The Sign Representation for Shephard Groups
Group Theory
2007-05-23 v1 Combinatorics
Geometric Topology
Representation Theory
Abstract
Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representation turns out to have many equivalent guises making it analogous to the sign representation of a finite Coxeter group. One of these guises is (up to a twist) the cohomology of the Milnor fiber for the isolated singularity at 0 in the hypersurface defined by any homogeneous invariant of minimal degree.
Cite
@article{arxiv.math/0011105,
title = {The Sign Representation for Shephard Groups},
author = {Peter Orlik and Victor Reiner and Anne V. Shepler},
journal= {arXiv preprint arXiv:math/0011105},
year = {2007}
}
Comments
Submitted. 13 pages