10 problems

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…

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 $\Sigma_{t_0}$ 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…

Let $\mathbf{d}_{i;t}^{B^0;t_0}$ be the denominator vectors of the cluster variables at a vertex $t$ relative to the initial vertex $t_0$. A collection of vectors is sign-coherent…

Let $\mathcal{A}$ be a cluster algebra with initial seed at $t_0$, initial cluster $(x_{1;t_0},\dots,x_{n;t_0})$, and let $\mathbf{d}_{\ell;t}^{B^0;t_0}=(d_1,\dots,d_n)$ be the den…

Let $z$ be a non-initial cluster variable in a cluster algebra, and let ${\bf d}$ be its $d$-vector, defined by writing $z=N(x_1,\dots,x_n)/x^{\bf d}$ with $N(x_1,\dots,x_n)$ not d…

Affine compatibility-degree conjecture. We have

Let $B$ be an exchange matrix, and let Theorem denote the source's theorem identifying the combinatorial Cambrian data with denominator vectors of cluster variables. Generalized Ca…

Let the strong $\mathbf g$-vector support conjecture be the assertion concerning equal $\mathbf g$-vectors, supports, and mutation sequences that do not exchange variables in the c…

Assume $B$ is acyclic. Let $\mathrm{Ex}_0(B)$ be the vertices of the exchange graph, let $I(v)$ index the cluster at $v$, and let $\mathbf d(x)$ denote the denominator vector of a…