Permutation Weights for Affine Lie Algebras

TL;DR

This paper extends the concept of permutation weights from finite to affine Lie algebras, enabling explicit classification of weights in affine Weyl orbits and simplifying calculations of weight multiplicities.

Contribution

It introduces a method to define permutation weights for affine Lie algebras and provides explicit rules for $A_N^{(1)}$, improving the calculation of weight multiplicities.

Findings

Permutation weights can be explicitly classified for affine Lie algebras.

The method simplifies the calculation of weight multiplicities using Weyl-Kac formulas.

Complete sets of weights at each depth M are determined, aiding representation theory applications.

Abstract

We show that permutation weights, which are previously introduced for finite Lie algebras, can be appropriately defined also for affine Lie algebras. This allows us to classify all the weights of an affine Weyl orbit explicitly. Let $\Lambda$ be a dominant weight of an affine Lie algebra $ G_N^{(r)} $ for r=1,2,3. At each and every order M of weight depths, the set $\wp_M(\Lambda)$ of permutation weights is formed out of a finite number of dominant weights of the finite Lie algebra $G_N$. In case of $A_N^{(1)}$ algebras, we give the rules to determine the elements of a $\wp_M(\Lambda)$ completely. As being a positive test of our proposal, we consider the problem of calculating weight multiplicities for affine Lie algebras and hence our discussions are based on explicit computations of Weyl-Kac character formula. It is known that weight multiplicities are provided by string functions which are defined to be formal power series $\sum_{M=0}^\infty C(M) q^M$ where the order M specifies the depth of weights contributing to C(M). In the conventional calculational schemes which are based on Kac-Peterson form of affine Weyl groups, Weyl-Kac formula includes a sum over a part of the whole root lattice and hence it is seen that the roots of the same length contribute in general to C(M) for several values of M. On the contrary, we will determine, for any fixed value of M, the complete set of weights having depth M and contributing only to C(M). For applications of Weyl-Kac formula, one must also know the signatures which correspond to weights within the Weyl orbits of strictly dominant weights. This is given by the aid of a properly defined index. Another emphasis is that the way of discussion adopted here gives us a possibility for extensions to other infinite dimensional Lie algebras beyond affine Lie algebras.