17 problems

For even $n$, let $\omega_0$ be the cluster DT-transformation corresponding to the longest element of the birational Weyl group. A maximal green sequence is a mutation sequence wit…

Red-path realization conjecture. Every maximal green sequence in $\mathcal{A}$ is induced by a red path in the wall and chamber structure of $\mathcal{A}$.

Let $Q$ and $Q'$ be finite quivers. For sequences ${\bf v}$ and ${\bf w}$ of vertices of $Q$ and $Q'$, respectively, let ${\bf v}\boxtimes{\bf w}$ be the corresponding sequence in…

Let $R$ be a symmetric quantum CGL extension, and consider localized quotients $(R/I)[E^{-1}]$, where $I$ is a torus-invariant prime ideal and $E$ is the collection of nonzero homo…

Let $R$ be the coordinate ring of a unipotent cell of a symmetric Kac–Moody algebra, equipped with the cluster algebra structure and certain seeds constructed by Geiß–Leclerc–Schrö…

Goddard–Wemyss conjecture. The partial order on $\widetilde{\mathsf{MG}}(\Lambda)$ considered in the paper coincides with the other two partial orders on the same underlying set.

Jordan--Hölder criterion. The maximal green sequences … have the same set of bricks if and only if they are equivalent.

Let $Q$ be a quiver with no oriented cycles. A maximal green sequence (MGS) is a sequence of green mutations starting with an extended exchange matrix $\tilde{B}={B\over I}$ and en…

Let $\Gamma_n$ be the $n$-cycle quiver, and let $\Phi^1$ denote the corresponding root set, with $|\Phi^1|=\binom{n+1}{2}-1$. A maximal green sequence for $\Gamma_n$ has a well-def…

Let $Q$ be a quiver, and consider all maximal green sequences for $Q$. Their lengths are integers. No Gap Conjecture. The set of lengths of all possible maximal green sequences for…

Igusa–Todorov's conjecture. There exists a tilted algebra $B$ such that $\Lambda\cong B\ltimes \operatorname{Ext}^2_B(DB,B)$ and the number of isomorphism classes of indecomposable…

Igusa–Todorov's conjecture. (a) The ordered $B$-modules form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maxima…

Berenstein–Geiss–Muller conjecture. The following are equivalent:

Let $\Lambda$ be a cluster-tilted algebra of finite type, and let $C$ be one of the associated tilted algebras. The Auslander–Reiten quiver of $\Lambda$ is used to order the $C$-mo…

Let $Q$ and $Q'$ be quivers such that $\Psi(Q)$ and $\Psi(Q')$ are finite and non-empty, and let $Q\oplus Q'$ denote their direct sum. Direct-sum finiteness conjecture. Then…

Let $Q$ be a minimal mutation-infinite quiver. Let $\widehat{\Psi}(Q)$ denote the subgraph associated with quivers admitting maximal green sequences, as in Theorem. Minimal mutatio…

Let $Q$ be a cluster quiver, meaning a finite connected quiver without loops or oriented cycles. A maximal green sequence is a green mutation sequence that cannot be extended by an…