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Alcoves corresponding to an affine Weyl group. (English) Zbl 0681.20032

Summary: In this paper, I study the alcoves of a Euclidean space \(E\) corresponding to an affine Weyl group \(W_a\). I give the coordinate form of an alcove of \(E\) and establish an explicit correspondence between the elements of \(W_a\) and the alcoves of \(E\). In particular, I characterize an alcove by a \(\Phi\)-tuple over \(\mathbb{Z}\) subject to certain conditions, where \(\Phi\) is the root system determined by \(W_a\).

MSC:

20H15 Other geometric groups, including crystallographic groups
20G15 Linear algebraic groups over arbitrary fields
20G05 Representation theory for linear algebraic groups
17B20 Simple, semisimple, reductive (super)algebras
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