Sums of squares of the Littlewood-Richardson coefficients and GL(n)-harmonic polynomials

We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coefficients. A "principal specialization" of the gradation is then related to the Hilbert series of the $\K$-invariant subring in the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.