Odd-dimensional orthogonal groups as amalgams of unitary groups.

Gramlich, Ralf; Horn, Max; Nickel, Werner

doi:10.1016/j.jalgebra.2007.03.035

Cited by 5 publications

(4 citation statements)

References 6 publications

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“…For q = 3, it implies that all residues of rank at least four are simply connected, leading to Main Theorem B. The cases n = 3 and 5 q 9 are dealt with in the second part [Part2] computationally. In the present paper we thus assume that q 11 if n = 3.…”

Section: Theoremmentioning

confidence: 99%

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Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

et al. 2007

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We extend the Phan theory described in [C. Bennett, R. Gramlich, C. Hoffman, S. Shpectorov, CurtisPhan-Tits theory, in: A.A. Ivanov, M.W. Liebeck, J. Saxl (Eds.), Groups, Combinatorics, and Geometry, World Scientific, River Edge, 2003, pp. 13-29] to the last remaining infinite series of classical Chevalley groups over finite fields. Namely, we prove that the twin buildings for the group Spin(2n + 1, q 2 ), q odd, admit a unique unitary flip and that the corresponding flipflop geometry is simply connected for almost all finite fields F q 2 . Applying standard methods from amalgam theory, this results in a characterization of central quotients of the group Spin(2n + 1, q) by a Phan system of rank one and rank two subgroups. In the present first part of a series of two articles we present simple connectedness results for sufficiently large fields or sufficiently large rank. To be precise, the result stated in the present paper is proved for all cases but n = 3 and q ∈ {3, 5, 7, 9}, the remaining cases are dealt with in the sequel [R. Gramlich, M. Horn, W. Nickel, Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 2: Machine computations, submitted for publication] computationally.

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Section: Theoremmentioning

confidence: 99%

“…For n = 3 and q = 3 there exists a counterexample to the conclusion of part (iii) of the theorem. Namely, the universal cover of is finite of degree 3 7 , see [Part2] for the details.…”

Section: Theoremmentioning

confidence: 99%

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

et al. 2007

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“…This result was later refined by Caprace [16]. Similar results on Curtis-Tits-Phan type amalgams have been obtained in [7,6,8,11,12,23,27,29,24]. For an overview of that subject see Köhl [26].…”

Section: Introductionmentioning

confidence: 64%

Curtis–Tits groups generalizing Kac–Moody groups of typeA˜n−1

Blok

Hoffman

2014

Journal of Algebra

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In [13] we define a Curtis-Tits group as a certain generalization of a Kac-Moody group. We distinguish between orientable and non-orientable Curtis-Tits groups and identify all orientable Curtis-Tits groups as Kac-Moody groups associated to twinbuildings.In the present paper we construct all orientable as well as non-orientable Curtis-Tits groups with diagram A n−1 (n ≥ 4) over a field k of size at least 4. The resulting groups are quite interesting in their own right. The orientable ones are related to Drinfeld's construction of vector bundles over a non-commutative projective line and to the classical groups over cyclic algebras. The non-orientable ones are related to expander graphs [14] and have symplectic, orthogonal and unitary groups as quotients.

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“…In Sect. 3, we analyze the amalgam of rank two parabolics for (n, q) = (3, 3) by applying methods from computer algebra, similar to those previously used by the author in [10,14] (for C n ) and [11] (for B n ; see also [4]). It is assumed throughout the paper that the reader is familiar with the concept of amalgams (refer to [17] for an introduction to the subject).…”

Section: Main Theoremmentioning

confidence: 99%

On the Phan system of the Schur cover of SU(4, 32)

Horn

2008

Des. Codes Cryptogr.

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This article is part of the program described in Bennett et al. (in: Ivanov et al. (ed.) Groups, combinatorics, and geometry, 2003) [3]. We study the Phan amalgams and their universal completions that occur for q = 3 in rank n = 3 for the diagram A 3 = D 3 , corresponding to SU(4, 3 2 ) ∼ = Spin − (6, 3). We show that the associated geometries admit universal 9-fold coverings, by showing that the universal completion of the Phan amalgam is the central extension of SU(4, 3 2 ) by its Schur multiplier. This information provides the last missing piece of information in the full classification of Phan amalgams and their universal completions for A n and D n .

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Odd-dimensional orthogonal groups as amalgams of unitary groups.

Cited by 5 publications

References 6 publications

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

Curtis–Tits groups generalizing Kac–Moody groups of typeA˜n−1

On the Phan system of the Schur cover of SU(4, 32)

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