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Discriminants 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 and let G ⊂ GL(V) be a finite reflection group. Let S be the C-algebra of polynomial functions on V with its usual G-module structure (gf)(v) = f{g-1v). Let R be the subalgebra of G-invariant polynomials. By Chevalley’s theorem there exists a set ℬ = {f1, …, fl} of homogeneous polynomials such that R = C[f1, …, fl]. We call ℬ a set of basic invariants or a basic set for G. The degrees di = deg fi are uniquely determined by G. We agree to number them so that d1 ≤ … ≤ di. The map τ: V/G → C1 defined by

is a bijection. Each reflection in G fixes some hyperplane in V.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 109 , March 1988 , pp. 23 - 45
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

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