Bruhat decomposition

Bruhat decomposition is a fundamental structural theorem in the theory of linear algebraic groups and Representation theory that expresses a reductive group as a union of double cosets indexed by a Weyl group. It connects constructions in Élie Cartan's theory of symmetric spaces, the work of François Bruhat, and structural results of Claude Chevalley, providing a bridge between combinatorial objects like Coxeter group elements and geometric objects like Flag variety strata. The decomposition underlies calculations in the contexts of Harish-Chandra modules, Borel–Weil–Bott theorem applications, and intersection theory on Schubert varietys.

Introduction

The decomposition was developed in the milieu of mid‑twentieth‑century studies by François Bruhat, with antecedents in the classification work of Élie Cartan and structural contributions by Claude Chevalley and Armand Borel. It concerns reductive groups such as GL(n), SL(n), SO(n), Sp(n), and their subgroups like Borel subgroups and Parabolic subgroups, using the discrete structure of the Weyl group (a special case of a Coxeter group) to index strata. The decomposition has ramifications in the study of Flag varietys, the geometry of Schubert varietys, and representation-theoretic constructs like Verma modules and the Kazhdan–Lusztig theory.

Statement and Formulation

Let G be a connected reductive linear algebraic group over an algebraically closed field and let B be a Borel subgroup of G. Fix a maximal torus T contained in B; the associated Weyl group W = N_G(T)/T is a finite Coxeter group generated by simple reflections corresponding to roots from the root system of G. The Bruhat decomposition asserts that G = ⋃_{w∈W} B w B, a disjoint union of double cosets indexed by W. Equivalently for a parabolic subgroup P one has G = ⋃_{w∈W^P} P w P where W^P denotes minimal coset representatives for W/W_P with W_P the Weyl group of the Levi factor of P. In classical matrix terms for GL(n), this corresponds to Gaussian elimination and the factorization into permutation matrices from the symmetric group S_n, triangular matrices in the Borel subgroup, and diagonal pieces in the maximal torus.

Bruhat Cells and Schubert Varieties

Each double coset B w B is a locally closed subset called a Bruhat cell; its closure in the Flag variety G/B is a Schubert variety, extensively studied in Algebraic geometry and Intersection theory. Schubert varieties are indexed by elements of W, and their geometry—singular loci, cohomology classes, and intersection numbers—ties into enumerative problems solved historically by methods of Hermann Schubert and structurally clarified by Alexander Grothendieck. The cell decomposition yields a CW‑type stratification of G/B used in computations of cohomology rings, characteristic classes, and sheaf cohomology computations central to the Borel–Weil–Bott theorem. Relations among Schubert classes are governed by combinatorial objects associated with Young tableaus in the Schubert calculus for GL(n) and by Bruhat order on W studied by Dimitry Kazhdan and George Lusztig in the context of Kazhdan–Lusztig polynomials.

Applications and Examples

In concrete examples, for GL(n), Bruhat decomposition corresponds to factorization into a permutation matrix from the symmetric group S_n and two invertible upper and lower triangular matrices in opposite Borel subgroups; this recovers classical matrix decompositions used in numerical linear algebra and in the theory of Determinants. For SL(2), the decomposition yields two Bruhat cells corresponding to the identity and the nontrivial element of the Weyl group of order two; this appears in the representation theory of SU(2), in harmonic analysis on p-adic groups studied by Jacques Tits, and in local zeta integral computations in the theory of Godement–Jacquet L‑functions. The decomposition is instrumental in the study of flag manifolds appearing in the geometry of Grassmannians, in problems from Schubert calculus and enumerative geometry tied to the work of Giambelli and Pieri, and in geometric representation theory contexts such as the construction of perverse sheaves in the framework developed by George Lusztig and Beilinson–Bernstein.

Generalizations and Variants

Generalizations include decompositions for reductive groups over nonalgebraically closed fields, p‑adic analytic groups used in the study of automorphic forms and the Langlands program by Robert Langlands, and affine versions involving Affine Weyl groups that lead to the affine Bruhat decomposition relevant to Kac–Moody algebra representations and the theory of Loop groups. There are parabolic Bruhat decompositions indexed by coset representatives for W/W_P, and microlocal or quantum analogues appearing in quantum group studies by Lusztig and in categorification programs of Mikhail Khovanov and Ostrik. Connections to the structure theory of Tits buildings, to the reduction theory of Armand Borel and Harish-Chandra, and to combinatorial models like the Gelfand–Cetlin patterns show the decomposition’s centrality across Representation theory, Algebraic geometry, and arithmetic aspects of Number theory.

Category: Algebraic groups