A REMARK ON THE PERMUTATION REPRESENTATIONS AFFORDED BY THE EMBEDDINGS OF IN

We show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

The complements in an affine group

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

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

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.