Dominance phenomena: mutation, scattering and cluster algebras

TL;DR

This paper investigates the dominance phenomena in cluster algebras, exploring how dominance relations between exchange matrices influence structures like mutation fans, scattering fans, and cluster algebra homomorphisms.

Contribution

The paper provides theorems demonstrating instances of dominance phenomena across various structures in cluster algebra theory.

Findings

Mutation-linear structures often preserve dominance relations.

Mutation fans and scattering fans are frequently refined under dominance.

Injective homomorphisms between cluster algebras often exist under dominance.

Abstract

An exchange matrix $B$ dominates an exchange matrix $B'$ if the signs of corresponding entries weakly agree, with the entry of $B$ always having weakly greater absolute value. When $B$ dominates $B'$, interesting things happen in many cases (but not always): the identity map between the associated mutation-linear structures is often mutation-linear; the mutation fan for $B$ often refines the mutation fan for $B'$; the scattering (diagram) fan for $B$ often refines the scattering fan for $B'$; and there is often an injective homomorphism from the principal-coefficients cluster algebra for $B'$ to the principal-coefficients cluster algebra for $B$, preserving $\mathbf{g}$-vectors and sending the set of cluster variables for $B'$ (or an analogous larger set) into the set of cluster variables for $B$ (or an analogous larger set). The scope of the description "often" is not the same in all four contexts and is not settled in any of them. In this paper, we prove theorems that provide examples of these dominance phenomena.