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

• Authors: Mercedes H. Rosas
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2001

The Kronecker product of two Schur functions isμ and isν, denoted by isμ * isν, 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 isλ in this product is denoted by γλμν, and corresponds to the multiplicity of the irreducible character χλ in χμχν.We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for isλ[iXY] to find closed formulas for the Kronecker coefficients γλμν when λ is an arbitrary shape and μ and ν are hook shapes or two-row shapes.Remmel (J.B. Remmel, iJ. Algebra 120 (1989), 100–118; iDiscrete Math. 99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, iBull. Belg. Math. Soc. Simon Stiven 1 (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.