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Báo cáo toán học: "Stability of Kronecker products of irreducible characters of the symmetric group"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Stability of Kronecker products of irreducible characters of the symmetric group. | Stability of Kronecker products of irreducible characters of the symmetric group Ernesto Vallejo1 Instituto de Matematicas Universidad Nacional Autonoma de Mexico Area de la Inv. Cient. 04510 Mexico D.F. evallejo@matem.unam.mx Submitted October 30 1998 Accepted September 6 1999 Primary classification 05E10 secondary classification 20C30 Abstract F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product x n- X2 . 0 x n-b 2 . of two irreducible characters of the symmetric group into irreducibles depends only on A A2 . and F f2 but not on n. In this note we prove a similar result given three partitions A F V of n we obtain a lower bound on n depending on A F Ũ for the stability of the multiplicity c A F n of x in xX 0 xM. Our proof is purely combinatorial. It uses a description of the c A F s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux. 1 Introduction. Let Xx denote the irreducible complex character of the symmetric group S n corresponding to the partition A. For any three partitions A F V of n we denote by c A F v hv 0 XM X i 1 the multiplicity of X in the Kronecker product Xx 0 XM. F. Murnaghan observed in 6 that the computation of the decompositon of the Kro-necker product X n-a x2 . 0 X n-b M2 . into irreducibles depends only on A A2 . and F f2 . but not on n. He gave fifty eight formulas for decompositions of Kro-necker products corresponding to the simplest choices of A and F. In fact his formulas are valid for arbitrary n only if one follows some rules to restore and discard disordered partitions appearing in them see comment on 6 p.762 . In this note we prove a similar result given three partitions A F V of n we obtain a lower bound on n depending on A F V for the stability of the coefficients c A F V . More precisely. Let A A2 . Ap F f2 . Fg V v2 . Vr be partitions of positive integers a b c respectively. For each n a A2 we consider the partition of n A n n a A2 . Ap . .