Inflations among quantum Grothendieck rings of type A

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 $\mathrm{A}_\infty$.