A two-dimensional pictorial presentation of Berele's insertion algorithm for symplectic tableaux

We give the first two-dimensional pictorial presentation of Berele's correspondence \cite{Berele}, an analogue of the Robinson-Schensted (R-S) correspondence \cite{Robinson, Schensted} for the symplectic group $Sp(2n, \Cpx )$. From the standpoint of representation theory, the R-S correspondence combinatorially describes the irreducible decomposition of the tensor powers of the natural representation of $GL(n,\Cpx)$. Berele's insertion algorithm gives the bijection that describes the irreducible decomposition of the tensor powers of the natural representation of $Sp(2n, \Cpx)$. Two-dimensional pictorial presentations of the R-S correspondence via local rules (first given by S. Fomin \cite{Fomin,FominGen}) 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.