Bruhat

Bruhat refers to concepts named after the French mathematician François Bruhat, particularly in the areas of algebraic groups, Coxeter groups, and related geometric structures. The term commonly denotes three interrelated ideas: the Bruhat order, the Bruhat decomposition, and Bruhat–Tits buildings, all of which play fundamental roles in modern representation theory, algebraic geometry, and number theory.

Bruhat order is a partial order on the elements of a Coxeter group W. It can be described in several equivalent ways, notably via the inclusion relations of Schubert varieties in the flag variety G/B or via a subword property of reduced expressions. Each element w has a length l(w), the minimum number of simple reflections needed to express w, and the order reflects how these expressions can be extended by simple reflections. The Bruhat order encodes containment relations among closures of certain algebraic cells and organizes the combinatorics of Weyl groups and flag varieties.

Bruhat decomposition is a structural result for connected reductive algebraic groups G over a field. If B is a Borel subgroup and W is the Weyl group of G, then G is a disjoint union of double cosets B w B as w runs over W. Each double coset is a Bruhat cell, whose dimension equals the length l(w). This decomposition yields a stratification of the flag variety G/B into Schubert cells and underpins many geometric and representation-theoretic constructions.

Bruhat–Tits buildings are geometric objects associated with reductive groups over non-Archimedean fields. These buildings have an apartment decomposition linked to maximal split tori and a chamber structure corresponding to Borel subgroups. They provide a useful framework for studying p-adic groups, harmonic analysis, and arithmetic geometry.