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Descent in Buildings (AM-190)$
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Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss

Print publication date: 2015

Print ISBN-13: 9780691166902

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691166902.001.0001

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Galois Involutions

Galois Involutions

(p.271) Chapter Thirty One Galois Involutions
Descent in Buildings (AM-190)

Bernhard M¨uhlherr

Holger P. Petersson

Richard M. Weiss

Princeton University Press

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe definition of a spherical building satisfying the Moufang condition and a Galois involution of Δ‎, described as an automorphism of Δ‎ of order 2 that is strictly semi-linear. It can be recalled that Δ‎ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. The chapter assumes that the building Δ‎ being discussed is as in 30.1 and that τ‎ is a Galois involution of Δ‎. It also considers the notation stating that the polar region of a root α‎ of Δ‎ is the unique residue of Δ‎ containing the arctic region of α‎.

Keywords:   fixed point, semi-linear automorphism, spherical building, Moufang condition, Galois involution, Coxeter diagram, polar region, root, residue, arctic region

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