The Kronecker product of Schur functions indexed by two-row shapes or hook shapes

Abstract

The Kronecker product of two Schur functions sµ and sν, denoted by sµ ∗ sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions µ and ν. The coefficient of sλ in this product is denoted by γ λ µν , and corresponds to the multiplicity of the irreducible character χ λ in χ µχ ν We use Sergeev’s Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for sλ[XY ] to find closed formulas for the Kronecker coefficients γ λ µν when λ is an arbitrary shape and µ and ν are hook shapes or two-row shapes. Remmel [9 J.B. Remmel, “A formula for the Kronecker product of Schur functions of hook shapes,” J. Algebra 120, 1989, pp. 100–118, 10 J.B. Remmel, “Formulas for the expansion of the Kronecker products S(m,n) ⊗ S(1p−r,r) and S(1k2 l) ⊗ S(1p−r,r) ,” Discrete Math. 99, 1992, pp. 265–287] and Remmel and Whitehead [11] J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. Simon Stevin 1, 1994, pp. 649–683. derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product.