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/9780691166902.003.0036
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits ...
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This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits indices, the canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram, and Moufang sets. It then presents a proposition about an involutory set, a quaternion division algebra, a root group sequence, and standard involution. It also describes Θ-orbits in S which are disjoint from A and which correspond to the vertices of the Coxeter diagram of Ξ and hence to the types of the panels of Ξ. Finally, it shows how it is possible in many cases to determine properties of the Moufang set and the Tits index for all exceptional Bruhat-Tits buildings of type other than Latin Capital Letter G with Tilde₂.Less
This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits indices, the canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram, and Moufang sets. It then presents a proposition about an involutory set, a quaternion division algebra, a root group sequence, and standard involution. It also describes Θ-orbits in S which are disjoint from A and which correspond to the vertices of the Coxeter diagram of Ξ and hence to the types of the panels of Ξ. Finally, it shows how it is possible in many cases to determine properties of the Moufang set and the Tits index for all exceptional Bruhat-Tits buildings of type other than Latin Capital Letter G with Tilde₂.
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/9780691166902.003.0034
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter deals with forms of residually pseudo-split buildings. The proof rests on the fact that in every case, there is a Galois action of Γ := GalL/K on ΔL whose fixed point building is ...
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This chapter deals with forms of residually pseudo-split buildings. The proof rests on the fact that in every case, there is a Galois action of Γ := GalL/K on ΔL whose fixed point building is isomorphic to Δ. A Tits index = (Π, Θ, A) is displayed by drawing the Coxeter diagram, bending edges where necessary so that vertices in the same Θ-orbit are conspicuously near to each other, and putting a circle around the set of vertices in each orbit of Θ disjoint from A. The chapter presents the main result showing that every exceptional Bruhat-Tits building of rank at least 3 but not of type G˜2 with Tilde₂ is the fixed point building of an unramified group of order 2 or 4 acting on a residually pseudo-split building.Less
This chapter deals with forms of residually pseudo-split buildings. The proof rests on the fact that in every case, there is a Galois action of Γ := GalL/K on ΔL whose fixed point building is isomorphic to Δ. A Tits index = (Π, Θ, A) is displayed by drawing the Coxeter diagram, bending edges where necessary so that vertices in the same Θ-orbit are conspicuously near to each other, and putting a circle around the set of vertices in each orbit of Θ disjoint from A. The chapter presents the main result showing that every exceptional Bruhat-Tits building of rank at least 3 but not of type G˜2 with Tilde₂ is the fixed point building of an unramified group of order 2 or 4 acting on a residually pseudo-split building.
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/9780691166902.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or ...
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This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.Less
This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.
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 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.
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/9780691166902.003.0035
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of ...
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This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of positive dimension, (K, V, q) is a regular quadratic space of positive Witt index, S is the vertex set of the Coxeter diagram, (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n greater than or equal to 3, S is the vertex set of the Coxeter diagram for some n greater than or equal to 3, and Dn.l,script small l is the Tits index of absolute type Dn for n greater than or equal to 3. The chapter also considers propositions dealing with regular quadratic spaces and hyperbolic quadratic spaces.Less
This chapter presents a few results about certain forms of orthogonal buildings. It begins with notations stating that V is a K-vector space of positive dimension, (K, V, q) is a quadratic space of positive dimension, (K, V, q) is a regular quadratic space of positive Witt index, S is the vertex set of the Coxeter diagram, (K, V, q) is a hyperbolic quadratic space of dimension 2n for some n greater than or equal to 3, S is the vertex set of the Coxeter diagram for some n greater than or equal to 3, and Dn.l,script small l is the Tits index of absolute type Dn for n greater than or equal to 3. The chapter also considers propositions dealing with regular quadratic spaces and hyperbolic quadratic spaces.
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/9780691166902.003.0027
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter shows that if Ξ is an affine building and Γ is a finite descent group of Ξ, then Γ is a descent group of Ξ∞ and (Ξ∞) is congruent to (Ξ∞). ΞΓ and Ξ can be viewed as metric ...
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This chapter shows that if Ξ is an affine building and Γ is a finite descent group of Ξ, then Γ is a descent group of Ξ∞ and (Ξ∞) is congruent to (Ξ∞). ΞΓ and Ξ can be viewed as metric spaces. The chapter first considers the assumptions that Π is an irreducible affine Coxeter diagram, Ξ is a thick building of type Ξ, Γis a finite descent group of Ξ, and Tits index �� = (Π, Θ, A). It then describes apartments that are endowed with reflection hyperplanes and reflection half-spaces before concluding with a theorem about a canonical isomorphism from the fixed point building ΞΓ to (ΞΓ).Less
This chapter shows that if Ξ is an affine building and Γ is a finite descent group of Ξ, then Γ is a descent group of Ξ∞ and (Ξ∞) is congruent to (Ξ∞). ΞΓ and Ξ can be viewed as metric spaces. The chapter first considers the assumptions that Π is an irreducible affine Coxeter diagram, Ξ is a thick building of type Ξ, Γis a finite descent group of Ξ, and Tits index �� = (Π, Θ, A). It then describes apartments that are endowed with reflection hyperplanes and reflection half-spaces before concluding with a theorem about a canonical isomorphism from the fixed point building ΞΓ to (ΞΓ).
