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Kronecker powers of harmonics, polynomial rings, and generalized principal evaluations

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Abstract

Our main goal is to compute the decomposition of arbitrary Kronecker powers of the Harmonics of \(S_n\). To do this, we give a new way of decomposing the character for the action of \(S_n\) on polynomial rings with k sets of n variables. There are two aspects to this decomposition. The first is algebraic, in which formulas can be given for certain restrictions from \(GL_n\) to \(S_n\) occurring in Schur-Weyl duality. The second is combinatorial. We give a generalization of the \({{\,\mathrm{{comaj}}\,}}\) statistic on permutations which includes the \({{\,\mathrm{{comaj}}\,}}\) statistic on standard tableaux. This statistic allows us to write a generalized principal evaluation for Schur functions and Gessel fundamental quasisymmetric functions.

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Acknowledgements

We must thank Nolan Wallach for suggesting the problem and for the numerous, helpful conversations. Also, thank you to Brendon Rhoades and Dun Qiu for the helpful comments and suggestions.

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  1. Department of Mathematics, University of Pennsylvania, Philadelphia, USA

    Marino Romero

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  1. Marino Romero
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The author was partially supported by the University of California President’s Postdoctoral Fellowship.

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Romero, M. Kronecker powers of harmonics, polynomial rings, and generalized principal evaluations. J Algebr Comb 57, 135–159 (2023). https://doi.org/10.1007/s10801-022-01159-6

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  • DOI: https://doi.org/10.1007/s10801-022-01159-6

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