The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes
نویسنده
چکیده
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 in χχ . 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 μν when λ is an arbitrary shape and μ and ν are hook shapes or two-row shapes. Remmel [9, 10] and Remmel and Whitehead [11] 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.
منابع مشابه
The power of symmetric functions in noncommutative variables
We show that the Kronecker coefficients indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. Join...
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