Double Centralizing Theorems for the Alternating Groups

Abstract

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 S n on V ⊗ n by permuting coordinates. In the super ( Z 2 graded) case V = V 0 ⊕ V 1 , a ± sign is added. 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 ⊂ 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 cross-product algebra S( A n , V ) ≅ S( S n , V ) ⊗ Z 2 .