On the decomposition of the tensor algebra of the classical Lie algebras
Let \(T_ n\) denote the tensor algebra of the Lie algebra \(gl_ n({\mathbb{C}})\), \(T^ f_ n\) its f-graded piece, \(R_{\infty}\) an appropriately defined sequence of irreducible representations \(R_ n\) of \(gl_ n({\mathbb{C}})\), \(m(R_ n,\psi,f)\) the multiplicity of \(R_ n\) in \(T^ f_{n \psi}\), the \(\psi\)-isotypic component for an irreducible character. Recently \textit{R. K. Gupta} and \textit{R. P. Stanley} got striking results on the limit behavior of the multiplicities as \(n\to \infty\) (their papers are not published yet) for the case of the symmetric algebra. The author shows that the limit exists and has a very simple expression. His results include the cases of the symmetric and exterior algebras. In section 5 he states analogous results for types \(B_ n\), \(C_ n\), \(D_ n\).
scientific article; zbMATH DE number 3922911
scientific article