440 problems

Let $\mathcal{A}$ be a cluster algebra, and let $x$ and $y$ be cluster variables that are not compatible, meaning that $x$ and $y$ do not occur together in any cluster. Then there…

Let $\mathcal{D}$ be a full-rank cluster algebra of finite type, and consider ratios of products of its cluster variables on the totally positive locus. Such a ratio is called boun…

For every $n\geq 1$, every initial seed with cluster $\mathbf{x}=(x_1,\ldots,x_n)$ in a cluster algebra of type $A_n$, and every cluster monomial $M$, write its Laurent expansion a…

Let $\mathbf{\Sigma}$ be any cluster pattern with initial point $t_0\in \mathbb{T}_n$, and let $F_{i;t}(\mathbf{y})$ be an $F$-polynomial in that pattern. Fomin–Zelevinsky's Lauren…

Let $\mathcal{A}$ be a cluster algebra with trivial coefficients, and fix a cluster $\mathbf{x}=(x_1,\dots,x_n)$. For every cluster monomial $z$, write its Laurent expansion unique…

Let $\mathscr{C}_{\ell}$ be the Hernandez–Leclerc monoidal subcategory and let $K_{0}(\mathscr{C}_{\ell})$ be its Grothendieck ring, identified with a cluster algebra. A simple mod…

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…

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

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 $\mathcal{G}$ be a simple complex Lie group, let $T=(\Gamma_{1},\Gamma_{2},\gamma)$ be a Belavin–Drinfeld triple, and let $k_T$ and $\mathcal{H}_T$ be the quantities defined by…

Let $B$ be an exchange matrix, and consider an extended exchange matrix in a $Y$-pattern with initial exchange matrix $B$ and principal coefficients. The rows indexed $n+1$ through…

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 $v\leq w$ be elements indexing an open Richardson variety $\mathring{\mathcal{R}}_{v,w}$, and let $\mathbf{w}$ be the chosen word defining Leclerc's seed…

Cluster-structure conjecture. The homogeneous coordinate ring equals the lifted cluster algebra:

Fock–Goncharov conjecture. The upper cluster algebra $\mathcal{U}$ possesses a basis parametrized by the tropical points.

Let $\frak{S}$ be a triangulable punctured surface. A congruent ${\rm SL}_3$-lamination is an element $\ell \in \mathscr{A}_{{\rm SL}_3,\frak{S}}(\mathbb{Z}^t)$, and…

Let $\mathbf{\Sigma}=\{\Sigma_t=(\mathbf{x}_t,\mathbf{y}_t,B_t)\mid t\in\mathbb{T}_n\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. An unlabeled cluster i…

Hernandez–Leclerc's conjecture. The cluster monomials of the cluster algebra are in bijection with the isomorphism classes of real simple objects in $\mathscr{C}^-$.

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…

Arborization conjecture. The forest form of a cluster monomial can be found from the web form via the arborization algorithm.

Let $\mathcal{A}$ be the cluster algebra identified with the ring of truncated $q$-characters of the category $\mathcal{C}^-$. A simple $U_q(\widehat{\mathfrak{g}})$-module $S$ is…

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…

Kontsevich's conjecture. For any $r_1,r_2\in\mathbb{Z}_{>0}$, all these iterations are given by noncommutative Laurent polynomials in $x$ and $y$.

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 $\mathcal{A}_{\Gamma_\infty'}$ be the cluster algebra associated with the quiver $\Gamma_\infty'$, and let $\mathcal{O}^{sh}$ be the shifted category appearing in the isomorphi…