18 problems

Acyclic Ext-quiver conjectures. 1. If $A$ is twisted fractionally Calabi–Yau, then $A$ has an acyclic Ext-quiver. 2. If $A$ is $d$-representation-finite, then $A$ has an acyclic Ex…

The paper studies Gale–Robinson quivers and the associated recurrence formulas, including the formula … Here the notation refers to the paper's construction of the sequence…

Let $Q$ be a good 3d $\mathcal{N}=4$ quiver theory, let $\mathcal{M}_C(Q)$ denote its Coulomb branch, and let $\mathcal{L}$ be the set of decay and fission products of $Q$, ordered…

Exponential-network reconstruction conjecture. The underlying quiver structure of $\mathcal{Q}$ is obtained from these calibrated paths: its vertices correspond to the basis states…

Smaller-quiver realization conjecture. Apart from the universal quiver of larger size, the whole permutohedra graph can also be obtained from a single quiver of a smaller size.

Let $Q$ be a quiver with a directed path … where $A$ and $B$ are distinguished vertices and the displayed arrows have multiplicities $x_0,x_1,\ldots,x_k$. A quiver $Q'$ is mutation…

Square tensor-product quiver conjecture. For suitable values of $m$, the quivers for many theories in this class can take the form of square tensor products, with diagrams

Let $B$ be a sign-skew-symmetric matrix, meaning that for every $i,j$, either $b_{ij}=b_{ji}=0$ or $b_{ij}b_{ji}<0$. A matrix is totally sign-skew-symmetric if every matrix obtaine…

Temperedness conjecture. Quiver A-polynomials are tempered for every choice of the adjacency matrix.

Temperedness conjecture. For any symmetric quiver, $A(x,y)$ is tempered.

Strong free product section conjecture. For every such $Q$, there exists a section $\sigma$ such that the quadruple $(L_Q,r,\sigma,B)$ satisfies the strong free product property.

Strong free product property conjecture. The map $\sigma'$ is a linear isomorphism, equivalently $(L_Q,r,\sigma,kQ_0[t,(t+q)^{-1}])$ satisfies the strong free product property.

Multiplicative preprojective algebra conjecture. For every non-Dynkin quiver, the corresponding properties—including 2-Calabi–Yau-ness, primeness, flatness in $q$, and the asserted…

Let $Q$ be a connected non-Dynkin quiver and let $q\in (\mathbb{C}^\times)^{Q_0}$. Write $\Lambda^q(Q)$ for the multiplicative preprojective algebra of $Q$ with parameter $q$. 2-Ca…

Let $\mathcal{Q}$ be an acyclic quiver, let $P_{\mathcal{Q}}$ be the relevant set of seeds, and for each $\sigma\in P_{\mathcal{Q}}$ let $\Gamma_{\sigma}$ be the set of isotopy cla…

Cospectral cluster quiver mutation conjecture. The quivers $Q',Q”\in\operatorname{Mut}(Q)$ are cospectral if and only if there exists a cluster quiver $R\in\operatorname{Mut}(Q)$ s…

Cospectral cluster quiver connectivity conjecture. Cospectral cluster quivers in $\operatorname{Mut}(Q)$ form a finite connected subgraph of the exchange graph, and one of them can…

Assume the notation above, and for each $bb\in\Lambda^+(n,r)$ let $\barbb$ be the dominant weight for $SL_n$ determined by $bb$. Let $w_{\barbb}$ be the unique element $x$ of the a…