Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A two-dimensional pictorial presentation of Berele’s insertion algorithm for symplectic tableaux. | A two-dimensional pictorial presentation of Berele s insertion algorithm for symplectic tableaux Tom Roby Department of Mathematics California State University Hayward CA 94542 USA troby@ Itaru Terada Graduate School of Mathematical Sciences University of Tokyo Komaba 3-8-1 Meguro-ku Tokyo 153-8914 Japan terada@ Submitted May 17 2004 Accepted Oct 13 2004 Published Jan 7 2005 Mathematics Subject Classifications 05E10 05E15 17B20 20G05 22E46 Abstract We give the first two-dimensional pictorial presentation of Berele s correspondence an analogue of the Robinson-Schensted R-S correspondence for the sym-plectic group Sp 2n C . From the standpoint of representation theory the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of GL n C . Berele s insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of Sp 2n C . Two-dimensional pictorial presentations of the R-S correspondence via local rules first given by S. Fomin and its many variants have proven very useful in understanding their properties and creating new generalizations. We hope our new presentation will be similarly useful. 1 Introduction Our purpose is to give a new presentation of Berele s correspondence. Berele s correspondence is a combinatorial construction devised by A. Berele in B as an Sp 2n C -analogue of one aspect of the Robinson-Schensted correspondence or the R-S THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R4 1 correspondence for short. The R-S correspondence describes the irreducible decomposition of the representation of the group GL n C on Cn f where f is a fixed positive integer derived from its natural action on the column vectors of Cn. Similarly Berele s correspondence describes the irreducible decomposition of the representation of the group Sp 2n C on C2ra 0f also derived from its natural action on the .
