34 problems

Let $Q$ be a quiver and let $\mu_k(Q)$ denote the quiver obtained by mutation at vertex $k$. The unrestricted red size $\operatorname{uRed}(Q)$ is the minimum red size over general…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. For each mutation index $i$, let $r_i$ denote the associated term and let $M$ be the set of relevant mon…

Let $Q$ and $Q'$ be finite quivers. For sequences ${\bf v}$ and ${\bf w}$ of vertices of $Q$ and $Q'$, respectively, let ${\bf v}\boxtimes{\bf w}$ be the corresponding sequence in…

Let $Q$ be a quiver of rank at least $4$, and let $[A_2]$ denote the finite-type $A_2$ quiver. Say that $[A_2]$ embeds into the mutation class of $Q$ when it occurs there under the…

Let $Q$ be a quiver with a reddening sequence, and let the associated permutation of a reddening sequence be the permutation induced by its action on the framing. An embedding of t…

Let $Q$ be a quiver, and let $\operatorname{uRed}(Q)$ denote its unrestricted red size, namely the maximal number of red vertices obtained after an arbitrary mutation sequence. Two…

Let $Q$ be a connected quiver on $n$ vertices that does not admit a reddening sequence. The red size $\operatorname{Red}(Q)$ is the maximal number of red vertices obtained after mu…

Let $Q$ be a quiver with a directed path … where $A$ and $B$ are distinguished vertices and the displayed arrows have multiplicities $x_0,x_1,\ldots,x_k$. A quiver $Q'$ is mutation…

Let $G$ and $G'$ be connected simple plabic graphs whose quivers $Q_G$ and $Q_{G'}$ are mutation equivalent. Let $L^{\operatorname{plab}}_G$ and $L^{\operatorname{plab}}_{G'}$ be t…

Let $Q$ be a connected quiver, let $V(Q)$ be its vertex set, and suppose that $Q$ does not admit a reddening sequence. A general maximal green sequence is a mutation sequence resul…

Let $(Q,W)$ and $(Q',W')$ be quivers with potential related by mutation at a vertex $k$, and let $d,d'$ and stability weights $\theta,\theta'$ be related by the mutation formulas i…

A real morsification is a real morsification of a real isolated plane curve singularity, and each such morsification has an associated quiver. Fomin–Pylyavskyy–Shustin conjecture.…

Let a mutation class be originating from an unpunctured surface or orbifold if it arises from an unpunctured surface or orbifold in the sense of the paper. A realization by reflect…

Let $Q$ be a framed quiver and consider the periodic mutation sequence … Assume that performing the mutations $v_1,\ldots,v_p$ on the base quiver returns it to the original quiver.…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. Bounded-coefficient conjecture for deformed $F$-polynomials. The polynomial $S_n$ is bounded: for each m…

Let $Q$ be a framed quiver and let $v_1,v_2,\ldots$ be a mutation sequence. Polynomiality conjecture for deformed $F$-polynomials. The deformed $F$-polynomial $S_n$ is a polynomial…

Let $\Gamma_n$ be the $n$-cycle quiver, and let $\Phi^1$ denote the corresponding root set, with $|\Phi^1|=\binom{n+1}{2}-1$. A maximal green sequence for $\Gamma_n$ has a well-def…

Let $Q$ be a quiver, and consider all maximal green sequences for $Q$. Their lengths are integers. No Gap Conjecture. The set of lengths of all possible maximal green sequences for…

Let $Q$ be a quiver in the class $\mathcal{P}$, and let $\mathcal{A}$ and $\mathcal{U}$ be the cluster algebra and upper cluster algebra associated with the cluster algebra defined…

Let $Q$ be a quiver in the class $\mathcal{P}'$, let $(\mathbf{x},\mathbf{y},Q)$ be a seed with quiver $Q$, and let $\mathcal{A}$ and $\mathcal{U}$ be the associated cluster algebr…

Let $Q$ be a quiver, and let $\mathcal{A}$ and $\mathcal{U}$ denote the associated cluster algebra and upper cluster algebra over a ground ring. A reddening-sequence criterion conj…

Let $Q$ be a quiver, and let $\mathcal{A}$ and $\mathcal{U}$ denote the associated cluster algebra and upper cluster algebra, respectively, over a specified ground ring. A maximal-…

Let $Q$ and $Q'$ be quivers such that $\Psi(Q)$ and $\Psi(Q')$ are finite and non-empty, and let $Q\oplus Q'$ denote their direct sum. Direct-sum finiteness conjecture. Then…

Let $Q$ be a minimal mutation-infinite quiver. Let $\widehat{\Psi}(Q)$ denote the subgraph associated with quivers admitting maximal green sequences, as in Theorem. Minimal mutatio…

Maximal green sequence equivalence conjecture. Various desirable properties of cluster algebras are equivalent to the existence of a maximal green sequence.