Inflations among quantum Grothendieck rings 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 ${\mathrm{A}}_{\infty }$.

arXiv commentv3: final version, 19 pages

References in corpus (6)

• Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras
• Cluster algebras and quantum affine algebras
• Quantum Grothendieck rings and derived Hall algebras
• Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras
• Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
• Isomorphisms among quantum Grothendieck rings and propagation of positivity