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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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Coxeter Groups

Coxeter Groups

Chapter:
(p.143) Chapter Nineteen Coxeter Groups
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/9780691166902.003.0019

This chapter develops a theory of descent for buildings by assembling various results about Coxeter groups. It begins with the notation stating that W is an arbitrary group with a distinguished set of generators S containing only elements of order 2, with MS denoting the free monoid on the set S and l: MS → ℕ denoting the length function. It then defines a Coxeter system and an automorphism of (W, S), which is an automorphism of the group W that stabilizes the set S, suggesting that there is a canonical isomorphism from Aut (W, S) to Aut(Π‎), where Π‎ is the associated Coxeter diagram with vertex set S. The chapter concludes with the proposition: Let α‎ be a root of Σ‎ and let T be the arctic region of α‎.

Keywords:   descent, building, Coxeter group, length function, Coxeter system, automorphism, canonical isomorphism, Coxeter diagram, root, arctic region

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