"Mutations in cluster algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 26번째 줄: | 26번째 줄: | ||
\end{cases} | \end{cases} | ||
$$ | $$ | ||
| + | or simply, | ||
| + | $$ | ||
| + | b_{ij}'= | ||
| + | \begin{cases} | ||
| + | -b_{ij}, & \text{if $k=i$ or $j$}\\ | ||
| + | b_{ij}+\operatorname{sgn}(b_{ik})[b_{ik}b_{kj}]_{+}, & \text{otherwise}\\ | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | where $[x]_{+}=\max(x,0)$ | ||
* Note that <math>\mu_k(B)</math> is skew-symmetrizable and <math>\mu_k^2=I</math>. | * Note that <math>\mu_k(B)</math> is skew-symmetrizable and <math>\mu_k^2=I</math>. | ||
| − | |||
==related items== | ==related items== | ||
* [[Quiver mutations]] | * [[Quiver mutations]] | ||
2013년 10월 13일 (일) 13:15 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
$$ b_{ij}'= \begin{cases} -b_{ij}, & \text{if $k=i$ or $j$}\\ b_{ij}, & \text{if $b_{ik}b_{kj}\leq 0$}\\ b_{ij}+b_{ik}b_{kj}, & \text{if $b_{ik},b_{kj}>0$}\\ b_{ij}-b_{ik}b_{kj}, & \text{if $b_{ik},b_{kj}<0$}\\ \end{cases} $$ or simply, $$ b_{ij}'= \begin{cases} -b_{ij}, & \text{if $k=i$ or $j$}\\ b_{ij}+\operatorname{sgn}(b_{ik})[b_{ik}b_{kj}]_{+}, & \text{otherwise}\\ \end{cases} $$ where $[x]_{+}=\max(x,0)$
- Note that \(\mu_k(B)\) is skew-symmetrizable and \(\mu_k^2=I\).