113 problems

Let $Q=(\bullet\rightrightarrows\bullet)$ be the self-dual $\tilde{A}_1$ quiver with its two vertices exchanged by the contravariant involution, and write $\tilde{A}_1^{u,v}$ for t…

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 $C=\{u_1,\ldots,u_n\}$ be a cluster in a cluster algebra of simply-laced type and rank $n$, with associated quiver $Q_C$ and category $\operatorname{Mod} Q_C$ defined by the sh…

Kac's conjectures. For every root $d\in\mathbf{N}I$, there exists a polynomial $P_d(t)$ with nonnegative integer coefficients such that

Let $Q$ be a quiver whose underlying unoriented graph is a disjoint union of Dynkin diagrams of type $A$, $D$, or $E$. A representation of $Q$ is called stable with respect to a we…

Path-category realization conjecture. For every spectroid $\mathcal{C}$, there exists a choice of $(\mathbb{X},\Gamma{/}{\sim})$ such that $\mathcal{C}$ is the $\Bbbk$-linear path…

Let $L$ be a link with $|L|$ components, and let $P^{\bigwedge^j}_L(q,a)$ denote its $\bigwedge^j$-colored HOMFLY-PT polynomial. Define … Let $P_Q(\overline{x})$ be the generating…

Let $Q$ be a quiver of symmetric Kac–Moody type. Let $\mathrm{KHA}^{\mathbb{C}^*}_{\Pi_Q}$ denote the preprojective K-theoretic Hall algebra of the preprojective algebra $\Pi_Q$, f…

Elgin–Reading–Stella's conjectures. For $i,j>0$, the coefficient $\tau(i,j)$ satisfies all of the following: it is a polynomial in $b$, $c$, and $g$; this polynomial has $g$ as a f…

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 $Q$ be a quiver, and for each dimension vector $\alpha$ let $\mathcal{V}_{\alpha}$ be its representation variety, with $G=\prod_i \operatorname{SL}_{\alpha(x_i)}(\mathbb{C})$ ac…

Let $Q$ be a quiver, let $d$ be an indivisible dimension vector such that … where $(d,e_i)$ is the relevant bilinear form and $Q_0$ is the vertex set. Let $\mathcal M_d^\chi(Q)$ be…

Let $Q$ be the quiver, let $d$ be the dimension vector, and let $\delta$ be as in Theorem; write $\operatorname{supp}\delta$ for its support, let $d_i$ denote the components of $d$…

Proper stratification conjecture. The preorder of is properly stratifying for $A_0(\vecn)$ in arbitrary characteristic.

Let $M$ and $N$ be dominant monomials such that $L(M)$ and $L(N)$ are real prime modules, and let $\mathfrak{e}(\mathbf{g}_M,\mathbf{g}_N)$ be their generic symmetrized $E$-invaria…

Let $\mathcal{A}$ be as above, let $M_1,M_2\in\mathcal{P}_\ell^+$ be dominant monomials such that $L(M_1)$ and $L(M_2)$ are real, and let $\mathbf{g}_{M_1},\mathbf{g}_{M_2}$ be the…

Let $\mathcal{A}$ be a cluster algebra with monoidal categorification $\mathscr{C}_\ell$ and additive categorification $(\mathcal{F},\widehat{T})$, let $\mathcal{C}$ be the stable…

Let $K>2$, and let $Y^{K}$ be the $C[Chbar_1,Chbar_2]$-linear algebra obtained by gluing the two copies of the double deformed current algebra $A^K$ constructed from observables on…

Closure-order conjecture. If

Fix $\alpha\geq1$, and let $\mathcal{A}_{\Pi_{(A_{2},\alpha)}}$ be the graded algebra associated with the preprojective construction for the quiver of type $A_{2}$ with multiplicit…

Let $Q$ be a quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq0}^{Q_{0}}\setminus\{0\}$ such that $(Q,\mathbf{d})$ has property (P). Let $\mu_{Q,\mathbf{d}}$ be the corresponding momen…

Let $Q$ be a quiver, let $\mathbf{r}\in\mathbb{Z}_{\geq0}^{Q_{0}}$ be a rank vector, and let $\alpha\geq1$. Let $A_{(Q,\alpha),\mathbf{r}}$ be the associated Kac polynomial for the…

Let $Q$ be a finite quiver without loops, let $E$ be a variety of representations of $Q$, and let $\mathcal{F}$ be an equivariant simple perverse sheaf on $E$. Let the singular sup…

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…