Book contents
- Frontmatter
- Contents
- Preface
- List of talks
- List of participants
- Representations of groups on finite simplicial complexes
- Coxeter groups and matroids
- Finite groups and geometries: A view on the present state and on the future
- Groups acting simply transitively on the vertices of a building of type Ãn
- Finite simple subgroups of semisimple complex Lie groups – a survey
- Flag-transitive extensions of buildings of type G2 and C3*
- Disconnected linear groups and restrictions of representations
- Products of conjugacy classes in algebraic groups and generators of dense subgroups
- Monodromy groups of polynomials
- Subgroups of exceptional algebraic groups
- The geometry of traces in Ree octagons
- Small rank exceptional Hurwitz groups
- The direct sum problem for chamber systems
- Embeddings and hyperplanes of Lie incidence geometries
- Intermediate subgroups in Chevalley groups
- On a certain class of Frattini extensions of finite Chevalley groups
- Economical generating sets for finite simple groups
The direct sum problem for chamber systems
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Contents
- Preface
- List of talks
- List of participants
- Representations of groups on finite simplicial complexes
- Coxeter groups and matroids
- Finite groups and geometries: A view on the present state and on the future
- Groups acting simply transitively on the vertices of a building of type Ãn
- Finite simple subgroups of semisimple complex Lie groups – a survey
- Flag-transitive extensions of buildings of type G2 and C3*
- Disconnected linear groups and restrictions of representations
- Products of conjugacy classes in algebraic groups and generators of dense subgroups
- Monodromy groups of polynomials
- Subgroups of exceptional algebraic groups
- The geometry of traces in Ree octagons
- Small rank exceptional Hurwitz groups
- The direct sum problem for chamber systems
- Embeddings and hyperplanes of Lie incidence geometries
- Intermediate subgroups in Chevalley groups
- On a certain class of Frattini extensions of finite Chevalley groups
- Economical generating sets for finite simple groups
Summary
Introduction
The Direct Sum Theorem for geometries states that a geometry belonging to a disconnected diagram is the direct sum of subgeometries corresponding to the connected components of that diagram. No analogous statement holds for chamber systems in general.
This situation has some uncomfortable consequences. For instance, we cannot reduce a classification problem for a class of chamber systems to cases with connected diagram, except when we have previously proved that the Direct Sum Theorem holds for the chamber systems of that class. Or, if we apply the celebrated criterion by Tits on rank 3 residues of spherical type to see if a given chamber system C belonging to a Coxeter diagram is covered by a building, we should check if the residues of C corresponding to disconnected rank 3 subdiagrams split as direct sums of subsystems of rank 1 or 2.
Unfortunately, some of the authors who have written on chamber systems seem to have been not awared of these problems. It would be stupid making a list of those who occasionaly said something wrong because of this oversight. I am not going to do that. Rather, I want to show that this situation is not really so bad as it might look. To support my optimistic opinion, I will show that in some important cases the counterexamples to the statement of the Direct Sum Theorem are quite sporadic, so that things can be kept under control in those cases.
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- Groups of Lie Type and their Geometries , pp. 185 - 214Publisher: Cambridge University PressPrint publication year: 1995