Double Centralizing Theorems for the Alternating Groups

Let 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 ( 2 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 = dim V1, S(An, V) is isomorphic to the cross-product algebra S(An, V) S(Sn, V) 2.