Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP1,P2 is defined by means of the Frobenius map by the formulaP1oP2=F(F−1P1)(F−1P2). WhenP1 andP2 are the Schur functionsSI,SJ then the resulting productSI oSJ is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofSI oSJ with a third Schur functionsSK gives the so called Kronecker coefficientcI,J,K=. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thecI,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of forSI a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result.