Abstract
The Kronecker product of two Schur functions s μ and s ν, denoted by s μ * s ν, 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 s λ in this product is denoted by γλ μν, 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 s λ[XY] to find closed formulas for the Kronecker coefficients γλ μν when λ is an arbitrary shape and μ and ν are hook shapes or two-row shapes.
Remmel (J.B. Remmel, J. Algebra 120 (1989), 100–118; Discrete Math. 99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. 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.
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Rosas, M.H. The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes. Journal of Algebraic Combinatorics 14, 153–173 (2001). https://doi.org/10.1023/A:1011942029902
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DOI: https://doi.org/10.1023/A:1011942029902