26 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…

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 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 $Q$ 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…

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

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 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 $d$…

Let $Q$ be a quiver, let $\alpha\geq1$, and let $\mathbf{r}=\underline{1}$ be the rank-one vector. Let $A_{(Q,\alpha),\mathbf{r}}$ be the corresponding Kac polynomial. Positivity c…

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 quiver, let $\mathbf{r}\in\mathbb{Z}_{\geq0}^{Q_0}$ be a rank vector, and let $\alpha\geq1$. Higher-depth positivity conjecture. … The conjecture extends positivity fr…

Let $Q$ be a quiver, let $boldsymbol{1}in author?$. For depth $alphageq1$, let $A{Q,boldsymbol{1},alpha}$ count locally free, absolutely indecomposable representations of rank…

Let $g$ be a non-negative integer and consider the $g$-loop quiver. Rodriguez-Villegas introduced a refinement of its Kac polynomials indexed by partitions; denote these refined Ka…

Let $Q$ be a quiver, let ${\mathcal B}_{\b0}$ be the slope-zero algebra graded by dimension vectors, and write … where $A_{Q,\boldsymbol{n}}(t)$ is the Kac polynomial. Define … and…

Continuity conjecture. The curves $G_{Q_{u\underline{m}},\operatorname{\mathbf{d}}}$ converge to a continuous curve as $u\to\infty$.

Zero-valuation conjecture. The valuation of $A_{Q_{\underline{n}},\operatorname{\mathbf{d}}}^{\flat}(q)$ is $0$.

Stabilization conjecture. The sequence

Let $g$ and $r$ be integers, let $C_{g,r}^{\mathrm{abs}}$ be the absolutely cuspidal polynomial for curves of genus $g$ and rank $r$, let $GSp(2g,\overline{\mathbb{Q}_l})$ be the g…

Let $\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\ne 0}\mathfrak{g}_\alpha$ be the Borcherds–Kac–Moody Lie algebra associated with a general Nakajima variety, with finite-dimen…

Okounkov's character conjecture. The character of $\mathbf{g}_Q$ is, up to a global grading shift,

For a quiver $Q$ and dimension vector $\mathbf{v}\in\mathbb{N}^I$, let $A_{\mathbf{v}}^{\flat}(t)\in\mathbb{Z}[t]$ be the polynomial counting absolutely indecomposable nilpotent or…

Let $\Gamma$ be a quiver with vertex set $\Gamma_0$, let $\R \Gamma_0$ be the associated real vector space, let $\N \Gamma_0$ be the monoid of dimension vectors, and let…

Positivity conjecture. For any $\underline{p}$ and any $\alpha$,

Motivic DT-invariant conjecture. The invariant $\Omega(\gamma)$ is essentially the same as the Kac polynomial $a_\gamma({\mathbb L})$; equivalently, it is related to the motivic DT…

Let $A_{g,r,d}(0)$ be the constant term of the polynomial counting absolutely indecomposable vector bundles of rank $r$ and degree $d$, and let $\mathfrak{h}^{\mathrm{sph}}$ be the…