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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.
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| More generally, any group with a [[(B,N) pair]] has a Bruhat decomposition.
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| ==Definitions==
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| *''G'' is a [[connected space|connected]], [[reductive group|reductive]] [[algebraic group]] over an [[algebraically closed field]].
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| *''B'' is a [[Borel subgroup]] of ''G''
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| *''W'' is a [[Weyl group]] of ''G'' corresponding to a maximal torus of ''B''.
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| The '''Bruhat decomposition''' of ''G'' is the decomposition
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| :<math>G=BWB =\coprod_{w\in W}BwB</math>
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| of ''G'' as a disjoint union of [[double coset]]s 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.)
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| == Examples ==
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| Let ''G'' be the [[general linear group]] '''GL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group ''W'' is isomorphic to the [[symmetric group]] ''S<sub>n</sub>'' 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 ''U<sub>1</sub>PU<sub>2</sub>'' where ''U<sub>1</sub>'' and ''U<sub>2</sub>'' are upper triangular, and ''P'' is a permutation matrix. Writing this as ''P = U<sub>1</sub><sup>-1</sup>AU<sub>2</sub><sup>-1</sup>'', 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 ''U<sub>1</sub><sup>-1</sup>'', and the column operations correspond to ''U<sub>2</sub><sup>-1</sup>''.
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| The [[special linear group]] '''SL'''<sub>n</sub> of invertible <math>n \times n</math> matrices with [[determinant]] 1 is a [[semisimple algebraic group|semisimple group]], and hence reductive. In this case, ''W'' is still isomorphic to the symmetric group ''S<sub>n</sub>''. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in '''SL'''<sub>n</sub>, 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 '''GL'''<sub>n</sub>.
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| == Geometry ==
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| The cells in the Bruhat decomposition correspond to the [[Schubert cell]] decomposition of Grassmannians. The dimension of the cells corresponds to the [[length function|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]].
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| ==Computations==
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| The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the ''q''-polynomial<ref>[http://math.ucr.edu/home/baez/week186.html This Week's Finds in Mathematical Physics, Week 186]</ref> of the associated [[Dynkin diagram]].
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| ==See also==
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| * [[Lie group decompositions]]
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| * [[Birkhoff factorization]], a special case of the Bruhat decomposition for affine groups.
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| ==Notes==
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| <references/>
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| ==References==
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| *[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
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| *[[Nicolas Bourbaki|Bourbaki, Nicolas]], ''Lie Groups and Lie Algebras: Chapters 4-6 (Elements of Mathematics)'', ISBN 3-540-42650-7
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| [[Category:Lie groups]]
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| [[Category:Algebraic groups]]
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