"Mutations in cluster algebras"의 두 판 사이의 차이
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imported>Pythagoras0 (새 문서: ==introduction== * A '''seed''' for A is an initial cluster <math>x=\{x_1,\cdots,x_n\}</math> and an <math>n\times n</math> skew-symmetrizable matrix $B$. (think of skew-symmetric) * ...) |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* A '''seed''' for A is an initial cluster <math>x=\{x_1,\cdots,x_n\}</math> and an <math>n\times n</math> skew-symmetrizable matrix $B$. (think of skew-symmetric) | * A '''seed''' for A is an initial cluster <math>x=\{x_1,\cdots,x_n\}</math> and an <math>n\times n</math> skew-symmetrizable matrix $B$. (think of skew-symmetric) | ||
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* Columns of $B$ encode exchange relations. | * Columns of $B$ encode exchange relations. | ||
| + | * From seed, we can mutate in each of n directions obtaining $n$ more seeds | ||
| + | * For mutation in the $k$-th direction, we obtain the new seed | ||
| + | :<math>\{\{x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n\}\cup\{x_k'\}, \mu_k(B)\}</math> | ||
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| + | ==exchange relation== | ||
* For <math>k\in \{1,2,\cdots, n\}</math>, | * For <math>k\in \{1,2,\cdots, n\}</math>, | ||
:<math>x_kx_k' = \prod_{b_{ik}>0} x_i^{b_{ik}}+\prod_{b_{ik}<0} x_i^{|b_{ik}|}</math> | :<math>x_kx_k' = \prod_{b_{ik}>0} x_i^{b_{ik}}+\prod_{b_{ik}<0} x_i^{|b_{ik}|}</math> | ||
| 8번째 줄: | 13번째 줄: | ||
* This is the mutation into the k-th direction | * This is the mutation into the k-th direction | ||
* (Fig3) | * (Fig3) | ||
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| + | ==matrix mutation== | ||
* Here <math>\mu_k(B)=(b_{ij}')</math> is a new matrix defined as | * Here <math>\mu_k(B)=(b_{ij}')</math> is a new matrix defined as | ||
$$ | $$ | ||
2013년 10월 13일 (일) 13:06 판
introduction
- A seed for A is an initial cluster \(x=\{x_1,\cdots,x_n\}\) and an \(n\times n\) skew-symmetrizable matrix $B$. (think of skew-symmetric)
- Columns of $B$ encode exchange relations.
- From seed, we can mutate in each of n directions obtaining $n$ more seeds
- For mutation in the $k$-th direction, we obtain the new seed
\[\{\{x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n\}\cup\{x_k'\}, \mu_k(B)\}\]
exchange relation
- For \(k\in \{1,2,\cdots, n\}\),
\[x_kx_k' = \prod_{b_{ik}>0} x_i^{b_{ik}}+\prod_{b_{ik}<0} x_i^{|b_{ik}|}\]
- This defines a new cluster variable \(x_k'\)
- This is the mutation into the k-th direction
- (Fig3)
matrix mutation
- Here \(\mu_k(B)=(b_{ij}')\) is a new matrix defined as
$$ b_{ij}'= \begin{cases} -b_{ij}, & \text{if $k=i$ or $j$}\\ b_{ij}, & \text{if $b_{ik}b_{kj}\leq 0$}\\ b_{ij}+b_{ik}b_{kj}, & \text{if $b_{ik},b_{kj}>0$}\\ b_{ij}-b_{ik}b_{kj}, & \text{if $b_{ik},b_{kj}<0$}\\ \end{cases} $$
- Note that \(\mu_k(B)\) is skew-symmetrizable and \(\mu_k^2=I\).