Double Centralizing Theorems for the Alternating Groups

Abstract

Let $V^{\otimes n}$ be the $n$-fold tensor product of a vector space $V.$ Following I. Schur we consider the action of the symmetric group $S_n$ on $V^{\otimes n}$ by permuting coordinates. In the `super' ($\Bbb Z_2$ graded) case $V=V_0\oplus V_1,$ a $\pm$ sign is added [BR]. These actions give rise to the corresponding Schur algebras S$(S_n,V).$ Here S$(S_n,V)$ is compared with S$(A_n,V),$ the Schur algebra corresponding to the alternating subgroup $A_n\subset S_n .$ While in the `classical' (signless) case these two Schur algebras are the same for $n$ large enough, it is proved that in the `super' case where $\dim V_0=\dim V_1, $ S$(A_n,V)$ is isomorphic to the crossed-product algebra S$(A_n,V)\cong$ S$(S_n,V)\times\Bbb Z_2 .$