Double Centralizing Theorems for the Alternating Groups
نویسنده
چکیده
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’ (Z2 graded) case V = V0 ⊕ V1, a ± sign is added [BR]. 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 dimV0 = dimV1 , S(An, V ) is isomorphic to the crossed–product algebra S(An, V ) ∼= S(Sn, V )× Z2 . Partially supported by ISF Grant 6629 and by Minerva Grant No. 8441. 1 2 §0. Introduction. Let V be a finite dimensional vector space over the field F = C of the complex numbers (in fact, we shall only need the fact that √ −1 ∈ F ), and let V ⊗n = V ⊗· · ·⊗V n times. The symmetric group Sn acts on V ⊗n (say, from the left) by permuting coordinates. This makes V ⊗n a left FSn module with the corresponding Schur algebra EndFSn(V ). Here FG denotes the group algebra of a group G. Formally, that action is given by a multiplicative homomorphism φ : Sn −→ EndF (V ) which extends linearly to an algebra homomorphism φ : FSn −→ EndF (V ) . Let An ⊂ Sn denote the alternating group, with the corresponding Schur algebra EndFAn(V ). The main purpose of this paper is to study and compare the pair of Schur algebras EndFSn(V ) ⊆ EndFAn(V ). We mention first the following phenomena: Theorem 1 (see Remark 1.9). Let dimV = k and consider the above (signless!) action of Sn on V ⊗n . If k n then φ(FAn) = φ(FSn), hence also EndFAn(V ) = EndFSn(V ). (0.1) In a sense, Theorem 1 shows an anomaly: even though |Sn| = 2|An| , nevertheless φ(FSn) = φ(FAn) , provided k 2 n . As indicated below, the incorporation of a ± sign to the above permutation action of Sn seems to be natural and to remove that anomaly. Such a sign permutation action is related to the representation theory of Lie superalgebras [BR] [Se]. We now briefly explain that Sn action, and this will allow us to formulate the main result of this paper, which is Theorem 2 below. Let V = V0 ⊕ V1 , dimV0 = k , dimV1 = l , and let Sn act on V ⊗n by permuting coordinates as before, but now, together with a ± sign; that sign is obtained by considering the elements of V0 as being central, and the non-zero elements of V1 as anti-commuting among themself. This is the sign-permutation action * of Sn on V ⊗n [BR §1], [Se]. This action determines the algebra homomorphism φ : FSn −→ EndF (V ). It endows V ⊗n with a new FSn module structure, which yields the corresponding (new!) Schur algebras EndFSn(V ) = Endφ∗(FSn)(V )= Bn , and EndFAn(V ) = Endφ∗(FAn)(V )= An , and clearly Bn ⊆ An. The main result of this paper is the following crossed product theorem. Theorem 2 (The Cross Product Theorem. See Theorem 1.1). Let V = V0 ⊕V1 with dimV0 = dimV1 and consider the above *-action of Sn on V . Then dimφ(FSn) = 2 dimφ (FAn) and dimAn = 2dimBn . (0.6)
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