Combinatorial operators for Kronecker powers of representations of $\S_n$

Abstract

We present combinatorial operators for the expansion of the Kronecker product of irreducible representations of the symmetric group. 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 {\it Kronecker tableaux}. Their combinatorial analysis leads to enumerative results for the multiplicity of any irreducible representation in the Kronecker powers of the form ${\c^{(n-1,1)}}^{\otimes k}$.