On Frenkel-Mukhin algorithm for q-character of quantum affine algebras

TL;DR

This paper examines the Frenkel-Mukhin algorithm for computing q-characters of quantum affine algebra representations, demonstrating its limitations through a specific counterexample where it fails to produce the correct q-character.

Contribution

The paper provides a critical analysis of the Frenkel-Mukhin algorithm, highlighting its failure in certain cases and contributing to a better understanding of its applicability.

Findings

The Frenkel-Mukhin algorithm does not always generate the correct q-character.

A specific counterexample where the algorithm fails is presented.

The limitations of the algorithm are clarified through this example.

Abstract

The q-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the q-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the q-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the q-character.