Fast recursion formula for weight multiplicities
Robert V. Moody, J. Patera · Bulletin of the American Mathematical Society · 1982
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.