Bruhat decomposition

introduction

• Given a Lie group \(G\) over \(\mathbb{C}\) and a Borel subgroup \(B\), there is famous Bruhat decomposition of the flag variety \(G/B\)
• \(G\) : connected reductive algebraic group over an algebraically closed field
• By allowing one to reduce many questions about \(G\) to questions about the Weyl group \(W\), Bruhat decomposition is indispensable for the understanding of both the structure and representations of \(G\)
• The order of a Chevalley group over a finite field was computed in [C1] (using Bruhat decomposition) in terms of the exponents of the Weyl group
• Bruhat order
• Weyl group action

example : general linear group

• \(G=GL_{n}\)
• \(B\) : upper triangular matrices in \(G\)
• \(B_{-}\) : lower triangular matrices in \(G\)
• \(W=S_{n}\) we can think of it as a subgroup of \(G\)
• Double cosets \(BwB\) and \(B_{-}wB_{-}\) are called Bruhat cells.

(B, N) pair

• A \((B, N)\) pair is a pair of subgroups \(B\) and \(N\) of a group \(G\) such that the following axioms hold:

1. \(G\) is generated by \(B\) and \(N\)
2. The intersection, \(T\), of \(B\) and \(N\) is a normal subgroup of N
3. The group \(W = N/T\) is generated by a set \(S\) of elements \(w_i\) of order 2, for \(i\) in some non-empty set \(I\)
4. If \(w_i\) is an element of \(S\) and \(w\) is any element of \(W\), then \(w_iBw\) is contained in the union of \(Bw_iwB\) and \(BwB\)
5. No generator \(w_i\) normalizes \(B\)

• we say \((B,N)\) form a \(BN\)-pair of \(G\), or that \((G,B,N,S)\) is a Tits system
• we call \(B\) the Borel subgroup of \(G\), and \(W=N/B\cap N\) the Weyl group associated with the Tits system
• the rank of the Tits system is defined to be \(|S|\)

why do we care?

• \((B, N)\) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs.
• Roughly speaking, it shows that all such groups are similar to the general linear group over a field
• BN-pairs can be used to prove that most groups of Lie type are simple

Bruhat decomposition theorem

thm

Let \(G\) be a group with a \(BN\)-pair. Then \[ G=BWB \] or, \[ G=\cup_{w\in W}BwB \] in which the union is disjoint, where \(BwB\) is taken to mean \(B\dot{w}B\) for any \(\dot{w}\in N\) with \(\dot{w}T=w\)

memo

• http://qchu.wordpress.com/2010/07/11/chevalley-bruhat-order/

related items

• Cartan decomposition of general linear groups

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZzFwSzhRYnRHalE/edit

encyclopedia

• http://en.wikipedia.org/wiki/(B,_N)_pair
• http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
• http://eom.springer.de/b/b017690.htm

expositions

• Lusztig, G. 2010. “Bruhat Decomposition and Applications.” arXiv:1006.5004 [math] (June 25). http://arxiv.org/abs/1006.5004.
• http://math.ucr.edu/home/baez/week186.html
• Bruhat decomposition via row reduction

articles

• [C1] Chevalley, C. 1955. “Sur Certains Groupes Simples.” The Tohoku Mathematical Journal. Second Series 7: 14–66.
• Bruhat, Fran\ccois. 1956. “Sur Les Représentations Induites Des Groupes de Lie.” Bulletin de La Société Mathématique de France 84: 97–205.

question and answers(Math Overflow)

• http://mathoverflow.net/questions/15438/a-slick-proof-of-the-bruhat-decomposition-for-gl-nk
• http://mathoverflow.net/questions/28569/is-there-a-morse-theory-proof-of-the-bruhat-decomposition
• http://mathoverflow.net/questions/168033/coxeter-groups-parabolic-subgroups/168035#168035
• http://mathoverflow.net/questions/188920/closure-relations-between-bruhat-cells-on-the-flag-variety/190961#190961

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