Dominance phenomena: mutation, scattering and cluster algebras

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.