## Pak, Igor and Panova, Greta and Vallejo, Ernesto - 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)
Kronecker coefficients: the tensor square conjecture and unimodality

Authors: Pak, Igor and Panova, Greta and Vallejo, Ernesto

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.

Source : oai:HAL:hal-01207577v1
Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Section: Proceedings
Published on: January 1, 2014
Submitted on: November 21, 2016
Keywords: Kronecker coefficients,symmetric group,irreducible representations,tensor square,unimodality,integer partitions,Gaussian coefficients,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]