"Mutations in cluster algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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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 | + | * 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 <math>B</math>. (think of skew-symmetric) |
| − | * Columns of | + | * Columns of <math>B</math> encode exchange relations. |
| − | * From seed, we can mutate in each of n directions obtaining | + | * From seed, we can mutate in each of n directions obtaining <math>n</math> more seeds |
| − | * For mutation in the | + | * For mutation in the <math>k</math>-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> | :<math>\{\{x_1,\cdots,x_{k-1},x_{k+1},\cdots,x_n\}\cup\{x_k'\}, \mu_k(B)\}</math> | ||
| 17번째 줄: | 17번째 줄: | ||
==matrix mutation== | ==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 | ||
| − | + | :<math> | |
b_{ij}'= | b_{ij}'= | ||
\begin{cases} | \begin{cases} | ||
| − | -b_{ij}, & \text{if | + | -b_{ij}, & \text{if </math>k=i<math> or </math>j<math>}\\ |
| − | b_{ij}, & \text{if | + | b_{ij}, & \text{if </math>b_{ik}b_{kj}\leq 0<math>}\\ |
| − | b_{ij}+b_{ik}b_{kj}, & \text{if | + | b_{ij}+b_{ik}b_{kj}, & \text{if </math>b_{ik},b_{kj}>0<math>}\\ |
| − | b_{ij}-b_{ik}b_{kj}, & \text{if | + | b_{ij}-b_{ik}b_{kj}, & \text{if </math>b_{ik},b_{kj}<0<math>}\\ |
\end{cases} | \end{cases} | ||
| − | + | </math> | |
or simply, | or simply, | ||
| − | + | :<math> | |
b_{ij}'= | b_{ij}'= | ||
\begin{cases} | \begin{cases} | ||
| − | -b_{ij}, & \text{if | + | -b_{ij}, & \text{if </math>k=i<math> or </math>j<math>}\\ |
b_{ij}+\operatorname{sgn}(b_{ik})[b_{ik}b_{kj}]_{+}, & \text{otherwise}\\ | b_{ij}+\operatorname{sgn}(b_{ik})[b_{ik}b_{kj}]_{+}, & \text{otherwise}\\ | ||
\end{cases} | \end{cases} | ||
| − | + | </math> | |
| − | where | + | where <math>[x]_{+}=\max(x,0)</math> |
* 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>. | ||
2020년 11월 13일 (금) 18:27 기준 최신판
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 \(\mu_k^2=I\).