59 problems

Let $\mathfrak g$ be a simple Lie algebra of simply-laced type $\mathbb{A},\mathbb{D},\mathbb{E}$, and let ${\mathcal B}_q^*$ be the basis dual to the canonical basis of…

Let $\Lambda_d$ be the relevant variety of $d$-dimensional modules over the preprojective algebra $\Lambda$, and let $Z$ be an irreducible component of $\Lambda_d$. Let $b_Z$ and…

Let $\Lambda$ be the preprojective algebra and let ${\mathcal C}$ be a cluster torsion class of $\operatorname{nil}(\Lambda)$. For a $\Lambda$-module $T$ in ${\mathcal C}$, conside…

Strong cluster-map conjecture. For every $w\in W$, the coordinate ring $\mathbb{C}[U^w]$ is a cluster algebra modelled by the strong cluster map $\varphi_w$ for the standard compon…

Connectedness problem. For every $w\in W$, the cluster tilting graph of $\operatorname{Sub}\nolimits\Lambda/I_w$ is connected.

Let $G$ be of simply-laced type $A$, $D$, or $E$, let $J$ be the chosen subset of simple roots, let $K$ be its complementary subset, and let $N_K$ be the corresponding unipotent su…

Let $G$ be a simple simply connected complex algebraic group of simply-laced type $A$, $D$, or $E$, and let $P$ be a parabolic subgroup with unipotent radical $N_P$. Let…

Let $\Gamma$ be a connected graph with adjacency matrix $A$, and let $P({\bf \Gamma})$ be the associated preprojective algebra, constructed from a choice of field $k$ and a modulat…

Let $\Lambda$ be the selfinjective preprojective algebra under consideration, let $T$ be a basic maximal rigid $\Lambda$-module, and let $\operatorname{End}_\Lambda(T)$ be its endo…

Let ${\mathcal T}_\Lambda$ be the graph whose vertices are the basic maximal rigid $\Lambda$-modules, with edges given by mutation. Let ${\mathcal T}_\Lambda^\circ$ be the connecte…

Let $\Lambda$ be the preprojective algebra of type $\mathbb{A}_4$. Let $Z_i\subset\Lambda_{{\mathbf V}_i}$ be irreducible components that are closures of the isomorphism classes of…

Let $Q$ be a Dynkin quiver of type $\mathbb{A}_n$, $\mathbb{D}_n$ or $\mathbb{E}_{6,7,8}$, let $\Lambda$ be its preprojective algebra, and let $R^+$ be the set of positive roots of…

Let $Q$ be a Dynkin quiver, let $\Lambda$ be its preprojective algebra, and let $C$ and $D$ be irreducible components of nilpotent varieties $\Lambda({\bf d})$. The quantity…

Let $X_{\mathrm{res}}$ be the resolution of the $A_n$ surface singularity, let $D_r$ be the indicated divisor, and let $\sigma$ be a stop. Let $s$ be a formal parameter, with coeff…

Let $X_{\mathrm{res}}$ be the resolution of the $A_n$ surface singularity, let $D_r$ be the indicated divisor, and let $s$ be a formal parameter with coefficient field…

Let $X_{\mathrm{Mil}}\smallsetminus D_m$ be the complement of a divisor in the $A_n$ Milnor fibre, and let the corresponding wrapped Fukaya and coherent-sheaf categories be the cat…

Let $S$ and $S'$ be sheets, regarded as objects of $\operatorname{Rep}(\Lambda_{\mathbb{I}})$. Elementary morphisms are those constructed in Proposition, and multi-elementary morph…

A sheet module is the image of a decorous submodule in a decorous quotient of a representable projective, as defined in the paper. The elementary and multi-elementary morphisms are…

Generic-torus conjecture. Theorems A and B of the cited reference should also hold under this weaker condition.

Let $\beta$ be a positive real root, and let $B_1$ and $B_2$ be shard modules of dimension $\beta$. Homomorphism conjecture. Then … This conjecture would imply that the dimension v…

Module-structure conjecture. For every preprojective algebra module $M$ of dimension vector $\nu$, the direct sum

Acyclicity conjecture. The theorem still holds if the acyclic assumption on $\Lambda$ is dropped.

Let $V^{(i)}_{k,q^p}$ denote the spectral shift of the Kirillov–Reshetikhin module $V^{(i)}_k$, and let $q^pK^{(i)}_k$ denote the corresponding grading shift of its generic kernel.…

Let $\mathfrak{O}(V^{(i)}_k,V^{(j)}_l)$ be the divisor of the normalized $R$-matrix denominator, and let $\dim_{q^{-1}}\mathop{\mathrm{ext}}^1_{\widetilde{\Pi}}$ denote the graded…

Let $\mathfrak{O}(M,N)$ be the divisor of the denominator of the normalized $R$-matrix between simple modules $M$ and $N$, and let…