Simple groups of Lie type

Carter, Roger W.; MacDonald, Ian G.; Segal, Graeme; Taylor, Michael

doi:10.1017/cbo9781139172882.006

Cited by 348 publications

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“…Let L be a simple complex Lie algebra of type F 4 and let {h r |r ∈ Δ} ∪ {e r |r ∈ Φ} be a Chevalley basis of L (in the sense of [2,Theorem 4…”

Section: Notation and Group-theoretical Properties Of 2 F 4 (Q 2 )mentioning

confidence: 99%

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Character Table of a Borel Subgroup of the Ree Groups ²F₄(q²)

Himstedt¹,

Huang²

2009

LMS J. Comput. Math.

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We compute the conjugacy classes and character table of a Borel subgroup of the Ree groups 2 F 4 (2 2n+1 ) for all n 1 and prove that these Borel subgroups are M -groups. We determine the degrees of the irreducible characters of the Sylow-2-subgroups of 2 F 4 (2 2n+1 ) and show that the IsaacsMalle-Navarro version of the McKay conjecture holds for 2 F 4 (2 2n+1 ) in characteristic 2. For most of the calculations we use CHEVIE.

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“…Let L be a simple complex Lie algebra of type F 4 and let {h r |r ∈ Δ} ∪ {e r |r ∈ Φ} be a Chevalley basis of L (in the sense of [2,Theorem 4…”

Section: Notation and Group-theoretical Properties Of 2 F 4 (Q 2 )mentioning

confidence: 99%

“…For z 1 , z 2 ∈ F × q 2 , u ∈ F q 2 , we have: For r ∈ Φ, let n r be the element in N := N G (T), given by [2,Lemma 6.4.4]. Then N is generated by T and the elements n r for r ∈ Δ.…”

Section: 21])mentioning

confidence: 99%

Character Table of a Borel Subgroup of the Ree Groups ²F₄(q²)

Himstedt¹,

Huang²

2009

LMS J. Comput. Math.

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“…In this section we describe the basic notation and some requisite facts concerning Chevalley groups; see [3,5,34,35,43].…”

Section: §2 Chevalley Groupsmentioning

confidence: 99%

“…Thus, for any ε ∈ K * we can define a K-character χ = χ ω,ε of the root lattice Q(Φ) by χ ω,ε (α) = ε (α,ω) . Now we can consider the diagonal automorphism associated with this character (see [34,43]). We denote by h ω (ε) an element conjugation by which realizes this diagonal automorphism.…”

Section: Introductionmentioning

confidence: 99%

“…Let B = U T be the standard Borel subgroup of the extended 1 simply connected Chevalley group G = G sc (Φ, K). Next, let N = N (Φ, K) be the subgroup of G generated by T and all w α (1), α ∈ Φ; see [3,36,43]. Almost always N coincides with the normalizer of T in G. The quotient group N /T is isomorphic to the Weyl group W = W (Φ) of the root system Φ.…”

Section: Introductionmentioning

confidence: 99%

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Weight elements of Chevalley groups

Vavilov

2008

St. Petersburg Math. J.

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Abstract. The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field -the weight elements. These are the conjugates of certain semisimple elements h ω (ε) of extended Chevalley groups G = G(Φ, K), where ω is a weight of the dual root system Φ ∨ and ε ∈ K * . In the adjoint case the h ω (ε)'s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of h ω (ε) are called weight elements of type ω. Various constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given x ∈ G all elements x(ε) = xh ω (ε)x −1 , ε ∈ K * , apart maybe from a finite number of them, lie in the same Bruhat coset BwB, where w is an involution of the Weyl group W = W (Φ). The elements h ω (ε) are particularly important when ω = i is a microweight of Φ ∨ . The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements x(ε) for the case where ω = i . It turns out that all nontrivial x(ε)'s lie in the same Bruhat coset BwB, where w is a product of reflections in pairwise strictly orthogonal roots γ 1 , . . . , γ r+s . Moreover, if among these roots r are long and s are short, then r + 2s does not exceed the width of the unipotent radical of the ith maximal parabolic subgroup in G. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Röhrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

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