Bruhat decomposition

introduction

• double Bruhat cells
• Bruhat order
• Weyl group action 
• The decomposition of G into strata G^{u,v} is 'good with respect to total positivity.

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.

 

double Bruhat cell (DBC)

• \(G^{u,v} =BuB\cap B_{-}vB_{-}\)
• \(G=\cup_{u,v\in W\times W} G^{u,v}\) (disjoint union)

 

 

realization of finite type cluster algebra

• Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. 0804.3303 (4월 21). http://arxiv.org/abs/0804.3303.
• \(\mathbb{C}[L^{c,c^{-1}}]\) is a cluster algebra of finite type. It has the same type as Cartan matrix.

 

type A_{n}

• (i) inite seed is given by x=(x_{[1,1]},\cdots,x_{[1,n]}), y=(y_1,\cdots,y_n), B=B(C)
• (ii) The set of cluster variables is \{x_{[i,j]}|1\leq i\leq j\leq n \}
• (iii) The exchange relations

$$x_{[i,k]}x_{[j,l]} = y_{j-1}y_{j}\cdots y_{k} x_{[i,j-2]}jx_{[i,j-2]}+x_{[i,l]}x_{[j,l]}$$ for $1\leq i\leq j-1\leq k\leq l-1\leq n$

• remark : $x_{[i,j]}$ corresponds to the diagonal between i and j in the triangulation of regular $(n+3)$-gon

 

example

• double Bruhat cell example

memo

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

computational resource

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

 

related items

• Total positivity

 

encyclopedia

• http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
• http://eom.springer.de/b/b017690.htm

expositions

• http://www-math.mit.edu/~gyuri/papers/bru1.pdf
• http://pages.uoregon.edu/dmoseley/talks/
  • Double Bruhat Cells
  • Cluster Structures on Double Bruhat Cells
• http://math.ucr.edu/home/baez/week186.html
• http://www.math.harvard.edu/~ryanr/bruhat_row-reduction.pdf

 

articles

• Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. 0804.3303 (4월 21). http://arxiv.org/abs/0804.3303.
  

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