Dominance order and monoidal categorification of cluster algebras

TL;DR

This paper explores the relationship between dominance orders in cluster algebras and partial orders from monoidal categorifications, focusing on quantum affine and quiver Hecke algebra representations.

Contribution

It establishes a compatibility framework between dominance order and categorification partial orders in specific algebraic contexts.

Findings

Compatibility between dominance order and categorification partial orders.

Application to quantum affine algebra representations.

Application to quiver Hecke algebra representations.

Abstract

We study a compatibility relationship between Qin's dominance order on a cluster algebra $\mathcal{A}$ and partial orderings arising from classifications of simple objects in a monoidal categorification $\mathcal{C}$ of $\mathcal{A}$. Our motivating example is Hernandez-Leclerc's monoidal categorification using representations of quantum affine algebras. In the framework of Kang-Kashiwara-Kim-Oh's monoidal categorification via representations of quiver Hecke algebras, we focus on the case of the category $R-gmod$ for a symmetric finite type $A$ quiver Hecke algebra using Kleshchev-Ram's classification of irreducible finite dimensional representations.