- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a ∈ Local Field
- Chapter Eight Quadratic Forms of Type E6, E7 and E8
- Chapter Nine Quadratic Forms of Type F4
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E6, E7 and E8
- Chapter Twelve Semi-ramified Quadrangles of Type E6, E7 and E8
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index

# Linear Automorphisms

# Linear Automorphisms

- Chapter:
- (p.251) Chapter Twenty Nine Linear Automorphisms
- Source:
- Descent in Buildings (AM-190)
- Author(s):
### Bernhard M¨uhlherr

### Holger P. Petersson

### Richard M. Weiss

- Publisher:
- Princeton University Press

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*₊, *U*₁, ..., *U*ₙ) is the root group sequence and *x*₁, ... , *x*ₙ 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 *E*⁶, *E*₇ or *E*₈. It also describes a notation dealing with the Moufang spherical building with Coxeter diagram Λ, an apartment of Δ, and a chamber of Σ.

*Keywords:*
linear automorphism, Moufang property, root group sequence, anisotropic pseudo-quadratic space, involutory set, quadratic space, Moufang spherical building, Coxeter diagram, apartment, chamber

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- Title Pages
- Dedication
- Epigraph
- Preface
- Chapter One Buildings
- Chapter Two Quadratic Forms
- Chapter Three Moufang Polygons
- Chapter Four Moufang Quadrangles
- Chapter Five Linked Tori, I
- Chapter Six Linked Tori, II
- Chapter Seven Quadratic Forms over a ∈ Local Field
- Chapter Eight Quadratic Forms of Type E6, E7 and E8
- Chapter Nine Quadratic Forms of Type F4
- Chapter Ten Residues
- Chapter Eleven Unramified Quadrangles of Type E6, E7 and E8
- Chapter Twelve Semi-ramified Quadrangles of Type E6, E7 and E8
- Chapter Thirteen Ramified Quadrangles of Type E6, E7 and E8
- Chapter Fourteen Quadrangles of Type E6, E7 and E8: Summary
- Chapter Fifteen Totally Wild Quadratic Forms of Type E7
- Chapter Sixteen Existence
- Chapter Seventeen Quadrangles of Type F4
- Chapter Eighteen The Other Bruhat-Tits Buildings
- Chapter Nineteen Coxeter Groups
- Chapter Twenty Tits Indices
- Chapter Twenty One Parallel Residues
- Chapter Twenty Two Fixed Point Buildings
- Chapter Twenty Three Subbuildings
- Chapter Twenty Four Moufang Structures
- Chapter Twenty Five Fixed Apartments
- Chapter Twenty Six The Standard Metric
- Chapter Twenty Seven Affine Fixed Point Buildings
- Chapter Twenty Eight Pseudo-Split Buildings
- Chapter Twenty Nine Linear Automorphisms
- Chapter Thirty Strictly Semi-linear Automorphisms
- Chapter Thirty One Galois Involutions
- Chapter Thirty Two Unramified Galois Involutions
- Chapter Thirty Three Residually Pseudo-Split Buildings
- Chapter Thirty Four Forms of Residually Pseudo-Split Buildings
- Chapter Thirty Five Orthogonal Buildings
- Chapter Thirty Six Indices for the Exceptional Bruhat-Tits Buildings
- Bibliography
- Index