The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

The Kronecker product of two Schur functions s and s, denoted by s * s, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions and . The coefficient of s in this product is denoted by , and corresponds to the multiplicity of the irreducible character in .We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for s[XY] to find closed formulas for the Kronecker coefficients when is an arbitrary shape and and are hook shapes or two-row shapes.Remmel (J.B. Remmel, J. Algebra120 (1989), 100–118; Discrete Math.99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. Belg. Math. Soc. Simon Stiven1 (1994), 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.