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

 

 

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

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