Mutation-acyclic quivers are totally proper

TL;DR

This paper proves that all mutation-acyclic quivers are totally proper, providing new criteria for mutation-acyclicity and extending the Markov invariant to all such quivers, with implications for classification.

Contribution

It establishes that mutation-acyclic quivers are totally proper, introducing new invariants and necessary conditions for mutation-acyclicity.

Findings

All mutation-acyclic quivers are totally proper.

A generalized Markov invariant applies to all mutation-acyclic quivers.

Finitely many acyclic quivers share the same Markov invariant.

Abstract

Totally proper quivers, introduced by S.~Fomin and the author arXiv:2406.03604, have many useful properties including powerful mutation invariants. We show that every mutation-acyclic quiver (i.e., a quiver that is mutation equivalent to an acyclic one) is totally proper. This yields new necessary conditions for a quiver to be mutation-acyclic. In particular, we show that a generalization of the Markov invariant for $3$-vertex quivers applies to all mutation-acyclic quivers. Only finitely many acyclic quivers share the same Markov invariant.