"Bruhat decomposition"의 두 판 사이의 차이
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2012년 10월 29일 (월) 08:05 판
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
history
encyclopedia
- http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
- http://eom.springer.de/b/b017690.htm
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/[1]
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
- [2]http://www-math.mit.edu/~gyuri/papers/bru1.pdf
- Double Bruhat Cells http://pages.uoregon.edu/dmoseley/talks/Lecture14.pdf
- Cluster Structures on Double Bruhat Cells http://pages.uoregon.edu/dmoseley/talks/Lecture15.pdf
- 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.
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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
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