19 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…

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 $Q$ be an acyclic quiver, and endow it with its standard cyclic ordering, as constructed in the cited observation. A cyclically ordered quiver is totally proper when every quiv…

Let $Q$ and $Q'$ be two $3$-vertex proper cyclically ordered quivers. A cosquare is the matrix invariant associated with the unipotent companion of a cyclically ordered quiver; wri…

Let $\gamma$ be a representation path of a mutation loop, let $\boldsymbol{\epsilon}_\gamma(w)$ denote its sign data at a point…

Greenness conjecture. The sequence of vertices of $\Gamma_{\overline{w}}$ mutated by $\mu_\bullet$ is a green sequence with respect to $\Gamma_{\overline{w}}$.

Let $B$ be a skew-symmetrizable matrix indexed by $mathcal I={1,ldots,n}$. For a mutation sequence $\boldsymbol{w}$, let $r_i^{\boldsymbol{w}}$ be the associated reflections, let…

Let $Q$ be a quiver, let $\boldsymbol{w}$ and $\boldsymbol{v}$ be mutation sequences, and suppose that both sequences lead from an initial labelled seed to the same labelled seed.…

Berenstein–Geiss–Muller conjecture. The following are equivalent:

Let $P$ and $P'$ be plabic graphs, and associate to each its quiver. Two plabic graphs are move equivalent when they are related by local moves and changes of the colors of boundar…

Let $(C,z)$ and $(C',z')$ be real isolated plane curve singularities, and choose real morsifications of them. Each morsification has an associated quiver; two quivers are mutation…

Integrality conjecture. The sequence $(b_n)_{n\in\mathbb{N}}$ is integer if and only if $p=1$; analogously, for the second recurrence in the preceding corollary, integrality holds…

Let $T_i$ denote the Lusztig isomorphisms associated with the simple roots of the quantum group, and let quiver mutations act on the corresponding quantum cluster structure. Luszti…

Quiver and mutation conjecture. For every semisimple, simply connected, complex Lie group $G$, there exists a quiver $Q_G$ and quiver mutations $\mu_G^{\operatorname{rot}}$ and…

The non-trivially valued mutation-periodicity conjecture. Corollary … and … , also hold when $Q$ is non-trivially valued.

Let ${\cal F}(\vec N;\zeta;q;1;1)$ be the generating function for numerical Coulomb-branch invariants, obtained by specializing the refined variables to $y=t=1$. Numerical mutation…

Let $\vec N$ be the truncated dimension vector obtained by omitting $N_k$, and define the generating functions … with an analogous function ${\cal F}'(\vec N;\zeta';q;y;t)$ for the…

Let a generalized quiver have basis vectors $\gamma_i$, DSZ pairings $\gamma_{ij}$, stability parameters $\zeta_i$, dimension vector $N_i$, and single-centered invariants…

Let $epsilon_{alpha beta}=3arr(alpha\rightarrowbeta)-3arr(beta\rightarrowalpha)$ denote the signed difference between the numbers of arrows from nodes $alpha$ to $beta$ and from…