*Bernhard Mühlherr, Holger P. Petersson, and Richard M. Weiss*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a ...
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This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with The Structure of Spherical Buildings and The Structure of Affine Buildings.Less

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with *The Structure of Spherical Buildings* and *The Structure of Affine Buildings*.

*Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0023
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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. ...
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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.Less

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 𝐓T_{Π}. 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.

*Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0028
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces a class of Moufang spherical buildings known as pseudo-split buildings and considers the notion of the field of definition of a spherical building satisfying the Moufang ...
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This chapter introduces a class of Moufang spherical buildings known as pseudo-split buildings and considers the notion of the field of definition of a spherical building satisfying the Moufang condition. It begins with the notation: Let Δ be an irreducible spherical building satisfying the Moufang condition, and let ℓ denote its rank (so ℓ is greater than or equal to 2 by definition). It then characterizes pseudo-split buildings as the spherical buildings which can be embedded in a split building of the same type. It also presents the proposition stating that every pseudo-split building is a subbuilding of a split building.Less

This chapter introduces a class of Moufang spherical buildings known as pseudo-split buildings and considers the notion of the field of definition of a spherical building satisfying the Moufang condition. It begins with the notation: Let Δ be an irreducible spherical building satisfying the Moufang condition, and let ℓ denote its rank (so ℓ is greater than or equal to 2 by definition). It then characterizes pseudo-split buildings as the spherical buildings which can be embedded in a split building of the same type. It also presents the proposition stating that every pseudo-split building is a subbuilding of a split building.