Abstract
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. Nous considérons deux aspects des coefficients de Kronecker dans le domaine de la Théorie des Représentations et le domaine Combinatoire. Nous considérons la conjecture suivante de Jan Saxl: le tenseur au carré de la représentation irréductible du groupe $S_n$ indexée par la partition $S_n (n= \left( \begin{array}{cc} k+1 \\ 2 \end{array} \right))$. Nous présentons une condition suffisante qui permet de déterminer si une représentation irréductible est une constituante d’un tenseur au carré. En utilisant ce résultat avec des résultats analytiques sur les partitions, nous prouvons la conjecture de Saxl pour plusieurs classes de partitions. Nous utilisons aussi les coefficients de Kronecker pour donner une nouvelle preuve et une généralisation de l’unimodalité des coefficients de Gauss ($q$-binomiaux) comme polynômes en $q$ et nous étendons cela à l’unimodalité stricte.
Highlights
The Kronecker product problem is a problem of computing the multiplicities, called Kronecker coefficients, g(λ, μ, ν) = χλ, χμ ⊗ χν of an irreducible character of Sn in the tensor product of two others
The significance of Kronecker coefficients has recently found a new meaning in Complexity Theory via a program designed to prove the “P vs NP” problem
They conjecture that for every n ≥ 5, there is an irreducible character χ of An whose tensor square χ ⊗ χ contains every irreducible character as a constituent.(i) Here is the symmetric group analogue of this conjecture: Conjecture 1.1 (Tensor square conjecture) For every n ≥ 3, n = 4, 9, there is a partition μ n, such that tensor square of the irreducible character χμ of Sn contains every irreducible character as a constituent
Summary
The Kronecker product problem is a problem of computing the multiplicities, called Kronecker coefficients, g(λ, μ, ν) = χλ, χμ ⊗ χν of an irreducible character of Sn in the tensor product of two others. These coefficients and the problem were introduced 75 years ago by Murnaghan following the discovery of the Littlewood-Richardson rule. [BO2, Bla, BOR, Ike, Reg, Rem, RW, Val, Val2] and references therein), it is universally agreed that “frustratingly little is known about them” [Bur]. We present two problems, one of representation theoretic and another of combinatorial nature, where we develop some tools and consider new aspects of this old problem. The work presented in this abstract is based on the results from [PPV], [PP-u], [PP-s], and partially on [Val3]
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