Abstract
We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the n × 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 subspace in the GL n -harmonic polynomials (in the sense of Kostant), where K denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.
To Nolan Wallach, who has influenced generations of mathematicians
This research was supported by the National Security Agency grant # H98230-09-0054.
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Harris, P.E., Willenbring, J.F. (2014). Sums of squares of Littlewood–Richardson coefficients and GL n -harmonic polynomials. In: Howe, R., Hunziker, M., Willenbring, J. (eds) Symmetry: Representation Theory and Its Applications. Progress in Mathematics, vol 257. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1590-3_11
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DOI: https://doi.org/10.1007/978-1-4939-1590-3_11
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