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/

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