Reflection groups acting on their hyperplanes

Details Reflection groups acting on their hyperplanes Reflection groups acting on their hyperplanes

2008-09-02

arxiv.org/abs/0809.0384v1

Mathematics

Representation Theory

15 pages

Scientific paper

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family.

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