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The Weyl group of a root system is a subgroup generated by reflections through the hyperplanes orthogonal to the roots.
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Weyl groups are Coxeter groups proof
I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed c …