Fixed Point Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Fundamental Theorem ◽  
Coxeter System ◽  
Coxeter Diagram ◽  
Vertex Set ◽  
Descent Group

This chapter presents the proof for the Fundamental Theorem of Descent in buildings: that if Γ‎ is a descent group, the set of residues of a building Δ‎ that are stabilized by a subgroup Γ‎ of Aut(Γ‎) forms a thick building. It begins with the hypothesis: Let Π‎ be an arbitrary Coxeter diagram, let S be the vertex set of Π‎ and let (W, S) be the corresponding Coxeter system. It then defines a Γ‎-residue and a Γ‎-chamber as well as a descent group of Δ‎ before concluding with the main result about the fixed point building of Γ‎.

Affine Fixed Point Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Canonical Isomorphism ◽  
Affine Building ◽  
Coxeter Diagram ◽  
Descent Group

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).

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

Tits Indices

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Coxeter Group ◽  
Coxeter System ◽  
Relative Rank ◽  
Coxeter Diagram ◽  
The Absolute ◽  
Relative Type

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.

Orthogonal Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Vector Space ◽  
Quadratic Space ◽  
Positive Dimension ◽  
Witt Index ◽  
Coxeter Diagram ◽  
Vertex Set ◽  
Dimension 2

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.

Unramified Galois Involutions

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Complete System ◽  
Canonical Isomorphism ◽  
Descent Group ◽  
Tits Building

This chapter describes the fixed point building of an automorphism of a Bruhat-Tits building Ξ‎ which induces an unramified Galois involution on the building at infinity Ξ‎∞. An element of G (for example, a Galois involution of Δ‎) is unramified if the subgroup of G it generates is unramified. Before presenting the main result, the chapter presents the notation stating that Δ‎ = Ξ‎∞ is the building at infinity of Ξ‎ with respect to its complete system of apartments and G = Aut(Δ‎), followed by definitions. The central theorem shows how an unramified Galois involution of Δ‎ is obtained. Here Γ‎ := τ‎ is a descent group of both Δ‎ and Ξ‎, there is a canonical isomorphism from Δ‎Γ‎ to (Ξ‎Γ‎), where Ξ‎Γ‎ and Ξ‎Γ‎ are the fixed point buildings.

Forms of Residually Pseudo-Split Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Galois Action ◽  
Coxeter Diagram ◽  
Tits Building

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.

The Other Bruhat-Tits Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
The Other ◽  
Quadratic Space ◽  
Coxeter Diagram ◽  
Vertex Set

This chapter summarizes the results about the residues of Bruhat-Tits buildings other than those associated with the exceptional Moufang quadrangles examined in previous chapters. It first considers cases, for which it assumes that Λ‎ is complete with respect to a discrete valuation in an appropriate sense. It then presents the Coxeter diagram of Ξ‎ and the vertex set S of such diagram, along with the J-residue of the building Ξ‎, which is called a gem if J is the complement in the S of a special vertex. The chapter also discusses the structure of the gems of Ξ‎ as well as cases in which the pseudo-quadratic space is defined to be ramified or unramified.

scholarly journals Generating infinite digraphs by derangements

2020 ◽  
Author(s):  
Daniel Horsley ◽  
Moharram Iradmusa ◽  
Cheryl E Praeger
Keyword(s):  
Fixed Point ◽  
Undirected Graph ◽  
Finite Sets ◽  
Generation Property ◽  
Vertex Set ◽  
De Bruijn

Abstract A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set V generates a digraph with vertex set V and arcs $(x,x^{\,\sigma})$ for x ∈ V and $\sigma\in\mathcal{S}$. We address the problem of characterizing those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in an earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined by properties of its finite sub-digraphs. The derangement generation property for a digraph is linked with the existence of a finite 1-factor cover for an associated bipartite (undirected) graph.

A Brouwer Type Coincidence Theorem and the Fundamental Theorem of Algebra

1984 ◽  
Vol 27 (4) ◽  
pp. 478-480 ◽  
Author(s):  
M. M. Dodson
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Brouwer’S Fixed Point Theorem ◽  
Coincidence Theorem ◽  
Fundamental Theorem Of Algebra ◽  
Natural Generalisation ◽  
Brouwer's Fixed Point Theorem

AbstractIt is shown that a coincidence theorem which is a natural generalisation of Brouwer's fixed point theorem also gives a short and simple proof of the fundamental theorem of algebra.

Digraphs associated with the kth power map on the quotient ring of polynomials over finite fields

2019 ◽  
Vol 18 (11) ◽  
pp. 1950218
Author(s):  
Amplify Sawkmie ◽  
Madan Mohan Singh
Keyword(s):  
Fixed Point ◽  
Finite Field ◽  
Quotient Ring ◽  
Monic Polynomial ◽  
Directed Edge ◽  
Polynomials Over Finite Fields ◽  
Vertex Set ◽  
Ring Of Polynomials ◽  
Polynomial Formula ◽  
Symmetric Property

Let [Formula: see text] be a monic polynomial over a finite field [Formula: see text], and [Formula: see text] an integer. A digraph [Formula: see text] is one whose vertex set is [Formula: see text] and for which there is a directed edge from a polynomial [Formula: see text] to [Formula: see text] if [Formula: see text] in [Formula: see text]. If [Formula: see text], where the [Formula: see text]’s are distinct monic irreducible polynomials over [Formula: see text], then [Formula: see text] can be factorized as [Formula: see text]. In this work, we investigate the structure of these power digraphs. The semiregularity property is examined, and its relationship with the symmetric property is established. In addition, we look into the uniqueness of factorization of trees attached to a fixed point.

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