Double Centralizing Theorems for the Alternating Groups

AbstractLet V⊗n be the n-fold tensor product of a vector space V. Following I. Schur we consider the action of the symmetric group Sn on V⊗n by permuting coordinates. In the super (Z2 graded) case V=V0⊕V1, a ± sign is added. These actions give rise to the corresponding Schur algebras S(Sn,V). Here S(Sn,V) is compared with S(An,V), the Schur algebra corresponding to the alternating subgroup An⊂Sn. 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 V0=dimV1,S(An,V) is isomorphic to the cross-product algebra S(An,V)≅S(Sn,V)⊗Z2