Bruhat decomposition

Examples

Let ${\displaystyle G}$ be the general linear group GL_n of invertible ${\displaystyle n\times n}$ matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group ${\displaystyle W}$ is isomorphic to the symmetric group ${\displaystyle S_n}$ on ${\displaystyle n}$ letters, with permutation matrices as representatives. In this case, we can take ${\displaystyle B}$ to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix ${\displaystyle A}$ as a product ${\displaystyle U_{1}P{U}_2}$ where ${\displaystyle U_1}$ and ${\displaystyle U_2}$ are upper triangular, and ${\displaystyle P}$ is a permutation matrix. Writing this as ${\displaystyle P=U_1^{-1}A{U}_2^{-1}}$, this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row ${\displaystyle i}$ (resp. column ${\displaystyle i}$) to row ${\displaystyle j}$ (resp. column ${\displaystyle j}$) if ${\displaystyle i>j}$ (resp. ${\displaystyle i<j}$). The row operations correspond to ${\displaystyle U_1^{-1}}$, and the column operations correspond to ${\displaystyle U_2^{-1}}$.

The special linear group SL_n of invertible ${\displaystyle n\times n}$ matrices with determinant ${\displaystyle 1}$ is a semisimple group, and hence reductive. In this case, ${\displaystyle W}$ is still isomorphic to the symmetric group ${\displaystyle S_n}$. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SL_n, we can take one of the nonzero elements to be ${\displaystyle -1}$ instead of ${\displaystyle 1}$. Here ${\displaystyle B}$ is the subgroup of upper triangular matrices with determinant ${\displaystyle 1}$, so the interpretation of Bruhat decomposition in this case is similar to the case of GL_n.

Geometry

The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of flag varieties. The dimension of the cells corresponds to the length of the word ${\displaystyle w}$ in the Weyl group. Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the fundamental class), and corresponds to the longest element of a Coxeter group.

Computations

The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ${\displaystyle q}$-polynomial of the associated Dynkin diagram.

Double Bruhat cells

With two opposite Borel subgroups, one may intersect the Bruhat cells for each of them, giving a further decomposition $${\displaystyle G=\underset{w_1,w_2\in W}{⨆}(B{w}_{1}B\cap B_{-}{w}_2{B}_{-}).}$$