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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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Subbuildings

Subbuildings

Chapter:
(p.195) Chapter Twenty Three Subbuildings
Source:
Descent in Buildings (AM-190)
Author(s):

Bernhard M¨uhlherr

Holger P. Petersson

Richard M. Weiss

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691166902.003.0023

This chapter presents a generalization of the typical notion of a subbuilding. It begins with the notation where Δ‎ is a building of type Π‎, and 𝐓 = (Π‎, Θ‎, A) is a Tits index of absolute type Π‎. A subbuilding of split type is a subbuilding of type 𝐓TΠ‎. The term “subbuilding” is used to refer to a subbuilding of split type of a residue. The chapter then considers apartments of a building that are classified as subbuildings, along with the notion of a thin 𝐓-building and a proposition in which 𝐓-buildings of Σ‎ are 𝐓-apartments. Finally, it describes a proposition where a Γ‎-residue of Δ‎ contains chambers of Σ‎.

Keywords:   subbuilding, Tits index, subbuilding of split type, residue, apartment, thin T-building, chamber

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