15 problems

Let ${\mathcal A}$ be a cluster algebra of rank $n$, with clusters consisting of $n$ cluster variables. Its exchange graph has clusters as vertices, up to permutation of cluster va…

Let ${\mathcal A}$ be a cluster algebra with initial exchange matrix $B$. Its exchange graph is the quotient of the regular tree of seeds by simultaneous permutations of cluster va…

Let $\mathbb{T}_n$ be the $n$-regular tree, and let $(\tilde B_t)_{t\in\mathbb{T}_n}$ be the family of $2n\times n$ extended exchange matrices for the principal-coefficients cluste…

Let $\mathcal{A}(\mathbf{x},\mathbf{y},B)$ be a cluster algebra with exchange matrix $B$. Its exchange graph is the graph whose vertices are seeds and whose edges join seeds relate…

Let $\mathcal{A}(\mathcal{S})$ be a cluster algebra with seed $(\tilde{\mathbf{x}},\tilde B)$, principal part $B$, rank $n$, clusters, and exchange graph whose vertices are seeds a…

Let $(\mathbf x, \mathbf p, B)$ denote a seed in the cluster algebra with coefficients, or let $(\mathbf x,B)$ denote a seed in the coefficient-free cluster algebra. A cluster is t…

Let $Q=T_{\boldsymbol{n},\boldsymbol{w}}$ and let $T^f_{\boldsymbol{n},\boldsymbol{w}}$ be the specially framed quiver obtained by adjoining the frozen vertices described in the pa…

Consider the exchange graphs of the graded cluster algebras associated with the affine exceptional types $\widetilde{E}_6$, $\widetilde{E}_7$, and $\widetilde{E}_8$. The paper desc…

Let $(Q,W)$ be a quiver with superpotential of degree $N$, and let $\Gamma$ be its associated Ginzburg differential graded algebra. Write $\operatorname{CT}(Q,W)$ for the cluster b…

Let an exchange graph be the graph whose vertices are seeds of a cluster algebra and whose edges correspond to mutations. Its fundamental group consists of homotopy classes of clos…

Exchange graph and seed conjectures. (a) The exchange graph of a cluster algebra with rank $n$ only depends on the initial exchange matrix; (b) every seed is uniquely determined by…

Let $d4d0$ and $d4d0'$ be connected cluster algebras, each of finite type or skew-symmetric finite mutation type. Let $d4e2=(\mathbf{x},B)$ and $d4e2'=(\mathbf{x}',B')$ be seeds of…

Let $u$ and $v$ be vertices of the exchange graph of a cluster algebra with principal coefficients, and let $H^u$ and $H^v$ denote the associated matrices. Rigidity conjecture for…

Let $u$ and $v$ be vertices of the exchange graph of a cluster algebra with principal coefficients. Two extended exchange matrices are equivalent when there is a bijection of their…

Let ${\mathcal X}$ be a set of cluster variables. A seed is a cluster together with its exchange data, and the exchange graph is the graph whose vertices are seeds and whose edges…