgroup Pieri formula Poincare series polynomial positive definite Proposition rank reduced decomposition Remark representation root systems satisfying Schubert calculus Schubert classes Schubert varieties subalgebra subspace Suppose Theorem vector bundle vector space Young tableau α α α ΕΣ