Essential dimension of double covers of symmetric and alternating groups

Abstract

±n and A n of symmetric groups S n and alternating groups A n , respectively. Representations of these groups are closely related to projective representations of S n and A n ; there is also a close relationship between these groups and spinor groups. We study the essential dimension ed( S ± n ) and ed( A n ). We show that over a base field of characteristic = 2, ed( S ± n ) and ed( A n ) grow exponentially with n, similar to ed(Spin n ). On the other case, in characteristic 2, they grow sublinearly, similar to ed(S n ) and ed(A n ). We give an application of our result in good characteristic to the theory of trace forms.