Fundamental Weights, Permutation Weights and Weyl Character Formula

TL;DR

This paper introduces permutation weights and their signatures in Weyl character formulas for finite Lie algebras, providing a new approach to calculating characters using fundamental weights and multiplicity rules.

Contribution

It presents a novel concept of permutation weights and their signatures, establishing a one-to-one correspondence with Weyl orbit elements, simplifying character calculations.

Findings

Permutation weights correspond uniquely to Weyl orbit elements.

Signatures of permutation weights are preserved under the correspondence.

Character calculations can be simplified using fundamental weights and Schur function rules.

Abstract

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(\Lambda^{++})$ of strictly dominant weights $\Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any $W(\Lambda^{++})$, there is a very peculiar subset $\wp(\Lambda^{++})$ for which we always have $$ dim\wp(\Lambda^{++})=dimW(G_N)/dimW(A_{N-1}) . $$ For any dominant weight $ \Lambda^+ $, the elements of $\wp(\Lambda^+)$ are called {\bf Permutation Weights}. It is shown that there is a one-to-one correspondence between elements of $\wp(\Lambda^{++})$ and $\wp(\rho)$ where $\rho$ is the Weyl vector of $G_N$. The concept of signature factor which enters in Weyl character formula can be relaxed in such a way that signatures are preserved under this one-to-one correspondence in the sense that corresponding permutation weights have the same signature. Once the permutation weights and their signatures are specified for a dominant $\Lambda^+$, calculation of the character $ChR(\Lambda^+)$ for irreducible representation $R(\Lambda^+)$ will then be provided by $A_N$ multiplicity rules governing generalized Schur functions. The main idea is again to express everything in terms of the so-called {\bf Fundamental Weights} with which we obtain a quite relevant specialization in applications of Weyl character formula.