53 problems

Let $(Q,\tau)$ be a general $\imath$quiver, allowing loops, and let the quantum symmetric pair and Hall algebra constructions associated with $(Q,\tau)$ be defined as in Theorem. H…

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…

Let $\Pi$ be the indexing set of partitions, let $|\emptyset\rangle$ be the empty-partition vector in the $q$-deformed Fock space, let $b_\lambda$ denote its canonical basis elemen…

Let $\triangle(n)$ be the cyclic quiver, let $\mathcal Z=\mathbb Z[\upsilon,\upsilon^{-1}]$, and let $\boldsymbol{\mathfrak D_{\!\vartriangle\!}}(n)$ be its double Ringel–Hall alge…

Let $\mathbf{s}_l \in \mathbb{Z}^l(s)$, and let $\mathbf{b}_{\lambda_l,\mathbf{s}_l}$ be the canonical-basis element indexed by $\lambda_l$ in the crystal component…

Let $\mathcal{C}$ be a hereditary finitary category. A derived functor is called friendly when it sends every object into two adjacent cohomological degrees. Automorphism conjectur…

Calabi–Yau Drinfeld-double conjecture. The assignment

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. An exact functor $F:D^b(\mathcal{C}_1)\to D^b(\mathcal{C}_2)$ is friendly…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. Their reduced Drinfeld doubles are denoted by…

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories, and let $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_i}$ denote their r…

Positivity conjecture. For any object $M$ and any exceptional object $K$ of ${\mathcal C}$, the integer $r(M,K)$ is nonnegative.

Let $n$ be the fixed positive integer in the construction, let $\mathbf{K}\simeq \Gamma\otimes\mathbb{C}[v,v^{-1}]$, and let $\langle\;,\;\rangle_G$ denote Green's scalar product.…

Let $\mathbf{U}^+$ be the $\mathbb{C}_v$-subalgebra of $\mathbf{U}_v(L\mathfrak{gl}_p)$ generated by $\mathbf{U}_0^+$ and $h_{0,l}$ for $l>0$, where $\mathbf{U}_0^+$ is generated b…

Positivity conjecture. For every $b\in\mathbf{C}^\imath$, all coefficients $f_c(v)$, $g_{b',c}(v)$ and $h_{c,c'}(v)$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$. In particular…

Positivity conjecture. For every $b\in\mathbf{C}^\imath$, all coefficients $f_c(v)$, $g_{b',c}(v)$ and $h_{c,c'}(v)$ belong to $\mathbb{N}[v^{\frac12},v^{-\frac12}]$. In particular…

Let $Q$ be a quiver, let … is a free $R$-module. The conjecture is used to obtain the integral-form isomorphism with the positive half of the Maulik–Okounkov quantum loop group; th…

Let $Q=(I,E)$ be a quiver. The cohomological Hall algebra and the $K$-theoretic Hall algebra associated with the preprojective or nilpotent quiver moduli have doubles, while the co…

Let $Q$ be the Jordan quiver, let ${\mathcal X}_a$ be the associated semi-nilpotent commuting stack, and suppose that there is a functor … from the Karoubi-completed trace of the a…

Let $C\in M_{\mathbb{I}\times\mathbb{I}}(\mathbb{Z})$ be a generalized Cartan matrix of finite type, let $W(C)$ be its Weyl group, and let $\mathbf{i}=(i_1i_2\cdots i_r)$ be a word…

Let $n\in\mathbb{N}$, let $(m,n)\in\mathbb{Z}\times\mathbb{N}$ be coprime, and let $d\geq 1$. For the equivariant derived category…

Subquotient conjecture. The explicit embedding for quantum groups in Theorem is a section of a subquotient. This conjecture is motivated by the corresponding result for Hall algebr…

Let $Q$ be an acyclic quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq 0}^{Q_0}$. For each $\mathbf{e}\leq\mathbf{d}$, let $[Q,\mathbf{e}]$ denote the relevant commuting-pair count in…

Let $g\geq 2$, let $\Sigma_g$ be a genus-$g$ surface, and let $p\colon \mathfrak{M}_{g,r}^{\operatorname{Betti}}\to \mathcal{M}_{g,r}^{\operatorname{Betti}}$ be the semisimplificat…

Let $(Q,\tau)$ be an acyclic $\imath$quiver, let $W_{\tau}$ be the restricted Weyl group, and let ${\rm Br}(W_{\tau})$ be the braid group generated by elements $t_i$ for…

Let $P$ be a preprojective representation and $I$ a preinjective representation. Let $\lambda$ be a partition, let $I_{\lambda}(x)$ denote the corresponding regular representation…