766 problems

A cluster algebra is an algebra generated by cluster variables grouped into clusters; a Laurent polynomial with positive coefficients is a Laurent polynomial whose coefficients are…

Let $cal C$ be a monoidal abelian length category, and let $cal A$ be a cluster algebra. A monoidal categorification of $cal A$ is an isomorphism of $bb Z$-algebras…

Let $V$ be a cluster variety and let $V^\vee$ be its mirror cluster variety. Write $V^\vee(\mathbb{Z}^T)$ for the tropical points of $V^\vee$. Fock–Goncharov's cluster duality conj…

Braid-group fixed-point conjecture. For every $i\in[d-1]$, if $(k,n)$ is $(3,9)$ or $(4,8)$, every unstable fixed point of $\sigma_i$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$ is a c…

Sign-coherence and normalization conjecture. (a) For every ${\bf w}\in\mathcal{T}$, all $C$-patterns ${\bf C}(B^{\bf w})$ and ${\bf C}((B^{\bf w})^{\top})$ are sign-coherent.

Let $B$ be an exchange matrix, and equip the corresponding cluster algebra with principal coefficients using the extended exchange matrix … For a mutation sequence, the $c$-vector…

Let $G$ be a simply-laced algebraic group, and let the cluster subalgebra constructed by Leclerc from modules of the preprojective algebra be a subalgebra of the coordinate ring of…

Let $G$ be a connected, simply connected, complex simple algebraic group, and let a Poisson bracket on $G$ belong to a Belavin–Drinfeld class. Gekhtman–Shapiro–Vainshtein conjectur…

Let a finite type simply laced Dynkin diagram be given, and consider its associated system of algebraic relations, called a Y-system. Zamolodchikov's periodicity conjecture. Soluti…

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 $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated gauge group $\mathsf{G}$, representation space $\mathsf{N}$, stability con…

Let the tree-level amplitudes be decomposed into terms arising from BCFW recursions, and let $\operatorname{Gr}_{4,n}$ be the Grassmannian with its cluster algebra structure. A col…

Monoidal categorification conjecture. If $C$ is symmetric, the pair

For an acyclic exchange matrix $B$ and a Coxeter element $c$, the source constructs a $c$-Cambrian fan and a collection of cones of cluster $\mathbf g$-vectors. Cambrian fan and…

Morier-Genoud–Ovsienko conjecture. The rank polynomial of $J(P)$ is unimodal.

Let $\sigma$ be a $d$-admissible generalized signature of size $n$ and dimension $d$, and let $R_\sigma=\mathbb{C}[V^\sigma]^{\operatorname{SL}(V)}$. Natural cluster structure conj…

Let $B$ be an exchange matrix. The scattering fan–mutation fan coincidence conjecture. The scattering fan $\operatorname{ScatFan}^T\!(B)$ coincides with the mutation fan…

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, and let $\mathcal{A}(\Sigma_{g,n})$ denote its cluster algebra. Non-finite generation conjecture. For $g\geq 2$,…

For an acyclic quiver, let the associated cluster algebra be defined from an acyclic seed, and let generic Caldero–Chapoton functions be the generic values of the Caldero–Chapoton…

Let $\mathcal G$ be a simple complex Lie group equipped with a factorizable Lie bialgebra structure, and let $T=(\Gamma_1,\Gamma_2,\gamma)$ be a Belavin–Drinfeld triple for its tan…

Let $\mathcal{A}$ be a cluster algebra, and let a cluster monomial mean a monomial in cluster variables all belonging to the same cluster. Strong positivity conjecture. Any cluster…

A cluster monomial is a monomial in cluster variables from one cluster, and its denominator vector is obtained from the denominator vectors of its variables with multiplicities. De…

Let $Q$ be an acyclic quiver. Let $\mathcal A(Q)$ be its acyclic cluster algebra and let $\mathcal G(Q)$ be the set of generic variables associated with $Q$. Generic-variable basis…

Let $D$ be a Postnikov diagram, let $Q_D$ be the associated ice quiver, and let $\mathsf{A_{MS}}$ be the cluster algebra defined by $Q_D$ after localising at the frozen cluster var…

For even $n$, let $\omega_0$ be the cluster DT-transformation corresponding to the longest element of the birational Weyl group. A maximal green sequence is a mutation sequence wit…