On exchange matrices from string diagrams

TL;DR

This paper characterizes the class of skew-symmetrizable matrices derived from string diagrams, showing they belong to a minimal mutation-closed class, with implications for Lie theory and cluster algebra structures.

Contribution

It proves that matrices from string diagrams are in the smallest class closed under mutations and source-sink extensions, linking them to Lie theory and cluster algebra applications.

Findings

Matrices from string diagrams are in class a0a0

Applicable to exchange matrices from various Lie-theoretic structures

Provides explanations for properties like reddening sequences

Abstract

Inspired by Fock-Goncharov's amalgamation procedure \cite{Fock-Goncharov-2006}, Shen-Weng introduced string diagrams in \cite{Shen-Weng-2021}, which are very useful to describe many interesting skew-symmetrizable matrices closely related with Lie theory. In this paper, we prove that the skew-symmetrizable matrices from string diagrams are in the smallest class $\mathcal P^\prime$ of skew-symmetrizable matrices containing the $1\times 1$ zero matrix and closed under mutations and source-sink extensions. This result applies to the exchange matrices of cluster algebras from double Bruhat cells, unipotent cells, double Bott-Samelson cells and so on. Our main result can be used to explain why many skew-symmetrizable matrices from Lie theory have reddening sequences. It can be also used to prove some interesting results regarding non-degenerate potentials on many quivers from Lie theory.