A formula for the Kronecker products of Schur functions of hook shapes

AbstractThe 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−1 P1)(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 scaler 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 I and J are hooks and K is arbitrary