Freezing operators in representation theory of quantum loop algebras

TL;DR

This paper proves a conjecture on simple $(q,t)$-characters for untwisted quantum loop algebras of classical types, introducing a bootstrapping method based on freezing operators to relate different types.

Contribution

It establishes the Hernandez conjecture for type C and a bijective correspondence between representations of different quantum loop algebras, using a novel bootstrapping approach.

Findings

Proved Hernandez conjecture for type C.

Established dimension-preserving bijection for types D.

Developed a bootstrapping method for $q$ and $(q,t)$-characters.

Abstract

We prove the Hernandez conjecture on the simple $(q,t)$-characters (an analog of the Kazhdan--Lusztig conjecture) for untwisted quantum loop algebras of classical type. This result is new in type $\mathrm{C}$. We also prove that the folding homomorphism, introduced by Hernandez, gives a dimension-preserving bijective correspondence between the finite-dimensional simple representations (in a skeletal subcategory) of untwisted quantum loop algebras of classical simply-laced type and those of the corresponding doubly-twisted quantum loop algebras. This result is new in type $\mathrm{D}$. In our approach, we develop a bootstrapping method for $q$ and $(q,t)$-characters, based on the freezing operator previously introduced in the context of cluster algebras by the second named author. This method allows us to reduce statements for general simple representations in all classical types to corresponding results on core subcategories in a uniform manner.