Math 610 Assignment 1

1. Let $b,c$ be positive integers. Let $\mathcal{A}_{(b,c)}$ to be the cluster algebra generated by $\{x_m : m \in \mathbb Z\}$ over $\mathbb Q$ subject to the following set of exchange relations, from initial cluster $\{x_1,x_2\}$. Check that $\mathcal{A}_{(b,c)}$ has finitely many variables if and only if $bc \le 3$.
   $$x_{m-1}x_{m+1} = \begin{cases} x_m^b + 1 & m \text{ odd} \\ x_m^c + 1 & m \text{ even}\end{cases} \qquad m\in\mathbb Z$$
2. Explain why this is equivalent to the condition that $\begin{bmatrix} 2 & -b \\ -c & 2\end{bmatrix}$ is the Cartan matrix of rank 2 root system.
3. Verify that in this case ($bc\le 3$) the non-initial cluster variables are parametrized by the positive roots of the corresponding root system.
4. Prove that the matrix $B_Q$ with $(i,j)$ entry given by $\# \{i\to j\} - \#\{ j\to i\})$ is anti-symmetric.
5. Check that quiver mutation is involutive. (Let $Q$ be a quiver, let $k$ be a vertex, denote mutation wrt $k$ by $\mu_k$ and verify that $\mu_k^2 Q = Q$.)
6. Check that seed mutation is involutive. (Let $(R,u)$ be a seed.)
7. Determine the exchange graph of $\bullet\to\bullet\to\bullet$ a quiver of type $A$ and rank 3.
8. Prove that the cluster structure $\mathcal{A}_Q$ does not depend on the choice of initial seed.
9. Let $n = 4$. Verify that the coordinate ring $A$ of the unipotent subgroup $N\subset \text{SL}(n+1,\mathbb C)$ is a type $D_6$ cluster algebra. More precisely, $A$ has the structure of the cluster-algebra-with-coefficients of type $\tilde Q$ over $A$, where $\tilde Q\sim D_6$ is the ice quiver