Decomposition of symmetric and exterior powers of the coadjoint representation of \({\mathfrak {gl}}(n)\) in the unstable case (Q1320680)

In the author's earlier papers a combinatorial expression for multiplicities of irreducible representations of \({\mathfrak gl}(n)\) in the exterior and symmetric powers \(S^ k{\mathfrak gl}(n)\) and \(\Lambda^ k{\mathfrak gl}(n)\) in the stable case (i.e., for \(k < n\)), was given. In this note we present a combinatorial expression for these multiplicities in the general case (for fields of characteristic zero). As a by-product, we obtain a natural interpretation of a known formula that expresses the multiplicities of irreducible representations in the space \(H\) of harmonic polynomials of the symmetric algebra \(S{\mathfrak gl}(n)\) via Kostka polynomials.