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Discriminants in the invariant theory of reflection groups. (English) Zbl 0614.20032
Let S be the algebra of polynomial functions on an \(\ell\)-dimensional vector space V/C. Let \(G\subset GL(V)\) be a finite unitary reflection group and let \(R=S^ G\) be the G-invariants. By Chevalley’s theorem, \(R=C[f_ 1,...,f_{\ell}]\) is a polynomial ring on a basic set \(B_ G=\{f_ 1,...,f_{\ell}\}\) of homogeneous generators. Let \(\Delta _ G(T_ 1,...,T_{\ell};B_ G)\) be the polynomial in the indeterminates \(T_ 1,...,T_{\ell}\) such that \(\Delta _ G(f_ 1,...,f_{\ell};B_ G)\) is the discriminant of the orbit map \(V\to V/G\). We show that if G is a Shepard group (defined in this paper) then there exists a Coxeter group \(W\subset GL(V)\) and basic sets \(B_ G\), \(B_ W\) such that \(\Delta _ G(T_ 1,...,T_{\ell},B_ G)\) and \(\Delta _ W(T_ 1,...,T_{\ell};B_ W)\) are equal, up to a constant multiple. Thus the discriminant loci for G and W are the same. Since the complement of the union of the reflecting hyperplanes for W is known to be a K(\(\pi\),1) space, this implies the same result for G. The equality of the discriminants is a consequence of a statement about discriminant matrices.

MSC:
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
51F15 Reflection groups, reflection geometries
15A72 Vector and tensor algebra, theory of invariants
14L30 Group actions on varieties or schemes (quotients)
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