On the decomposition of the tensor algebra of the classical Lie algebras

Abstract

Let g n be the Lie algebra gl n ( C), let S( g n) be the symmetric algebra of g n , and let T( g n) be the tensor algebra of g n . In a recent paper, R. K. Gupta studied certain sequences of representations R ∞ = ( R n ) ∞ n = 1 , where R n is a representation of g n . These sequences have the property that every irreducible representation occurring in S( g n) is in exactly one of these sequences. Fixing f, she considers s( R ∞, f) which is the limit on n of the multiplicity of R n in S f ( g n), the fth-graded piece of S( g n). She and R. P. Stanley independently showed that the limit s( R ∞, f) exists and is given by an amazingly elegant formula. They call s( R ∞, f) the stable multiplicity of R n in S f ( g n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R ∞ for all of the classical Lie algebras g n are studied, and a simple formula for the stable multiplicity m( R) ∞, ψ, f, g ∞) of R n in the ψ-isotypic component of T f ( g n), where ψ is any irreducible character of the symmetric group tS f , is obtained. As in the work of Gupta and Stanley, the expressions for m( R) ∞, ψ, f, g ∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of g n . As a byproduct of our proof, a “stable” decomposition of every isotypic component of T( g n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S( g n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f < n the combinatorial decomposition of T f ( g n) projects onto Kostant's decomposition of S f ( g n).