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Buildings of Spherical Type and Finite BN-Pairs pp 60–74Cite as

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Reduction

Reduction

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Part of the Lecture Notes in Mathematics book series (LNM,volume 386)

Abstract

In this section 4, Δ and Δ′ denote two buildings of rank at least 3, whose Weyl complexes are finite and irreducible. For A ∈ Δ or Δ′, and i ∈ \( \mathop N\limits_ = \) such that i ≥ codim A, we denote by Ei(A) the set of all chambers having a face of codimension i in common with A. The purpose of this section is to establish the following two theorems:

  1. 4.1.1.

    THEOREM. Let Σ ⊂ Δ be an apartment and C ∈ Σ be a chamber. Then, any isomorphism ϕ : Δ → Δ′ is entirely determined by its restriction to E1(C) ∪ Σ.

  2. 4.1.2.

    THEOREM. Let C ∈ Δ and C′ ∈ Δ′ be chambers. Then, every adjacence-preserving bijection ϕ : E2(C) → E2(C′) extends to an isomorphism of Δ onto Δ′.

Keywords

  • Convex Hull
  • Induction Hypothesis
  • Admissible Pair
  • Coherence Relation
  • Unicity Assumption

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1974 Springer-Verlag Berlin Heidelberg

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(1974). Reduction. In: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol 386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38349-9_4

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  • DOI: https://doi.org/10.1007/978-3-540-38349-9_4

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