Combinatorial Operators for Kronecker Powers of Representations of Sn

Alain Goupil ; Cedric Chauve

Séminaire lotharingien de combinatoire, Tome 54 (2006-2007)

Résumé

We present combinatorial operators for the expansion of the Kronecker product of irreducible representations of the symmetric group S_n. These combinatorial operators are defined in the ring of symmetric functions and act on the Schur functions basis. This leads to a combinatorial description of the Kronecker powers of the irreducible representations indexed with the partition (n-1,1) which specializes the concept of oscillating tableaux in Young's lattice previously defined by S. Sundaram. We call our specialization Kronecker tableaux. Their combinatorial analysis leads to enumerative results for the multiplicity of irreducible representations in the Kronecker powers of the forms \chi^{(n-1,1)\otimes k} and P^{\otimes k} where P is the permutation representation of S_n.