Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A λ-ring Frobenius Characteristic for G Sn. | A A-ring Frobenius Characteristic for G X Sn Anthony Mendes Department of Mathematics California Polytechnic State University San Luis Obispo CA 93407. USA aamendes@ Jeffrey Remmel Department of Mathematics University of California San Diego La Jolla CA 92093-0112. USA jremmel@ Jennifer Wagner University of Minnesota School of Mathematics 127 Vincent Hall 206 Church Street SE Minneapolis MN 55455. USA wagner@ Submitted Apr 21 2003 Accepted Jul 1 2004 Published Sep 3 2004 MR Subject Classifications 05E10 20C15 Abstract A A-ring version of a Frobenius characteristic for groups of the form GI Sn is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of G Ỉ Sn into its irreducible components along with generalizations of the Murnaghan-Nakayama rule the Hall inner product and the reproducing kernel for G Ỉ Sn. 1 Introduction Let G be a finite group and let Sn be the symmetric group on n letters. In the early 1930 s Specht described the irreducible representations of the wreath product G 1 Sn in his dissertation 16 but did not describe an analog of the Frobenius characteristic for the symmetric group. Since then there have been numerous accounts of the representation theory of G 1 Sn 6 7 . Most have not attempted to generalize the Frobenius map although at least one has 10 . In 10 Macdonald gives a generalization of Schur s theory of polynomial functors before showing that a specialization of that theory naturally leads to Specht s results on the representations of G 1 Sn. Macdonald s version of the Frobenius map for G 1 Sn is not the same as the Frobenius map in this paper but it is shown to have some of the same properties. In particular Macdonald verifies a sort of Frobenius reciprocity. These results are reproduced in 11 . Our presentation of the Frobenius map for G1 Sn can essentially be
