R-S correspondence for \((\mathbb Z_{2} \times \mathbb Z_{2}) \wr S_{n}\) and Klein-4 diagram algebras (Q1010825)

Summary: In [PS] a new family of subalgebras of the extended \({\mathbb Z}_2\)-vertex colored algebras, called Klein-4 diagram algebras, are studied. These algebras are the centralizer algebras of \(G_n:=({\mathbb Z}_2 \times {\mathbb Z}_2) \wr S_n\) 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.