Geodesic
Kronecker coefficients: the tensor square conjecture and unimodality
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014).

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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. 
@article{DMTCS_2014_special_265_a13,
     author = {Pak, Igor and Panova, Greta and Vallejo, Ernesto},
     title = {Kronecker coefficients: the tensor square conjecture and unimodality},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
     year = {2014},
     doi = {10.46298/dmtcs.2388},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2388/}
}
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%A Vallejo, Ernesto
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%J Discrete mathematics & theoretical computer science
%D 2014
%V DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
%I mathdoc
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%F DMTCS_2014_special_265_a13
Pak, Igor; Panova, Greta; Vallejo, Ernesto. Kronecker coefficients: the tensor square conjecture and unimodality. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi : 10.46298/dmtcs.2388. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2388/

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