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.
Similar content being viewed by others
References
Garsia, A.M., Remmel, J.: Sur le produit de Kronecker des Polynomes symmetriques. Comptes Rendus Acad. Sc. Paris (to appear)
Garsia A.M., Remmel J.: Symmetric functions and raising operators. Linear and Multilinear Algebra10, 15–43 (1981)
Gessel, I.M.: Multipartitep-partitions and inner products of Schur functions. Contemp. Math.34, 289–302 (1984)
James, G.D., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics 16. Reading: Addison-Wesley 1981
Knuth, D.E.: Permutations, matrices, and generalized Young tableaux, Pacific J. Math.,34, 709–727 (1970).
Knuth, D.E.: The Art of Computer Programming.Sorting and Searching vol. 3. Reading: Addison-Wesley 1968
Lascoux, A.: Produit de Kronecker des representations du group symmetrique, in Séminaire d'Algebrè P. Dubreil et M.-P. Malliavin. (Proceedings, Paris 1979 In: Lecture Notes in Mathematics 795, pp. 319–329. Berlin-Heidelberg-New York: Springer-Verlag 1980
Lascoux, A., Schützenberger, M.P.: Le monoide plaxique. Quad. Ricerca Scientifica C.N.R.,109, 129–156 (1981)
Littlewood, D.E. The Kronecker product of symmetric group representations. J. London Math. Soc. (1),31, 89–93 (1956).
Littlewood, D.E.: Pleythysm and the inner product ofS-functions. J. London Math. Soc. (1),32, 18–22 (1957)
Littlewood, D.E.: The Theory of Group Characters and Matrix Representations of Groups (2nd ed.) Oxford: Clarendon Press 1950
Macdonald, I.G.: Symmetric Functions and Hall Polynomials. New York: Oxford University Press 1979
Remmel, J., Whitney R.: Multiplying Schur functions. J. Algorithms5, 471–487 (1984)
Robinson, G. de B.: On the representations of the symmetric group. Amer. J. Math.,60, 754–760 (1938)
Schensted, C.: Longest increasing and decreasing subsequences. Canad. J. Math.,13, 179–191 (1961)
Schützenberger, M.P.: La correspondence de Robinson. In: Combinatoire et Représentation du Groupe Symmetrique (Actes Table Ronde du C.N.R.S., Strasbourg, 1976), Lecture Notes in Mathematics 579, pp 59–113. Berlin-Heidelberg-New York: Springer-Verlag 1977
Schützenberger, M.P.: Promotions des morphismes d'ensembles ordonnes. Discrete Math.,2, 73–94 (1972)
Schützenberger, M.P.: Proprietes nouvelles des tableaux de Young, Seminaire Delange-Pisot-Poutiou. In: Theorie des nombres, 19e anneé 1977/78 no. 20. p. 14
Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra,41, 255–268 (1976)
Stanley, R.: Theory and applications of plane partitions, Parts 1 & 2. Stud. Appl. Math.,50, 167–188 and 258–279 (1971)
Author information
Authors and Affiliations
Additional information
Work supported by NSF grant at the University of California, San Diego.
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/BF02582950