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We consider the example from invariant theory concerning the conjugationaction of the general linear group on several copies of the $n \times n$matrices, and examine a symmetric function which stably describes the Hilbertseries for the invariant ring with respect to the multigradation by degree. Theterms of this Hilbert series may be described as a sum of squares ofLittlewood-Richardson coefficients. A "principal specialization" of thegradation is then related to the Hilbert series of the $\K$-invariant subringin the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonalembedding of a product of general linear groups. We also consider otherspecializations of this Hilbert series.

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