Mutation-acyclic quivers are totally proper

TL;DR

This work proves that mutation-acyclic quivers are totally proper within the COQ framework, enabling a robust mutation-invariant via unipotent companions and their cosquares. The authors construct canonical cyclic orderings and unipotent companions for acyclic seeds, show mutation compatibility through matrix congruences, and derive invariants such as the Alexander polynomial and Markov invariant that persist under mutation. These results yield new criteria to certify mutation-acyclicity, finite-classifications for acyclic quivers with fixed invariants, and practical tools for detecting non-acyclicity, with broad implications for cluster-algebra combinatorics. The approach synthesizes COQ structure, quasi-Cartan companions, and linear-algebra techniques, and is complemented by independent proofs in the literature.

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.

Figures (14)

• Figure 1: An (unlabeled) triangular grid quiver with $4$ vertices on each side. Readers may recognize it as the default quiver in B. Keller's mutation applet KellerApp.
• Figure 2:
• Figure 3: The black circle vertices $v_i$ support an acyclic quiver $Q$. With the additional square (frozen) vertices $v_i'$, we get the principal framing $\widehat{Q}$.
• Figure 4: Two quivers $Q$ and $Q'$. The cycle $(v_1 - v_2 - v_3 - v_4 - v_5 - v_6 - v_1)$ in $K_Q$ is chordless and forward-oriented, while in $K_{Q'}$ it is not chordless. The chordless cycles $(v_1 - v_2 - v_3 - v_6 - v_1)$ and $(v_3 - v_4 - v_5 - v_6 - v_3)$ in $K_{Q'}$ are not oriented.
• Figure 5: The $4!/4 = 6$ distinct COQs on the $4$-vertex acyclic quiver with vertices and arrows $v_1 \stackrel{}{\rightarrow} v_2 \stackrel{2}{\rightarrow} v_4$, $v_1 \stackrel{}{\rightarrow} v_3 \stackrel {}{\rightarrow} v_4$, and $v_1 \stackrel 3 {\rightarrow} v_4$.

Theorems & Definitions (107)

• Theorem 1.1
• Remark 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Definition 2.7
• Definition 2.8