On the Kronecker product of Schur functions of two row shapes

Abstract

The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is dened by means of the Frobenius map by the formula P1P2 = F (F 1 P1)(F 1 P2). When P1 and P2 are Schur functions s and s respectively, then the resulting product s s is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams and . Taking the scalar product of ss with a third Schur function s gives the so-called Kronecker coecient g =hss;si which gives the multiplicity of the representation corresponding to in the tensor product. In this paper, we prove a number of results about the coecients g when both and are partitions with only two parts, or partitions whose largest part is of size two. We derive an explicit formula for g and give its maximum value.