in mathematics, the bruhat decomposition (introduced by franÃ§ois bruhat for classical groups and by claude chevalley in general) g = bwb into cells can be regarded as a general expression of the principle of gaussâjordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matricesâbut with exceptional cases. it is related to the schubert cell decomposition of grassmannians: see weyl group for this. more generally, any group with a (b,n) pair has a bruhat decomposition. ==definitions== *g is a connected, reductive algebraic group over an algebraically closed field. *b is a borel subgroup of g *w is a weyl group of g corresponding to a maximal torus of b. the bruhat decomposition of g is the decomposition :g=bwb =\coprod_{w\in w}bwb of g as a disjoint union of double cosets of b parameterized by the elements of the weyl group w. (note that although w is not in general a subgroup of g, the coset wb is still well defined.) == examples == let g be the general linear group gln of invertible n \times n matrices with entries in some algebraically closed field, which is a reductive group. then the weyl group w is isomorphic to the symmetric group sn on n letters, with permutation matrices as representatives. in this case, we can take b to be the subgroup of upper triangular invertible matrices, so bruhat decomposition says that one can write any invertible matrix a as a product u1pu2 where u1 and u2 are upper triangular, and p is a permutation matrix. writing this as p = u1-1au2-1, 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 i (resp. column i) to row j (resp. column j) if i>j (resp. i<j). the row operations correspond to u1-1, and the column operations correspond to u2-1. the special linear group sln of invertible n \times n matrices with determinant 1 is a semisimple group, and hence reductive. in this case, w is still isomorphic to the symmetric group sn. however, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in sln, we can take one of the nonzero elements to be -1 instead of 1. here b is the subgroup of upper triangular matrices with determinant 1, so the interpretation of bruhat decomposition in this case is similar to the case of gln. == geometry == the cells in the bruhat decomposition correspond to the schubert cell decomposition of grassmannians. the dimension of the cells corresponds to the length of the word w in the weyl group. poincarÃ© duality constrains the topology o
Algebraic_groups Lie_groups