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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the Kronecker Product s(n−p,p) ∗ sλ. | On the Kronecker Product S n pp sA C.M. Ballantine College of the Holy Cross Worcester MA 01610 cballant@holycross.edu R.C. Orellana Dartmouth College Hanover NH 03755 Rosa.C.Orellana@Dartmouth.edu Submitted Oct 17 2004 Accepted Jun 1 2005 Published Jun 14 2005 Mathematics Subject Classifications 05E10 20C30 Abstract The Kronecker product of two Schur functions sx and sM denoted sx sp is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of n X and 1 respectively. The coefficient gx P v of sv in sx sp is equal to the multiplicity of the irreducible representation indexed by V in the tensor product. In this paper we give an algorithm for expanding the Kronecker product S n-P P sx if Al X2 2p. As a consequence of this algorithm we obtain a formula for g n-p p x v in terms of the Littlewood-Richardson coefficients which does not involve cancellations. Another consequence of our algorithm is that if A1 X2 2p then every Kronecker coefficient in S n-P P sx is independent of n in other words g n-p p x v is stable for all V. Introduction Let xx and Ý be the irreducible characters of Sn the symmetric group on n letters indexed by the partitions A and j of n. The Kronecker product xxý is dehned by xxx w xx w xM w for all w 2 Sn. Hence xxx is the character that corresponds to the diagonal action of Sn on the tensor product of the irreducible representations indexed by A and j. Then we have xxx- X xv v n the electronic journal of combinatorics 12 2005 R28 1 where gX w is the multiplicity of Ý in xXX . Hence the gX w are non-negative integers. By means of the Frobenius map one can dehne the Kronecker internal product on the Schur symmetric functions by Sx Sa 52 gX wSv v n A reasonable formula for decomposing the Kronecker product is unavailable although the problem has been studied since the early twentieth century. In recent years Lascoux La Remmel R-1 Remmel and Whitehead RWd and Rosas
