Quadrangles of Type F4

This chapter deals with the case that the building at infinity Λ‎ of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type F₄. It begins with the hypothesis stating that Λ‎ = (K, L, q) is a quadratic space of type F₄, K is complete with respect to a discrete valuation ν‎ and F is closed with respect to ν‎, Λ‎ is the Moufang quadrangle corresponding to a root group sequence, and R₀ and R₁ as the two residues of Ξ‎. The chapter also considers the theorem supposing that Λ‎ is of type F₄ and that R₀ and R₁ are not both indifferent, and claims that both cases really occur. Finally, it presents the proposition that R₀ and R₁ are both indifferent if and only if q is totally wild.

Residues

10.23943/princeton/9780691166902.003.0010 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Euclidean Plane ◽

Complete System ◽

Quadratic Space ◽

Affine Building ◽

Root Group ◽

Tits Building ◽

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of typeF₄ over a field K that is complete with respect to a discrete valuation, and if in the F4-case the subfield F is closed with respect to this valuation and if Δ‎ is the corresponding Moufang quadrangle of type Eℓ or F₄, then there always exists a unique affine building Ξ‎ such that Δ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments. The chapter also considers the standard embedding of the apartment A in the Euclidean plane which takes the intersection of A and R to the set of eight triangles containing the origin. Finally, it describes a Moufang polygon with two root group sequences.

Unramified Quadrangles of Type E6, E7 and E8

10.23943/princeton/9780691166902.003.0011 ◽

2017 ◽

Cited By ~ 2

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Quadratic Space ◽

Root Group ◽

Tits Building ◽

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ‎, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ‎ is an unramified quadrangle if the proposition δ‎Ψ‎ = 2 holds. It also explains two other propositions: Ξ‎ is a semi-ramified quadrangle if δ‎Λ‎ = 1 and δ‎Ψ‎ = 2 holds, and a ramified quadrangle if δ‎Λ‎ = δ‎Ψ‎ = 1 holds.

Quadrangles of Type E6, E7 and E8: Summary

10.23943/princeton/9780691166902.003.0014 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Space ◽

The Third ◽

First Case ◽

Root Group ◽

Tits Building ◽

This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.

Moufang Quadrangles

10.23943/princeton/9780691166902.003.0004 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Abelian Groups ◽

Field Extension ◽

Quadratic Space ◽

Root Group ◽

This chapter proves various results about Moufang quadrangles. It first considers the notions of a proper involutory set, a proper indifferent set, and a proper anisotropic pseudo-quadratic space. It then shows that the root group sequence Ω‎ is isomorphic to a root group sequence of exactly one of six types relating to some proper involutory set, some non-trivial anisotropic quadratic space, some proper indifferent set, some proper anisotropic pseudo-quadratic space, and some quadratic space. It also describes the degree of a finite purely inseparable field extension as a power of the characteristic, an isomorphism from a root group sequence of Δ‎ to the Moufang quadrangle, and abelian and non-abelian groups.

Strictly Semi-linear Automorphisms

10.23943/princeton/9780691166902.003.0030 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Fixed Points ◽

Division Algebra ◽

Quadratic Space ◽

Spherical Building ◽

Linear Automorphism ◽

Coxeter Diagram ◽

Non Linear ◽

Root Group

This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ‎ is a Moufang spherical building and Π‎ is the Coxeter diagram of Δ‎; here the chapter fixes an apartment Σ‎ of Δ‎ and a root α‎ of Σ‎. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.

Linear Automorphisms

10.23943/princeton/9780691166902.003.0029 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Quadratic Space ◽

Spherical Building ◽

Parameter System ◽

Linear Automorphism ◽

Coxeter Diagram ◽

Root Group

This chapter considers the notion of a linear automorphism of an arbitrary spherical building satisfying the Moufang property. It begins with the notation whereby Ω‎ = (U₊, U₁, ..., Uₙ) is the root group sequence and x₁, ... , xₙ the isomorphisms obtained by applying the recipe in [60, 16.x] for x = 1, 2, 3, ... or 9 to a parameter system Λ‎ of the suitable type (and for suitable n) and Δ‎ is the corresponding Moufang n-gon. The chapter proceeds by looking at cases where Λ‎ is a proper anisotropic pseudo-quadratic space defined over an involutory set or a quadratic space of type E⁶, E₇ or E₈. It also describes a notation dealing with the Moufang spherical building with Coxeter diagram Λ‎, an apartment of Δ‎, and a chamber of Σ‎.

Semi-ramified Quadrangles of Type E6, E7 and E8

10.23943/princeton/9780691166902.003.0012 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Space ◽

Basic Proposition ◽

Tits Building ◽

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang semi-ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a semi-ramified quadrangle if δ‎Λ‎ = 1 and δ‎Ψ‎ = 2 holds. The chapter first considers the theorem supposing that ℓ = 6, that δ‎Λ‎ = 1 and δ‎Ψ‎ = 2, and that the Moufang residues R0 and R1 are not both indifferent. This is followed by cases ℓ = 7 and ℓ = 8 as well as theorems concerning an anisotropic pseudo-quadratic space, a quaternion division algebra, standard involution, a proper involutory set, and isotropic and anisotropic quadratic spaces.

Moufang Polygons

10.23943/princeton/9780691166902.003.0003 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Generalized Quadrangle ◽

Quadratic Space ◽

Root Group ◽

Basic Facts

This chapter introduces some basic facts about Moufang polygons and root group sequences. For each root group sequence Ω‎, there is a unique Moufang polygon Δ‎ such that Ω‎ is isomorphic to a root group sequence of Δ‎. The classification of Moufang n-gons states that, up to isomorphism, there are no other Moufang polygons. The chapter also considers the notion of an isomorphism of root group sequences and the notion of an anti-isomorphism of root group sequences. It concludes with an example involving a non-trivial anisotropic quadratic space and a generalized quadrangle with a root group sequence.

Residually Pseudo-Split Buildings

10.23943/princeton/9780691166902.003.0033 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Fixed Point ◽

Division Algebra ◽

Residue Field ◽

Quadratic Extension ◽

Splitting Field ◽

Quadratic Space ◽

Descent Group ◽

Tits Building ◽

This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified extension such that the residue field is a pseudo-splitting field. It then considers a wild quaternion or octonion division algebra and the existence of an unramified quadratic extension L/K such that L is a splitting field of the quaternion division algebra. It also discusses the properties of an unramified extension L/K and shows that every exceptional Bruhat-Tits building is the fixed point building of a strictly semi-linear descent group of a residually pseudo-split building.

Indices for the Exceptional Bruhat-Tits Buildings

10.23943/princeton/9780691166902.003.0036 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Capital Letter ◽

Coxeter Diagram ◽

Root Group ◽

Relative Type ◽

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.