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Shuffles of permutations and the Kronecker product

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Abstract

The Kronecker product of two homogeneous symmetric polynomialsP 1,P 2 is defined by means of the Frobenius map by the formulaP 1oP 2=F(F −1 P 1)(F −1 P 2). WhenP 1 andP 2 are the Schur functionsS I ,S J then the resulting productS I oS J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS I oS J with a third Schur functionsS K gives the so called Kronecker coefficientc I,J,K =<S I oS J ,S K >. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <S I oS J ,S K > forS I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 92093, La Jolla, CA, USA

    A. M. Garsia & J. Remmel

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  1. A. M. Garsia
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  2. J. Remmel
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Work supported by NSF grant at the University of California, San Diego.

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Garsia, A.M., Remmel, J. Shuffles of permutations and the Kronecker product. Graphs and Combinatorics 1, 217–263 (1985). https://doi.org/10.1007/BF02582950

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