On multiplicity problems for finite-dimensional representations of hyper loop algebras

TL;DR

This paper investigates multiplicity issues in finite-dimensional representations of hyper loop algebras over non-algebraically closed fields, providing formulas for characters, multiplicities, and coefficients.

Contribution

It introduces explicit formulas for l-characters, Jordan-Holder multiplicities, and Clebsch-Gordan coefficients in this context, advancing understanding of representation theory for hyper loop algebras.

Findings

Formulas for l-characters of simple objects

Calculations of Jordan-Holder multiplicities of Weyl modules

Expressions for Clebsch-Gordan coefficients

Abstract

Given a hyper loop algebra over a non-algebraically closed field, we address multiplicity problems in the underlying abelian tensor category of finite-dimensional representations. Namely, we give formulas for the l-characters of the simple objects, the Jordan-Holder multiplicities of the Weyl modules, and the Clebsch-Gordan coefficients.