226 problems

Let $Q=(I,E)$ be a quiver, let $(\alpha,\alpha)$ denote the associated symmetric bilinear form, and let $A_\alpha(q)\in\mathbb{Z}[q]$ be the Kac polynomial associated to a positive…

Reineke's conjecture. For every standard Coxeter element $c$, there exists $\gamma\in V^*$ such that $Z_{\rho,\gamma}$ is totally stable for $\mathcal{Q}^c$, meaning that every ind…

Let $Q$ be an arbitrary finite quiver and fix a dimension vector $\mathbf{d}\in\mathbf{N}^{Q_0}$. Let $\mathtt{a}_{Q,\mathbf{d}}(q)\in\mathbf{Z}[q]$ be Kac's polynomial, counting i…

Let $Q$ be a symmetric quiver, and let $\mathcal{A}_Q$ be the associated supercommutative quadratic algebra. An algebra is Koszul when its ground-field trivial module admits a line…

Let $Q$ be a quiver without oriented cycles, and let a root of $Q$ mean a root of its associated quadratic form. An indecomposable tree module is an indecomposable representation w…

Let $Q$ be the quiver under consideration, let $\mathfrak{g}$ be the Lie algebra spanned by the coefficients of the classical $R$-matrix, and let the graded multiplicity of a root…

Let $\mathcal{C}=(C_j)_{1\le j\le k}$ be a tuple of conjugacy classes of $\operatorname{GL}_n$, defining a star-shaped quiver $Q$, a deformation parameter $\mathbf{q}$, and a dimen…

Let $A_{Q,\alpha}$ and $B_{\mu_Q,\alpha}$ be the quiver-representation and moment-map counting polynomials whose normalised limits, when they exist, are rational fractions in $q$.…

Let $Q$ be a quiver and let $\mathbf{d}\in\mathbf{N}^{Q_0}$ be a dimension vector. The polynomial $C^{\mathrm{abs}}_{Q,\mathbf{d}}(q)$ is the absolutely cuspidal polynomial associa…

Lusztig's crystal-graph conjecture. The operation $*$ on irreducible components of the nilpotent varieties gives rise to Kashiwara's crystal graph.

Let $n$ be a positive integer. Consider the set of roots associated with non-self-intersecting curves in an $n$-punctured disc, and the set of real Schur roots of acyclic valued qu…

Valuation conjecture. Provided $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}(q)\neq 0$, its valuation is

Superfluity conjecture. In the proposition above, condition (3) is superfluous.

Let $Q$ be a quiver, let $\mathbf{d}$ be a dimension vector, let $C_{Q,\mathbf{d}}^{\mathrm{abs}}(t)$ be the absolutely cuspidal polynomial, and let $\mathbf{k}$ be a finite field.…

Let $Q$ be a quiver, let $\mathbf{d}$ be a dimension vector, and let $C_{Q,\mathbf{d}}^{\mathrm{abs}}(t)$ be the polynomial obtained from the graded Borcherds-algebra construction…

Let $Q$ be a valued acyclic quiver, and consider the associated quantum cluster algebra over a finite field. Rupel's quantum Caldero–Chapoton map assigns quantum Laurent polynomial…

Let $\Lambda_V$ be the representation variety under consideration, let $G_V$ act on it, and let $\Lambda_b$ be an irreducible component indexed by a canonical-basis element $b$. A…

Projective-cover and homological-dimension conjecture. The category $Qui_\lambda({\cal A})$ should contain the projective covers and have homological dimension at most $N$.

First Kac conjecture. For any $\alpha\in Q_+$,

Conjectural character formulas. The character formula known for the part

Let $G\subset \operatorname{GL}(n,\Bbbk)$ be a finite subgroup. For all $\theta$ in the rational character space, let $\mathcal{M}_\theta=Z{\operatorname{/\tslash}}_\theta H$ be th…

Let $T$ be a maximal torus in $G$, let $\mu$ be an orbit of representations of an arbitrary quiver of type $A$, and let $\overline{\mu}$ denote its closure. A K-theoretic lace diag…

Let $C$ be a cluster of a cluster algebra of Dynkin type $D$ or $E$, let $(\mathcal{Q}_C,I_C)$ be its associated quiver with relations, and let $\nu$ be the number of positive root…

Let $C$ be a cluster, let $(\mathcal{Q}_C,I_C)$ be its associated quiver with relations, and let $T$ be the corresponding tilting object in the cluster category $\mathcal{C}$. Writ…

Let $Q$ be a quiver, let $\beta$ be a Schur root, and let $C(Q,\beta)$ be the associated cone. Proposition … is minimal; equivalently, the facets of $C(Q,\beta)$ are in one-to-one…