Inflations among quantum Grothendieck rings of type A
Ryo Fujita
TL;DR
The paper constructs a family of injective quantum inflations between the quantum Grothendieck rings $K_t(\mathscr{C}_n)$ of type $A$ across ranks, parameterized by height functions and monotone inclusions, and proves these maps preserve the canonical $(q,t)$-character bases. In the classical limit $t\to1$, these inflations recover Brito--Chari’s inflation, providing a quantum analog and validating their conjecture for simple classes; the construction relies on the bosonic extension presentation of the HL framework and uses explicit embeddings between the corresponding quantum tori. Furthermore, the authors provide a categorification of these inflations via quiver Hecke algebras of type $A_\infty$, establishing graded exact monoidal functors $F_\nu$ between localization categories that reflect the inflation at the level of simple objects. The results unify and extend connectivity between different Dynkin types, offer a robust method to transport canonical-basis information across ranks, and open a pathway to categorified realizations through infinite-type quiver Hecke algebras. Collectively, this work advances understanding of quantum affine representations, canonical bases, and categorification in 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 $\mathrm{A}_\infty$.