The sign representation for shephard groups

Peter Orlik, Victor Reiner, Anne V. Shepler

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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.

Original languageEnglish (US)
Pages (from-to)477-492
Number of pages16
JournalMathematische Annalen
Volume322
Issue number3
DOIs
StatePublished - Dec 1 2002

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    Orlik, P., Reiner, V., & Shepler, A. V. (2002). The sign representation for shephard groups. Mathematische Annalen, 322(3), 477-492. https://doi.org/10.1007/s002080200001