1982/01/01 by Robert V. Moody, J. Patera · 3 citations
Computer Science · Mathematics · #Polynomial and algebraic computation #Advanced Algebra and Geometry #Computability, Logic, AI Algorithms
paper · pdf · doi:10.1090/s0273-0979-1982-15021-2
openalex publication_date 1982/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07
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 100 and for ranks -10 and dimensions ~ 10 4 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 E s . 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 = a t , ..., a ; the set of simple roots. Let Q and P be the root and weight lattices respectively spanning the real vector space FC *. 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.