Bruhat decomposition

introduction

• $G$ : 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 

Bruhat cell

• $G=GL_{n}$
• $B$ : 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.

thm

$$ G=BWB $$

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/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