"Bruhat decomposition"의 두 판 사이의 차이
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==introduction== | ==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. | ||
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==Bruhat cell== | ==Bruhat cell== | ||
| − | + | * G=GL_{n} | |
| − | G=GL_{n} | + | * B : upper triangular matrices \in G |
| − | + | * B_{_} : lower triangular matrices in G | |
| − | B : upper triangular matrices \in G | + | * W=S_{n} we can think of it as a subgroup of G |
| − | + | * Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells. | |
| − | B_{_} : lower triangular matrices in G | ||
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| − | W=S_{n} we can think of it as a subgroup of G | ||
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| − | Double cosets <math>BwB</math> and <math>B_{-}wB_{-}</math> are called Bruhat cells. | ||
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(iii) The exchange relations | (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 | + | $x_{[i,k]}x_{[j,l]} = y_{j-1}y_{j}\cdots y_{k} x_{[i,j-2]}jx_{[i,j-2]}+x_{[i,l]}x_{[j,l]}$ for $1\leq i\leq j-1\leq k\leq l-1\leq n$ |
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| − | *remark | + | * remark |
x_{[i,j]} corresponds to the diagonal between i and j in the triangulation of regular (n+3)-gon | x_{[i,j]} corresponds to the diagonal between i and j in the triangulation of regular (n+3)-gon | ||
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* [[double Bruhat cell example]] | * [[double Bruhat cell example]] | ||
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* http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group | * http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group | ||
* http://eom.springer.de/b/b017690.htm | * http://eom.springer.de/b/b017690.htm | ||
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==expositions== | ==expositions== | ||
| 138번째 줄: | 95번째 줄: | ||
==articles== | ==articles== | ||
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* Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. <em>0804.3303</em> (4월 21). http://arxiv.org/abs/0804.3303.<br> | * Yang, Shih-Wei, 와/과Andrei Zelevinsky. 2008. “Cluster algebras of finite type via Coxeter elements and principal minors”. <em>0804.3303</em> (4월 21). http://arxiv.org/abs/0804.3303.<br> | ||
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| 156번째 줄: | 104번째 줄: | ||
* http://mathoverflow.net/questions/15438/a-slick-proof-of-the-bruhat-decomposition-for-gl-nk | * 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/28569/is-there-a-morse-theory-proof-of-the-bruhat-decomposition | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:cluster algebra]] | [[분류:cluster algebra]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 6월 26일 (수) 12:43 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
encyclopedia
- http://en.wikipedia.org/wiki/Longest_element_of_a_Coxeter_group
- http://eom.springer.de/b/b017690.htm
expositions
- [1]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.