Discriminants in the invariant theory of reflection groups
- 1 March 1988
- journal article
- research article
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- Vol. 109, 23-45
- https://doi.org/10.1017/s0027763000002749
Abstract
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 = {f 1, …, fl } of homogeneous polynomials such that R = C[f 1, …, f l]. We call ℬ a set of basic invariants or a basic set for G. The degrees d i = deg f i are uniquely determined by G. We agree to number them so that d 1 ≤ … ≤ di . The map τ: V/G → C1 defined by is a bijection. Each reflection in G fixes some hyperplane in V.Keywords
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