Fast recursion formula for weight multiplicities

Abstract

The purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple Lie algebras over C. Until now information about weight multiplicities for all but some special cases [1 ,2] has had to be found from the recursion formulas of Freudenthal [3] or Racah [4] . Typically these formulas become too laborious to use for hand computations for ranks ^ 5 and dimensions ^ 1 0 0 and for ranks — 10 and dimensions ~ 10 on a large computer [5, 6 ] . With the proposed method the multiplicities can routinely be calculated, even by hand, for dimensions far exceeding these. As an example we present a summary of calculations [7] of all multiplicities in the first sixteen irreducible representations of Es. Let ($ be a semisimple Lie algebra over C with root system A and Weyl group W relative to a Cartan subalgebra §. Let A be the positive roots with respect to some ordering and II = {at, . . . , a;} the set of simple roots. Let Q and P be the root and weight lattices respectively spanning the real vector space F C § * . If X C P we denote by X + the set of dominant elements of X relative to n. Let M be an irreducible ($ -module with highest weight A and weight system £2. An important feature of the approach is the direct determination of £2 + + without computing outside the dominant chamber. Since every W-orbit is represented by one weight X £ £2 + + of the same multiplicity, it suffices to compute such X's. The recursion formula for computing the multiplicities is a modification (Proposition 4) of the Freudenthal formula in which the Weyl group has been exploited to collapse it as much as possible. After describing the procedure, we present the E% example. Finally the necessary proofs are given.