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Séminaire Lotharingien de Combinatoire, B54j (2006), 13 pp.

# Alain Goupil and Cedric Chauve

# 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 *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}.

Received: September 30, 2005.
Revised: July 6, 2006.
Accepted: July 7, 2006.

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