Inflations among quantum Grothendieck rings of type A

TL;DR

This paper constructs injective homomorphisms between quantum Grothendieck rings of type A modules, confirming a conjecture and linking to classical limits and categorification via quiver Hecke algebras.

Contribution

It introduces new injective homomorphisms among quantum Grothendieck rings of type A, respecting canonical bases and categorified through quiver Hecke algebras.

Findings

Homomorphisms respect canonical bases of simple $(q,t)$-characters.

Specializes to classical inflation among Grothendieck rings.

Provides categorification via quiver Hecke algebras of type A_.

Abstract

We introduce a collection of injective homomorphisms among the quantum Grothendieck rings of finite-dimensional modules over the quantum loop algebras of type $\mathrm{A}$. In the classical limit, it specializes to the inflation among the usual Grothendieck rings studied by Brito-Chari [J. Reine Angew. Math. 804, 2023]. We show that our homomorphisms respect the canonical bases formed by the simple $(q,t)$-characters, which in particular verifies a conjecture of Brito-Chari in loc. cit. We also discuss a categorification of our homomorphisms using the quiver Hecke algebras of type $\mathrm{A}_\infty$.