Exchange matrices of I-boxes

TL;DR

This paper investigates the exchange matrices associated with admissible chains of i-boxes, establishing their role as exchange matrices in cluster algebra seeds within certain categorifications, and provides explicit constructions of mutation relations.

Contribution

It introduces a general framework for defining exchange matrices from maximal commuting families of i-boxes and proves their correspondence to cluster algebra seeds in specific categories.

Findings

The exchange matrix matches the seed in cluster algebra structures.

Explicit short exact sequences represent mutation relations.

The framework applies to modules over quantum affine and quiver Hecke algebras.

Abstract

Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations.