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By Brian Lehmann

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Much of the theory of buildings centers around classification theorems, and the analysis we have done is adequate preparation to start work in this direction. The most fundamental classification theorem is that of spherical buildings. 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).

By using the classification of spherical buildings, we can again reduce the problem of classifying affine buildings to a more manageable question to one about root groups and labeling schemes. These classification theorems come in useful for working with Lie groups or manifolds: if we can construct a building based on the object in question, our classification theorem will help us determine its properties. As the close link between buildings and BN-pairs suggests, the theory of buildings can be used to derive many more useful theorems in group theory.

Furthermore, we can choose b so that all diagonal entries are 1 and all other entries in b are 0. Then, the resulting matrix wb w−1 will have the required form. Thus, the final condition of BN-pairs holds, and (B, N ) is indeed a BN-pair for the general linear group. This demonstrates that GLn has two BN-pairs that are essentially equivalent. Areas for Further Study Having developed a basic theory of the structure of buildings, we now discuss briefly several possibilities for further study. Much of the theory of buildings centers around classification theorems, and the analysis we have done is adequate preparation to start work in this direction.

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