Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: R-S correspondence for (Z2 × Z2) Sn and Klein-4 diagram algebras. | R-S correspondence for Z2 X Z2 o Sn and Klein-4 diagram algebras M. Parvathi and B. Sivakumar Ramanujan Institute for Advanced Study in Mathematics University of Madras Chennai-600 005 India Submitted Nov 20 2007 Accepted Jul 22 2008 Published Jul 28 2008 Mathematics Subject Classifications 05A05 20C99 Abstract In PS a new family of subalgebras of the extended Z2-vertex colored algebras called Klein-4 diagram algebras are studied. These algebras are the centralizer algebras of Gn Z2 X Z2 o Sn when it acts on V where V is the signed permutation module for Gn. In this paper we give the Robinson-Schensted correspondence for Gn on 4-partitions of n which gives a bijective proof of the identity P n f A 2 4nn where f A is the degree of the corresponding representation indexed by A for Gn. We give proof of the identity 2knk P A erc f A mkA where the sum is over 4-partitions which index the irreducible appearing in the decomposition of V0k and mk is the multiplicity of the irreducible Gn-module indexed by A . Also we develop an R-S correspondence for the Klein-4 diagram algebras by giving a bijection between the diagrams in the basis and pairs of vacillating tableau of same shape. 1 Introduction The Bratteli diagrams of the group algebras arising out of the sequence of wreath product groups ZI o Sn and Z2 X Z2 o Sn are the same. The structures associated with the wreath product ZI o Sn are well studied. This motivated us to study the centralizer of wreath product of the Klein-4 group with Sn in PS and we obtained a new family of subalgebras of the extended Z2-vertex colored algebras called Klein-4 diagram algebras. These algebras are the centralizer algebras of Gn Z2 X Z2 o Sn when it acts on V where V is the signed permutation module for Gn and are denoted by Rk n . The second author was supported through SRF from CSIR New Delhi. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R98 1 The partition algebras were .
