\title[The sign representation for Shephard groups]
{The sign representation for Shephard groups}


\author[Peter Orlik]{Peter Orlik}
\address{Dept.\  of Mathematics\\
         University of Wisconsin\\
         Madison, WI 53706}

\author[Victor Reiner]{Victor Reiner}
\address{School of Mathematics\\
         University of Minnesota\\
         Minneapolis, MN 55455}

\author[Anne V. Shepler]{Anne V. Shepler}
\address{Dept.\ of Mathematics\\
         University of California at Santa Cruz\\
         Santa Cruz, CA 96054}


\keywords{Coxeter group, unitary reflection group, Shephard group,
regular complex polytope, arrangement of hyperplanes, Milnor fiber}

\thanks{Work of second author partially supported by
NSF grant DMS-9877047.  Work of third author partially supported by
NSF grant DMS-9971099.\\


\begin{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.

\end{abstract}