Mutations in cluster algebras
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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 \(\mu_k^2=I\).