Essential<i>p</i>-dimension of algebraic groups whose connected component is a torus

Lötscher, Roland; MacDonald, Mark L.; Meyer, Aurel; Reichstein, Zinovy

doi:10.2140/ant.2013.7.1817

Cited by 8 publications

(5 citation statements)

References 22 publications

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“…The kernel of ν is a finite group of order prime to p; it is contained in the maximal torus T of G. From now on we will replace G by G = G/ Ker(ν). All other G-actions we will construct (including the linear G-action on W ) will factor through G. In the end we will show that ed(W ; p) = ed(G; p); once again, this is enough because ed(G; p) = η(G) = η(G) = ed(G; p) by [14,Proposition 2.4]. In other words, from now on we may (and will) assume that the G-action on V is faithful.…”

Section: Proof Of Theorem 12mentioning

confidence: 95%

“…Let G be an algebraic group over k such that the connected component T = G 0 is a torus, and the component group F = G/T is a finite p-group, as in (1.1). By [14,Lemma 5.3], there exists a finite p-subgroup F ′ ⊂ G such that π| F ′ : F ′ → F is surjective. We will refer to F ′ as a "quasi-splitting subgroup" for G. We will denote the subgroup generated by F ′ and T [n] by G n .…”

Section: The Groups G Nmentioning

confidence: 99%

“…Informally speaking, the lower bound of (1.2) is the strongest lower bound on ed(G; p) one can hope to prove by the methods of [11], [13], and [14]. In the case, where the upper and lower bounds of (1.2) diverge, Conjecture 1.1 calls for a new approach.…”

Section: Introductionmentioning

confidence: 96%

“…We denote by η(G) (respectively, ρ(G)) the smallest dimension of a p-faithful (respectively, p-generically free) representation. R. Lötscher, M. MacDonald, A. Meyer, and the first author [14,Theorem 1.1] showed that the essential p-dimension ed(G; p) satisfies the inequalities…”

Section: Introductionmentioning

confidence: 99%

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On the essential dimension of an algebraic group whose connected component is a torus

Reichstein¹,

Scavia²

2020

Preprint

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Let p be a prime integer, k be a p-closed field of characteristic = p, T be a torus defined over k, F be a finite p-group, and 1 → T → G → F → 1 be an exact sequence of algebraic groups. In this paper we study the essential dimension ed(G; p) of G at p. R. Lötscher, M. MacDonald, A. Meyer, and the first author showed thatwhere V and W range over the p-faithful and p-generically free k-representations of G, respectively. This generalizes the formulas for the essential dimension at p of a finite p-group due to N. Karpenko and A. Merkurjev (here T = {1}) and of a torus, due to Lötscher et al. (here F = {1}). In both of these cases every p-generically free representation of G is p-faithful, so the upper and lower bounds on ed(G; p) given above coincide. In general there is a gap between these bounds. Lötscher et al. conjectured that the upper bound is, in fact, sharp; that is, ed(G; p) = min dim(W ) − dim(G), where W ranges over the p-generically free representations, as above. We prove this conjecture in the case, where F is diagonalizable. Moreover, we give an explicit way to compute min dim(W ) in this case. As an application of our main theorem we compute ed(G; p), where G is the normalizer of a split maximal torus in a split simple algebraic group, in all previously inaccessible cases.

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Section: Proof Of Theorem 12mentioning

confidence: 95%

Section: The Groups G Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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On the essential dimension of an algebraic group whose connected component is a torus

Reichstein¹,

Scavia²

2020

Preprint

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“…[GR09]) and by Lötscher, MacDonald, Meyer and Reichstein (cf. [LMMR13]) in the study of essential dimension for linear algebraic groups. In the latter, an important issue is to control the prime numbers dividing the order of the finite group obtained.…”

Section: Introductionmentioning

confidence: 99%

Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

Arteche

2015

Preprint

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For a finite smooth algebraic group F over a field k and a smooth algebraic group Ḡ over the separable closure of k, we define the notion of F -kernel in Ḡand we associate to it a set of nonabelian 2-cohomology. We use this to study extensions of F by an arbitrary smooth k-group G. We show in particular that any such extension comes from an extension of finite k-groups when k is perfect and we give explicit bounds on the order of these finite groups when G is linear. We prove moreover some finiteness results on these sets.

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Essential dimension and canonical dimension of gerbes banded by groups of multiplicative type

Lötscher

2015

Math. Z.

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

Abstract: Abstract. Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G 0 is a torus.

Cited by 8 publications

References 22 publications

On the essential dimension of an algebraic group whose connected component is a torus

On the essential dimension of an algebraic group whose connected component is a torus

Extensions of algebraic groups with finite quotient and nonabelian 2-cohomology

Essential dimension and canonical dimension of gerbes banded by groups of multiplicative type

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