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By Brian Lehmann

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This classification is well-known; the set of all finite Coxeter groups is composed of four infinite classes along with six additional exceptional groups (or seven, depending on how you count). Thus, every spherical building will have a type that falls into one of these classes. The problem can be reduced further in the following manner. Much of the information about the structure of a building is contained in the different J-residues for |J| = 2. In fact, by including a labeling scheme on the chambers, it is possible in many cases to collate all of the important information about the building into a foundation consisting of these J-residues and the labeling information.

This classification is well-known; the set of all finite Coxeter groups is composed of four infinite classes along with six additional exceptional groups (or seven, depending on how you count). Thus, every spherical building will have a type that falls into one of these classes. The problem can be reduced further in the following manner. Much of the information about the structure of a building is contained in the different J-residues for |J| = 2. In fact, by including a labeling scheme on the chambers, it is possible in many cases to collate all of the important information about the building into a foundation consisting of these J-residues and the labeling information.

This theorem describes all spherical buildings whose Coxeter groups have rank at least 3. Of course, the natural first step is to classify finite Coxeter groups. This classification is well-known; the set of all finite Coxeter groups is composed of four infinite classes along with six additional exceptional groups (or seven, depending on how you count). Thus, every spherical building will have a type that falls into one of these classes. The problem can be reduced further in the following manner.

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