Essential dimension of finite p-groups

Abstract

Abstract. We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F .The notion of the essential dimension ed(G) of a finite group G over a field F was introduced in [5]. The integer ed(G) is equal to the smallest number of algebraically independent parameters required to define a Galois G-algebras over any field extension of F . If V is a faithful linear representation of G over Prop. 4.15]). The essential dimension of G can be smaller than dim(V ) for every faithful representation V of G over F . For example, we have ed(Z/3Z) = 1 over Q or any field F of characteristic 3 (cf. [2, Cor. 7.5]) and ed(S 3 ) = 1 over C (cf. [5, Th. 6.5]).In this paper we prove that if G is a p-group and F is a field of characteristic different from p containing p-th roots of unity, then ed(G) coincides with the least dimension of a faithful representation of G over F (cf. Theorem 4.1).We also compute the essential p-dimension of a p-group G introduced in [15]. We show that ed p (G) = ed(G) over a field F containing p-th roots of unity.In the paper the word "scheme" means a separated scheme of finite type over a field and "variety" an integral scheme. Acknowledgment:We are grateful to Zinovy Reichstein for useful conversations and comments.

Essentialp-dimension of algebraic groups whose connected component is a torus

“…Our argument below is a variant of the proof of [KM,Theorem 5.1], where G is assumed to be a (constant) finite p-group and C = C(G) (recall that C(G) is defined at the beginning of Section 4).…”

Essential p-dimension of the normalizer of a maximal torus

“…By Lemma 1.10 this gives us a p-faithful N H -representation, V Λ . Let V be a faithful H-representation of dimension ed(W ; p) (which is possible by [KM08], extending the base field if necessary). Then, as in [MR09, Lemma 3.2], V Λ × V is a p-generically free N H -representation, and thus we have the required upper bound on ed(N ; p) = ed(N H ; p).…”

Maximal representation dimension of finite p-groups

“…The lemma can be deduced from [7,Remark 4.7] or [8, Theorem 1.2]; for the sake of completeness we give a self-contained proof.…”

“…We are motivated by a recent result of N. Karpenko and A. Merkurjev [7,Theorem 4.1], which states that if G is a finite p-group then the essential dimension of G is equal to rdimðGÞ. For a detailed discussion of the notion of essential dimension for finite groups (which will not be used in this paper), see [1] or [6, §8].…”