Mutation classes of skew-symmetrizable 3x3 matrices

TL;DR

This paper classifies mutation classes of 3x3 skew-symmetrizable matrices, introduces a new numerical invariant, and generalizes previous results in matrix mutation theory.

Contribution

It provides a complete classification of mutation classes for 3x3 matrices and introduces a novel invariant applicable to matrices of any size.

Findings

Complete set of representatives for 3x3 mutation classes

New numerical invariant for matrix mutations

Generalization of previous classification results

Abstract

In this paper, we determine representatives for the mutation classes of skew-symmetrizable 3x3 matrices and associated graphs using a natural minimality condition, generalizing and strengthening results of Beineke-Brustle-Hille and Felikson-Shapiro-Tumarkin. Furthermore, we obtain a new numerical invariant for the mutation operation on skew-symmetrizable matrices of arbitrary size.