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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Outer automorphisms of adjoint groups of type $\mathsf {D}$ and nonrational adjoint groups of outer type $\mathsf {A}$

Authors: Demba Barry and Jean-Pierre Tignol
Journal: Trans. Amer. Math. Soc. 372 (2019), 2613-2630
MSC (2010): Primary 11E57; Secondary 20G15
Published electronically: May 7, 2019
MathSciNet review: 3988587
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Abstract: For a classical group $G$ of type $\mathsf {D}_n$ over a field $k$ of characteristic different from $2$, we show the existence of a finitely generated regular extension of $k$ over which $G$ admits outer automorphisms. Using this result and a construction of groups of type $\mathsf {A}$ from groups of type $\mathsf {D}$, we construct new examples of groups of type $^2\mathsf {A}_n$ with $n\equiv 3\bmod 4$ and the first examples of type $^2\mathsf {A}_n$ with $n\equiv 1\bmod 4$ $(n\geq 5)$ that are not $R$-trivial, hence not rational (nor stably rational).

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Demba Barry
Affiliation: Faculté des Sciences et Techniques de Bamako, BP: E3206 Bamako, Mali; and Departement Wiskunde–Informatica, Universiteit Antwerpen, 2000 Antwerpen, Belgium
MR Author ID: 1054810

Jean-Pierre Tignol
Affiliation: Université catholique de Louvain, ICTEAM Institute, Avenue G. Lemaître 4, Box L4.05.01, B-1348 Louvain-la-Neuve, Belgium
MR Author ID: 172580

Received by editor(s): January 21, 2018
Received by editor(s) in revised form: June 4, 2018
Published electronically: May 7, 2019
Additional Notes: The first author gratefully acknowledges support from the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory).
The second author acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grant no. J.0149.17
Article copyright: © Copyright 2019 American Mathematical Society