Permutation Weights for A_r^(1) Lie Algebras

TL;DR

This paper introduces a novel reformulation of the Weyl-Kac character formula for affine Lie algebras using permutation weights and Schur functions, providing a finite and structured approach to calculating characters.

Contribution

It presents a new formulation of the Weyl-Kac character formula in terms of permutation weights and Schur functions, simplifying computations for affine Lie algebras.

Findings

Finite permutation weights at each weight depth are identified.

Reformulation enables expression of affine signatures via an index based on fundamental weights.

Provides a structured method for character calculations in affine Lie algebras.

Abstract

When it is based on Kac-Peterson form of Affine Weyl Groups, Weyl-Kac character formula could be formulated in terms of Theta functions and a sum over finite Weyl groups. We, instead, give a reformulation in terms of Schur functions which are determined by the so-called Permutation Weights and there are only a finite number of permutation weights at each and every order of weight depths . Affine signatures are expressed in terms of an index which by definition is based on a decomposition of horizontal weights in terms of some Fundamental Weights.