Vol. 305, No. 2, 2020

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Dominance order and monoidal categorification of cluster algebras

Elie Casbi

Vol. 305 (2020), No. 2, 473–537
DOI: 10.2140/pjm.2020.305.473
Abstract

We study a compatibility relationship between Qin’s dominance order on a cluster algebra $\mathsc{𝒜}$ and partial orderings arising from classifications of simple objects in a monoidal categorification $\mathsc{𝒞}$ of $\mathsc{𝒜}$. Our motivating example is Hernandez and Leclerc’s monoidal categorification using representations of quantum affine algebras. In the framework of Kang, Kashiwara, Kim and 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 ${A}_{n}$ quiver Hecke algebra using Kleshchev and Ram’s classification of irreducible finite-dimensional representations.

Keywords
quiver Hecke algebras, categorification of cluster algebras, dominance order
Mathematical Subject Classification 2010
Primary: 16G99, 16T30