28 problems

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 $\Gamma$ 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 $Q$ 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 $X_{\mathrm{res}}$ 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 $S$ 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…

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…

Fix a Serre subcategory $\mathcal{S}$, a stability condition $\upzeta$, and a slope $\uptheta$. Let $\operatorname{\mathcal{S}ta}^{G,\upzeta}_{\Pi_Q,\uptheta}$ be the direct sum of…

Let $W$ be a Weyl group, let $w\in W$, and let $\mathcal{F}(w)$ be its torsion-free class. A brick is an object with division endomorphism ring. Brick occurrence characterization.…

Let $W$ be a Weyl group, let $w\in W$, and let $B_1,\dots,B_l$ be a brick sequence of the torsion-free class $\mathcal{F}(w)$. Suppose that … for $1\leq i<m<j\leq l$. Brick-sequenc…

Strong free product property conjecture. The map $\sigma'$ is a linear isomorphism, equivalently $(L_Q,r,\sigma,kQ_0[t,(t+q)^{-1}])$ satisfies the strong free product property.

Let $Q$ be a quiver, let $W$ be its Weyl group, let $\Pi$ be the associated preprojective algebra, and for $w\in W$ let $I_w$ be the corresponding ideal. For a full subcategory…

Suppose that $Q$ is an affine quiver, let $W$ be its Weyl group, let $\Pi$ be the associated preprojective algebra, and let $I_w$ be the ideal associated to $w\in W$. Let…

Let $Q$ be a quiver, let $W$ be its Weyl group, let $\Pi$ be the associated preprojective algebra, and for $w\in W$ let $I_w$ be the corresponding ideal. Write…

Let $w$ be an element of the Weyl group, let ${\mathcal C}_w$ be the associated category, and let ${\mathcal B}(w)$ be 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 $X=B_K^-d4f G$ be the partial flag variety, let $X_{>0}$ be its totally positive part, and let $N_K$ be the embedded open cell. Let … be a basic complete rigid $\Lambda$-module…