Minimal affinizations as projective objects

TL;DR

This paper proves that certain specialized quantum affine modules are projective, providing a uniform character formula, and explores conjectures related to minimal affinization and q-characters.

Contribution

It establishes the projectivity of specialized Kirillov-Reshetikhin modules and proposes a uniform character formula, extending to conjectures on minimal affinization and Jacobi-Trudi determinants.

Findings

Specialization at q=1 yields projective modules in classical types.

Provides a uniform character formula for Kirillov-Reshetikhin modules.

Verifies the Nakai-Nakanishi conjecture in specific cases.

Abstract

We prove that the specialization to q=1 of a Kirillov-Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov-Reshetikhin modules. We conjecture that these results holds for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q-characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi-Trudi determinants.