A Two-Dimensional Pictorial Presentation of Berele's Insertion Algorithm for Symplectic Tableaux

Abstract

We give the first two-dimensional pictorial presentation of Berele's correspondence, an analogue of the Robinson-Schensted (R-S) correspondence for the symplectic group $Sp(2n, {\Bbb C})$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,{\Bbb C})$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n,{\Bbb C})$. 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.