Bruhat decomposition

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In mathematics, the Bruhat decomposition (introduced by François 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 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 Grassmannians: see Weyl group for this.

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

Definitions[edit]

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[edit]

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[edit]

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. 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[edit]

The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the q-polynomial[1] of the associated Dynkin diagram.

See also[edit]

Notes[edit]

  1. ^ This Week's Finds in Mathematical Physics, Week 186

References[edit]