Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras

TL;DR

This paper demonstrates that the Grothendieck ring of a subcategory of representations of Borel subalgebras in quantum affine algebras has a cluster algebra structure, linking representation theory with combinatorial mutation relations.

Contribution

It establishes a cluster algebra structure on the Grothendieck ring of a subcategory of representations of Borel subalgebras in quantum affine algebras, with initial seeds from prefundamental representations.

Findings

Grothendieck ring has a cluster algebra structure of infinite rank

Baxter relations interpreted as Fomin-Zelevinsky mutation relations

Initial seed consists of prefundamental representations

Abstract

Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the structure of a cluster algebra of infinite rank, with an initial seed consisting of prefundamental representations. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin-Zelevinsky mutation relations.