9 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 $B$ be an exchange matrix, and equip the corresponding cluster algebra with principal coefficients using the extended exchange matrix … For a mutation sequence, the $c$-vector…

Contiguity conjecture. The transjective differences are contiguous in every affine type. Consequently, filling fails precisely when $m_r>m_t$; determining $m_t$ and $m_r$ from the…

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.

Admissible-curve conjecture. For any mutation sequence $\boldsymbol{w}$, there exist non-self-intersecting admissible curves $\eta_i^{\boldsymbol{w}}$ such that

Seed dependence conjecture. The $L$-matrix depends, up to signs of its row vectors, only on the seed; equivalently, mutation sequences leading to the same seed yield the same $L$-m…

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