Eventual sign coherence

TL;DR

This paper proves that in skew-symmetric cluster algebras, random mutation sequences almost surely lead to sign-coherent c-vectors, confirming a conjecture about asymptotic sign coherence.

Contribution

It establishes the asymptotic sign coherence conjecture for skew-symmetric cluster algebras of arbitrary rank, with high probability, and introduces a new class of brog quivers.

Findings

Random mutation sequences almost surely become sign-coherent.

The conjecture holds in full generality for many quiver families.

Results confirm the asymptotic sign coherence in broad cases.

Abstract

The sign coherence of $c$-vectors is one of the fundamental theorems of cluster algebras with principal coefficients. In 2019, Gekhtman and Nakanishi posed the asymptotic sign coherence conjecture for arbitrary cluster algebras of geometric type, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence. We prove that their conjecture holds almost always for skew-symmetric cluster algebras of arbitrary rank. That is, we prove that with probability $1$, the sequence of $c$-vectors obtained by random mutation of an arbitrary quiver eventually becomes sign-coherent. Our results also establish the conjecture in full generality for many families of quivers by studying a new class of brog quivers.