Sums of squares of Littlewood–Richardson coefficients and GLn-harmonic polynomials

  • Pamela E. HarrisEmail author
  • Jeb F. Willenbring
Part of the Progress in Mathematics book series (PM, volume 257)


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.


Littlewood–Richardson coefficients Harmonic functions 

Mathematics Subject Classification

05E05 17B10 22E46 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.MSCS DepartmentMarquette UniversityMilwaukeeUSA
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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