R-S Correspondence for $({\Bbb Z}_2 \times {\Bbb Z}_2) \wr S_n$ and Klein-$4$ Diagram Algebras

Abstract

In [PS] a new family of subalgebras of the extended ${\Bbb Z}_2$-vertex colored algebras, called Klein-$4$ diagram algebras, are studied. These algebras are the centralizer algebras of $G_n:=({\Bbb Z}_2 \times {\Bbb Z}_2) \wr S_n$ when it acts on $V^{\otimes k},$ where $V$ is the signed permutation module for $G_n.$ In this paper we give the Robinson-Schensted correspondence for $G_n$ on $4$-partitions of $n,$ which gives a bijective proof of the identity $\sum_{[\lambda] \vdash n } (f^{[\lambda]})^2 = 4^n n!,$ where $f^{[\lambda]}$ is the degree of the corresponding representation indexed by $[\lambda]$ for $G_n.$ We give proof of the identity $2^kn^k = \sum_{[\lambda] \in \Gamma_{n,k}^G} f^{[\lambda]} m_{k}^{[\lambda]}$ where the sum is over $4$-partitions which index the irreducible $G_n$-modules appearing in the decomposition of $V^{\otimes k} $ and $m_{k}^{[\lambda]}$ is the multiplicity of the irreducible $G_n$-module indexed by $[\lambda ].$ 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.