The maximal subgroups of positive dimension in exceptional algebraic groups

Liebeck, Martin W.; Seitz, Gary M.

doi:10.1090/memo/0802

Cited by 72 publications

(96 citation statements)

References 42 publications

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“…To apply this, a painstaking analysis of the function ζ M (s) is required, leading to a bound on ζ M (1) in terms of the index of M in G (see Theorem 5.1 below). This bound is combined with recent results on maximal subgroups [26] to complete the proof of Theorem 1.6. Theorem 1.6 is new even for surface groups, where it takes the following form.…”

Section: ])mentioning

confidence: 82%

“…For the remaining types, the result follows from [26,Corollary 4]. Note that the log log q term comes from the subfield subgroups G(q 1/r ), where r is a prime divisor of log p q.…”

Section: Proof Of Theorem 51mentioning

confidence: 90%

“…For this case we require the following recent result on maximal subgroups of G taken from [26]. Lemma 6.3 Let G = G(q) be an exceptional simple group of Lie type over F q , and let M be a maximal subgroup of G. Then there are absolute constants c 1 , c 2 such that one of the following holds:…”

Section: Proof Of Theorem 51mentioning

confidence: 99%

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Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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Let Γ be a Fuchsian group of genus at least 2 (at least 3 if Γ is non-oriented). We study the spaces of homomorphisms from Γ to finite simple groups G, and derive a number of applications concerning random generation and representation varieties.Precise asymptotic estimates for |Hom(Γ, G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G| μ(Γ)+1+o(1) , where μ(Γ) is the measure of Γ. We then prove that a randomly chosen homomorphism from Γ to G is surjective with probability tending to 1 as |G| → ∞. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties Hom(Γ,Ḡ), whereḠ is GL n (K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic.A key ingredient of our approach is character theory, involving the study of the 'zeta function' ζ G (s) = χ(1) −s , where the sum is over all irreducible complex characters χ of G.

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Section: ])mentioning

confidence: 82%

“…For the remaining types, the result follows from [26,Corollary 4]. Note that the log log q term comes from the subfield subgroups G(q 1/r ), where r is a prime divisor of log p q.…”

Section: Proof Of Theorem 51mentioning

confidence: 90%

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Fuchsian groups, finite simple groups and representation varieties

Liebeck

Shalev

2005

Invent. math.

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“…In the proof of Corollary 3 we need the classification of maximal subgroups of the algebraic group G ¼ G 2 ðKÞ, from [14]. (ii) a subsystem subgroup of maximal rank;…”

Section: Now Definementioning

confidence: 99%

The reductive subgroups of G 2

Stewart¹

2010

Journal of Group Theory

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Abstract. Let G :¼ G 2 ðKÞ be a simple algebraic group of type G 2 defined over an algebraically closed field K of characteristic p > 0. Let s denote a standard Frobenius automorphism of G such that G s G G 2 ðqÞ with q d 4. In this paper we find all reductive subgroups of G and quasisimple subgroups of G s in the defining characteristic. Our results extend the complete reducibility results of [13, Theorem 1].

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“…This was achieved in [Sei87] for the classical algebraic groups and, under some fairly mild restrictions on the characteristic p of k in [Sei91] for the exceptional algebraic groups. Later, in work by Liebeck and Seitz [LS04], the latter classification (for exceptional algebraic groups) was completed to cover all characteristics and extended to all maximal, closed, positive dimensional subgroups (not necessarily connected). All these positive characteristic results rely on work of Donna Testerman [Tes88], [Tes89], [Tes92] which, particularly, classify and construct subgroups of type A 1 .…”

Section: Introductionmentioning

confidence: 99%

Maximal subalgebras of Cartan type in the exceptional Lie algebras

Herpel

Stewart

2015

Sel. Math. New Ser.

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Abstract. In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for g. In this paper we deal with what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of g and give an explicit classification of when it is maximal in g.

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The maximal subgroups of positive dimension in exceptional algebraic groups

Cited by 72 publications

References 42 publications

Fuchsian groups, finite simple groups and representation varieties

Fuchsian groups, finite simple groups and representation varieties

The reductive subgroups of G 2

Maximal subalgebras of Cartan type in the exceptional Lie algebras

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