A new method of weight multiplicities evaluation for semi-simple Lie algebras

TL;DR

This paper introduces a novel direct method for calculating weight multiplicities in semi-simple Lie algebras, avoiding Weyl group summations and recursion, resulting in faster computations demonstrated through Fortran implementations.

Contribution

A new direct algorithm for weight multiplicity calculation that is faster and does not rely on Weyl group or recursive formulas, improving computational efficiency.

Findings

The new method is significantly faster than traditional formulas.

The algorithm is implemented in Fortran and ready for practical use.

Numerous examples demonstrate the method's effectiveness.

Abstract

In most applications of semi-simple Lie groups and algebras representation theory, calculating weight multiplicities is one of the most often used and effort consuming operations. The existing tools were created many years ago by Kostant and Freudenthal. The celebrated Kostant weight multiplicity formula uses summation over the Weyl group of values of Kostant partition function, and the Freudenthal formula is recurrent. In this paper, a new way for calculating weight multiplicities is presented. The method does not employ the Weyl group and is direct, not recurrent. The algorithm realized in accordance with this method is much faster than those realized with the previously employed techniques. Many examples of programs realized in Fortran language are given. They are ready for compilation and execution on desktop and laptop computers.