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The Hessian map in the invariant theory of reflection groups

Published online by Cambridge University Press:  22 January 2016

Peter Orlik  and
Louis Solomon
Peter Orlik
Affiliation:
University of Wisconsin, Madison, WI, 53706, U.S.A.
Louis Solomon
Affiliation:
University of Wisconsin, Madison, WI, 53706, U.S.A.
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Let V be a complex vector space of dimension l. Let S be the C-algebra of polynomial functions on V. Let Ders be the S-module of derivations of S and let Ωs = Homs (Ders, S) be the dual S-module of differential 1-forms. Let {ei} be a basis for V and let {xi} be the dual basis for V. Then {Di = ∂/∂xi and {dxi} are bases for Ders and Ωs as S-modules. If f ∈ S, define a map Hess (f): Ders → Ωs by

Then Hess (f) is an S-module homomorphism which does not depend on the choice of basis for V.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 109 , March 1988 , pp. 1 - 21
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

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