Related papers: Notes on completely reducible subcomplexes of sphe…
We generalize a result of Serre's to show that if every vertex of some fixed type of a convex subcomplex of an irreducible spherical building has an opposite, then the subcomplex is completely reducible.
Let B be a thick spherical building equipped with its natural CAT(1) metric and let M be a proper, convex subset of B. If M is open or if M is a closed ball of radius pi/2, then the maximal subcomplex supported by the complement of M is…
Completely reducible subcomplexes of spherical buildings was defined by J.P. Serre and are used in studying subgroups of reductive algebraic groups. We begin the study of completely reducible subcomplexes of twin buildings and how they may…
We prove that two dimensional convex subsets of spherical buildings are either buildings or have a center.
In this paper we study convex subcomplexes of spherical buildings. We pay special attention to fixed point sets of type-preserving isometries of spherical buildings. This sets are also convex subcomplexes of the natural polyhedral structure…
We provide the spherical systems of the wonderful reductive subgroups of any reductive group.
We prove that a convex subcomplex of a spherical building of type F4 or E6 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Our approach is differential-geometric and based on the theory of metric spaces with…
A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural…
In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--R\"{o}hrle, the usual notion of $G$-complete reducibility can be…
We study convex subsets of buildings, discuss some structural features and derive several characterizations of buildings.
This is a complete classification of the complex forms of quaternionic symmetric spaces
We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.
We prove equivalence of certain axiom sets for affine buildings. Along the lines a purely combinatorial proof of the existence of a spherical building at infinity is given. As a corollary we obtain that ``being an affine building'' is…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
In this article we describe the general and special projectivity groups for all irreducible residues of all thick, irreducible, spherical buildings of type $ \mathsf{B_{n}}$, $ \mathsf{C_{n}}$ and $\mathsf{F_4}$, and rank at least 3. This…
In this paper we classify the the epimorphisms of irreducible spherical Moufang buildings (of rank at least 2) defined over a field. As an application we characterize indecomposable epimorphisms of these buildings as those epimorphisms…
We classify thick irreducible 3-spherical twin buildings of rank at least 3 in which every panel contains at least 6 chambers. Together with the Main result of [11] we obtain a classification of thick irreducible 3-spherical twin buildings.
We give a generalization of subspace codes by means of codes of modules over finite commutative chain rings. We define a new class of Sperner codes and use results from extremal combinatorics to prove the optimality of such codes in…
Let $\Delta$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $\Delta$ can be given the structure of a topological building that is compact and…
In this paper we show that a convex subcomplex of a spherical building of type E6, E7 or E8 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Together with previous results of M\"uhlherr-Tits, and Leeb and the…