On the Kronecker Product of Schur Functions of Two Row Shapes

The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is defined by means of the Frobenius map by the formula P1 ⊗ P2 = F (F−1P1)(F−1P2). When P1 and P2 are Schur functions sλ and sμ respectively, then the resulting product sλ ⊗ sμ is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams λ and μ. Taking the scalar product of sλ ⊗ sμ with a third Schur function sν gives the so-called Kronecker coefficient gλμν = 〈sλ ⊗ sμ, sν〉 which gives the multiplicity of the representation corresponding to ν in the tensor product. In this paper, we prove a number of results about the coefficients gλμν when both λ and μ are partitions with only two parts, or partitions whose largest part is of size two. We derive an explicit formula for gλμν and give its maximum value.