The Sign Representation for Shephard Groups
نویسندگان
چکیده
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.
منابع مشابه
The Sign Representation for Shephard Groups Peter Orlik Victor Reiner and Anne V Shepler Dedicated to Louis Solomon on His Seventieth Birthday
Shephard groups are unitary re ection groups arising as the sym metries of regular complex polytopes For a Shephard group we identify the representation carried by the principal ideal in the coinvariant algebra gener ated by the image of the product of all linear forms de ning re ecting hyper planes This representation turns out to have many equivalent guises making it analogous to the sign rep...
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We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.
متن کاملm at h . A C ] 2 4 M ay 2 00 8 EXTENDING THE COINVARIANT THEOREMS OF CHEVALLEY , SHEPHARD – TODD , MITCHELL , AND SPRINGER
We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.
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