8 problems

Let $Q$ be a connected quiver with at least one frozen vertex, and let $\mathbf M$ be a reduced (weakly) balanced and monotone mutation sequence. Write $Q^{(j)}_{\mathbf M}$ for th…

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…

Asymptotic sign coherence conjecture. There exists a sequence of strict sign vectors

Let $\mathcal A(\Sigma)$ be a cluster algebra with principal coefficients at $\Sigma=(X,Y,B)$, where $B$ is an $n\times n$ sign-skew-symmetric integer matrix. For a cluster $X'$ an…

Signed-column conjecture. For all $t\in\mathbb{T}_n$, the matrix $D_t^{B_0;t_0}$ has signed columns.

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…