8 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 $Q$ 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…

Let $Q$ be a connected quiver with no frozen vertices, and let $\mathbf M$ be a monotone and (weakly) balanced mutation sequence for the principal framing of $Q$. For…

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…

Let $B\in\mathrm{M}_{n}(\mathbb{R})$ be a skew-symmetrizable matrix with skew-symmetrizer $D=\mathrm{diag}(d_1,\dots,d_n)$. Suppose that $B$ satisfies the sign-coherent property, m…

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

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