Formulas for the expansion of the Kronecker products S(m, n) ⊗ S(1p−r, r) and S(1k2l) ⊗ S(1p−r, r)

AbstractThe Kronecker product of two homogeneous symmetric polynomials P1 and P2 is defined by means of the Frobenius map F by the formula P1 ⊗ P2=F((F−1P1)(F−1P2)). When P1 and P2 are Schur functions SI and SJ respectively, then the resulting product SI ⊗ SJ is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the diagrams I and J. Taking the scalar product of SI ⊗ SJ with a third Schur function SK gives the so called Kronecker coefficient CIJK=〈SI ⊗ SJ, SK〉 which gives the multiplicity of the representation corresponding to K in the tensor product. In this paper, we give an explicit formula for CIJK where K is an arbitrary shape, J is a hook shape, and I is either a shape with two parts or a shape whose largest part is of size 2