On character generators for simple Lie algebras

Abstract

We study character generating functions (character generators) of simple Lie algebras. The expression due to Patera and Sharp, derived from the Weyl character formula, is first reviewed. A new general formula is then found. It makes clear the distinct roles of "outside" and "inside" elements of the integrity basis, and helps determine their quadratic incompatibilities. We review, analyze and extend the results obtained by Gaskell using the Demazure character formulas. We find that the fundamental generalized-poset graphs underlying the character generators can be deduced from such calculations. These graphs, introduced by Baclawski and Towber, can be simplified for the purposes of constructing the character generator. The generating functions can be written easily using the simplified versions, and associated Demazure expressions. The ranktwo algebras are treated in detail, but we believe our results are indicative of those for general simple Lie algebras.

Generating Functions for Orthogonal Polynomials of A2, C2 and G2

“…The lowest orthogonal polynomials (25) of variables x 1 , x 2 , given by relations (23) and (24), are for the first kind U (0,0) λ of the following explicit form:…”

“…Developed into the power series, the coefficients of the series provide answers to infinite number of computational problems involving the same Lie group [18,[23][24][25]. A practical difficulty often is the complexity of the generating functions for the higher ranks of G. So far, the generating functions practically for all problems are explicitly derived by hand computation.…”

On the generating function of weight multiplicities for the representations of the Lie algebra C2

“…11,12 If the z-variables are expanded in terms of the x-ones, and a multi-index exponential notation L λ = L p 1 1 L p 2 2 · · · L p r r is used, our generating function G(t i ; z j ) can be identified with the generating function…”

Generating functions and multiplicity formulas: The case of rank two simple Lie algebras

“…As it often happens whith many issues having to do with Lie algebra representations, one of the most efficient tools available to tackle the question of multiplicities is the theory of characters. In a recent paper [7], we have presented a general method for computing the generating function of the characters of simple Lie algebras by means on the theory of the quantum trigonometric Calogero-Sutherland system [8]- [11] (see also [12,13] for other approaches to that problem). One advantage of the method described in [7] is its simplicity: it requires some acquaintance with the Calogero-Sutherland theory, especially its treatment by means of Weyl-invariant variables but, apart from that, the computations involved are quite elementary.…”

“…Substitution of the decompositions (12), (13) in the generating function (14), allows to identify the relevant poles for computing the function H. They are…”