7 problems

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 ${\bf SC}$ be the class of real skew-symmetrizable exchange matrices satisfying the sign-coherent property, and let ${\bf B}(B)$ denote the mutation-equivalence class of $B$. T…

Let $B\in{\bf SC}$ be a real exchange matrix with skew-symmetrizer $D=\mathrm{diag}(d_1,\dots,d_n)$, where ${\bf SC}$ denotes the class of skew-symmetrizable matrices satisfying th…

Positive dual c-vector uniqueness conjecture. Any positive dual $\boldsymbol{c}$-vector supports exactly one extremal exchangeable pair.

Root conjecture. Every $c$-vector of the cluster algebra $\mathcal{A}_\bullet(B)$ is a root of $\Delta(A(B))$.

Sign-coherence conjecture. (i) Any $c$-vector of $\mathcal{A}_\bullet(B)$ is a nonzero vector, and its components are either all non-negative or all non-positive. (ii) Any non-init…

Sign-coherence and constant-term conjectures. (i) Every column of $C'$ is a nonzero vector, and its nonzero components are either all positive or all negative. (ii) Every $F$-polyn…