Essential p-dimension of the normalizer of a maximal torus

Abstract

Abstract. We compute the exact value for the essential p-dimension of the normalizer of a split maximal torus for most simple connected linear algebraic groups. These values give new upper bounds on the essential p-dimension of some simple groups, including some exceptional groups.For each connected simple algebraic group, we also give an upper bound on the essential p-dimension of any torus contained in that group. These results are achieved by a detailed case-by-case analysis.

Pseudo-reflection groups and essential dimension

“…Theorem 1. 3(a) answers an open question posed in [26,Remark 5.2]. The proof of this part relies on a geometric construction suggested to us by Dolgachev. We now recall that ed(G) is the minimal dimension of a versal G-variety and ed(G; p) is the minimal dimension of a p-versal G-variety; see [12, Remark 2.5; 40, Section 5].…”

“…The absolute essential dimension ed(S n ) is largely unknown. In characteristic zero, we know only that max p ed(S n ; p) = n 2 n + 1 2 ed(S n ) n − 3 (1.3) for any n 6; see [3,11,26]. We know even less about ed(S n ) in prime characteristic.…”

“…It is thus natural to try to compute ed(G) and ed(G; p), where G is a finite pseudo-reflection group, and p is a prime. The first steps in this direction were taken by MacDonald [26,Section 5.1], who computed ed(G; p) for all primes p and all irreducible Weyl groups G. He also computed ed(G) for every irreducible Weyl group G except for G = S n and G = W (E 6 ), the Weyl group of the root system of type E 6 . His proofs are based on case-by-case analysis.…”

“…The case where G = S n is the symmetric group is of particular interest because it relates to classical questions in the theory of polynomials; see [BR97,BR99]. Here the relative essential dimension is known exactly for every prime p, n + 1 2 ed(S n ) n − 3 for any n 6; see [BR97], [Dun10] and [Mac11]. We know even less about ed(S n ) in prime characteristic.…”

Upper Bounds for the Essential Dimension of E7

“…Also, ed(E 7 ; 3) = 3 ( [GR09] or [Ga09]). For p = 2 we have 7 ≤ ed(E 7 ; 2) ≤ 33 ([RY00], [CS06], [Mac11]). In this paper we improve the upper bound.…”