We introduce a collection of injective homomorphisms among the quantum Grothendieck rings of finite-dimensional modules over the quantum loop algebras of type \(\textrm{A}\) . In the classical limit, it specializes to the inflation among the usual Grothendieck rings studied by Brito–Chari (J Reine Angew Math 804:221–262, 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 \(\textrm{A}_\infty \) .
