| Summary: | 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.
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