Igor Pak ; Greta Panova ; Ernesto Vallejo
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Kronecker coefficients: the tensor square conjecture and unimodality
dmtcs:2388 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2014,
DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
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https://doi.org/10.46298/dmtcs.2388
Kronecker coefficients: the tensor square conjecture and unimodalityArticle
Authors: Igor Pak 1; Greta Panova 1; Ernesto Vallejo 2
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Igor Pak;Greta Panova;Ernesto Vallejo
1 Department of Mathematics [UCLA]
2 Centro de Ciencias Matematicas [Mexico]
We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the $S_n$-irreducible representation indexed by the staircase partition contains every irreducible representation of $S_n$. We present a sufficient condition allowing to determine whether an irreducible representation is a constituent of a tensor square and using this result together with some analytic statements on partitions we prove Saxl conjecture for several partition classes. We also use Kronecker coefficients to give a new proof and a generalization of the unimodality of Gaussian ($q$-binomial) coefficients as polynomials in $q$, and extend this to strict unimodality.