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The \(\widehat W\)-orbit of \(\rho\), Kostant’s formula for powers of the Euler product and affine Weyl groups as permutations of \(\mathbb Z\). (English) Zbl 1160.17007

Summary: Let an affine Weyl group \(\widehat W\) act as a group of affine transformations on a real vector space \(V\). We analyze the \(\widehat W\)-orbit of a regular element in \(V\) and deduce applications to Kostant’s formula for powers of the Euler product and to the representations of \(\widehat W\) as permutations of the integers.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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