Mutations in cluster algebras

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 \(\mu_k^2=I\).

related items

• Quiver mutations