In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ${\displaystyle G=BWB}$ of certain algebraic groups ${\displaystyle G=BWB}$ into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.

More generally, any group with a (B, N) pair has a Bruhat decomposition.

• ${\displaystyle G}$ is a connected, reductive algebraic group over an algebraically closed field.
• ${\displaystyle B}$ is a Borel subgroup of ${\displaystyle G}$
• ${\displaystyle W}$ is a Weyl group of ${\displaystyle G}$ corresponding to a maximal torus of ${\displaystyle B}$.

The Bruhat decomposition of ${\displaystyle G}$ is the decomposition

${\displaystyle G=BWB=\bigsqcup _{w\in W}BwB}$

of ${\displaystyle G}$ as a disjoint union of double cosets of ${\displaystyle B}$ parameterized by the elements of the Weyl group ${\displaystyle W}$. (Note that although ${\displaystyle W}$ is not in general a subgroup of ${\displaystyle G}$, the coset ${\displaystyle wB}$ is still well defined because the maximal torus is contained in ${\displaystyle B}$.)

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 ${\displaystyle U_{1}PU_{2}}$ where ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ are upper triangular, and ${\displaystyle P}$ is a permutation matrix. Writing this as ${\displaystyle P=U_{1}^{-1}AU_{2}^{-1}}$, 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.

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.

The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ${\displaystyle q}$-polynomial^[1] of the associated Dynkin diagram.

With two opposite Borel subgroups, one may intersect the Bruhat cells for each of them, giving a further decomposition $${\displaystyle G=\bigsqcup _{w_{1},w_{2}\in W}(Bw_{1}B\cap B_{-}w_{2}B_{-}).}$$

• Lie group decompositions
• Birkhoff factorization, a special case of the Bruhat decomposition for affine groups.
• Cluster algebra

• Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag, 1991. ISBN 0-387-97370-2.
• Bourbaki, Nicolas, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics), Springer-Verlag, 2008. ISBN 3-540-42650-7