The stability of the Kronecker product of Schur functions

International audience In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For $n$ large enough, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree $n$ do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n when this stable expansion is reached. We also compute two new bounds for the stabilization of a particular coefficient of such a product. Given partitions $\alpha$ and $\beta$, we give bounds for all the parts of any partition $\gamma$ such that the corresponding Kronecker coefficient is nonzero. Finally, we also show that the reduced Kronecker coefficients are structure coefficients for the Heisenberg product introduced by Aguiar, Ferrer and Moreira. Dans les années 30 Murnaghan a découvert une propriété de stabilité pour le produit de Kronecker de fonctions de Schur. En degré assez grand, les valeurs des coefficients qui aparaissent dans le produit de Kronecker de deux fonctions de Schur ne dépendent pas de la première part des partitions en indice, mais seulement des parts suivantes. Dans ce travail nous calculons la valeur exacte du degré à partir duquel ce développement stable est atteint. Nous calculons aussi deux nouvelles bornes supérieures pour la stabilisation d'un coefficient particulier d'un tel produit. Nous donnons en outre, pour $\alpha$ et $\beta$ fixés, des bornes supérieures pour toutes les parts des partition $\gamma$ rendant le coefficient de Kronecker d'indices $\alpha$, $\beta$, $\gamma$ non―nul. Finalement, nous identifions les coefficients de Kronecker réduits comme des constantes de structures pour le produit de Heisenberg de fonctions symétriques défini par Aguiar, Ferrer et Moreira.

On the Kronecker Product $s_{(n-p,p)}\ast s_{\lambda}$

The Kronecker product of two Schur functions $s_{\lambda}$ and $s_{\mu}$, denoted $s_{\lambda}\ast s_{\mu}$, is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of $n$, $\lambda$ and $\mu$, respectively. The coefficient, $g_{\lambda,\mu,\nu}$, of $s_{\nu}$ in $s_{\lambda}\ast s_{\mu}$ is equal to the multiplicity of the irreducible representation indexed by $\nu$ in the tensor product. In this paper we give an algorithm for expanding the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$ if $\lambda_1-\lambda_2\geq 2p$. As a consequence of this algorithm we obtain a formula for $g_{(n-p,p), \lambda ,\nu}$ in terms of the Littlewood-Richardson coefficients which does not involve cancellations. Another consequence of our algorithm is that if $\lambda_1-\lambda_2\geq 2p$ then every Kronecker coefficient in $s_{(n-p,p)}\ast s_{\lambda}$ is independent of $n$, in other words, $g_{(n-p,p),\lambda,\nu}$ is stable for all $\nu$.