438 problems

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…

Let $\beta=\beta^1\beta^2$ be the braid and let $r_1$ and $w$ be as in the construction of the open subset $\mathcal{U}_{r_1,w}(\beta)\subseteq X(\beta)$. Assume that…

Let $T\in\operatorname{SSYT}(k,[n])$ be a tableau whose distinct entries are $a_1<\ldots<a_r$. Let $n'\ge n$ and let $f:\{a_1,\ldots,a_r\}\to[n']$ satisfy $f(a_1)<\ldots<f(a_r)$; w…

For positive integers $k,n,r$, let $N_{k,n,r}$ denote the number of cluster variables of rank $r$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$. Cluster-variable counting conjecture. The…

Let $\eta$ be a braid word and let $X(\eta)$ denote the corresponding braid variety. A cluster algebra structure conjecture asserts that the coordinate ring of any braid variety…

Let $\mathcal{C}_{\epsilon^\mathbb{Z}}$ be the monoidal category associated to $\mathfrak{g}=\mathfrak{sl}_k$ at a parameter $\epsilon\in\mathbb{C}^*$ such that $\epsilon^2$ is an…

Let $a_n$ be the cluster variables associated with the extending vertex, and let $\tilde{K}_n$ denote the quantity whose period-two behavior is being conjectured. Period-two formul…

Let $\tilde{K}_n$ be the determinant introduced above, namely … Here the variables $a_n$ and $h_n$ are the cluster variables associated with the extending vertex and its neighborin…

Let $b,c$ be positive integers, and let $X[a_1,a_2]$ denote the quantum greedy element of the rank $2$ quantum cluster algebra $\mathcal{A}_v(b,c)$ associated with…

Let $n$ be a positive integer, and let $P$ be a linear polynomial in $x_1,\ldots,x_n$ with positive coefficients. Say that $P$ generates a period 1 seed when it generates a seed of…

Fomin–Zelevinsky's conjectures. The following statements hold:

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 ${\cal D}$ be the cluster double, with Langlands dual ${\cal D}^{\vee}$, and let ${\mathbb Z}_+\{S\}$ denote the semigroup of finitely supported nonnegative-integer-valued func…

Strong cluster-map conjecture. For every $w\in W$, the coordinate ring $\mathbb{C}[U^w]$ is a cluster algebra modelled by the strong cluster map $\varphi_w$ for the standard compon…

Let $G$ be a simple simply connected complex algebraic group of simply-laced type $A$, $D$, or $E$, let $J$ be a subset of the simple roots, and let $K$ be its complementary subset…

Let $G$ be of simply-laced type $A$, $D$, or $E$, let $J$ be the chosen subset of simple roots, let $K$ be its complementary subset, and let $N_K$ be the corresponding unipotent su…

Let $\mathcal{A}=\mathcal{A}(\mathcal{S})$ be a cluster algebra, let $\mathcal{X}$ be its set of cluster variables, and let the cluster complex be the simplicial complex on…

Let $\mathcal{A}=\mathcal{A}(\mathcal{S})$ be a normalized skew-symmetrizable cluster algebra, where $\mathcal{S}$ is a mutation equivalence class of seeds, and let a cluster mean…

Let $W$ be a finite Coxeter group, let $c$ be a Coxeter element, and let a $c$-cluster $C$ have upper roots among its roots. Associate to $C$ the intersection of the hyperplanes or…

Let $W$ be a finite Coxeter group of rank $n$, let $c$ be a Coxeter element, and let $(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_{n-k})$ be a $c$-cluster, where the $\alpha_i$…

Fix $B^0,t_0,\ell,t$ such that the cluster variable $x_{\ell;t}$ is not in the initial cluster at $t_0$, and write its denominator vector as…

Let $t_0\xrightarrow{k}t_1$ be adjacent vertices of $\mathbb{T}_n$, let $B^1=\mu_k(B^0)$, and fix $t\in\mathbb{T}_n$ and $\mathbf{a}\in\mathbb{Z}_{\geq0}^n$. Write…

Let $\mathcal{A}$ be a cluster algebra with initial seed at $t_0$, and let $\mathbf{g}_{\mathbf{a};t}^{B^0;t_0}$ be the $\mathbf{g}$-vector of a cluster monomial $x_{\mathbf{a};t}$…

Let $\Sigma_{t_0}$ be an initial seed, and let $\mathbf{d}_{\mathbf{a};t}^{B^0;t_0}$ denote the denominator vector of a cluster monomial at vertex $t$. Denominator-vector injectivi…