zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI
[1] DOI: 10.4153/CJM-1954-028-3 · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3
[2] DOI: 10.1007/BF01390173 · Zbl 0287.20043 · doi:10.1007/BF01390173
[3] Proc. London Math. Soc 2 pp 82– (1952)
[4] DOI: 10.1080/00927878008822464 · Zbl 0428.14020 · doi:10.1080/00927878008822464
[5] Math. Ann 33 pp 317– (1888)
[6] Nagoya Math. J 109 pp 1– (1988) · Zbl 0614.20033 · doi:10.1017/S0027763000002737
[7] Complex Reflection Groups (1975)
[8] Lecture Notes in Math pp 193– (1981)
[9] Vorlesungen Über das Ikosaeder (1884)
[10] DOI: 10.1007/BF01390316 · Zbl 0452.20050 · doi:10.1007/BF01390316
[11] The Algebra of Invariants (1903)
[12] Papers College of Arts and Sciences, Univ. Tokyo 33 pp 1– (1983)
[13] DOI: 10.1007/BF01406236 · Zbl 0238.20034 · doi:10.1007/BF01406236
[14] Ann. of Math 61 (1968)
[15] Regular Complex Polytopes (1974) · Zbl 0296.50009
[16] Abhandlungen Math. Sem. Univ. Hamburg 31 pp 127– (1967)
[17] DOI: 10.1007/BF01650052 · Zbl 0112.12903 · doi:10.1007/BF01650052
[18] DOI: 10.2969/jmsj/02830447 · Zbl 0326.57015 · doi:10.2969/jmsj/02830447
[19] The microlocal structure of weighted homogeneous polynomials associated with Coxeter systems I, II 2 pp 193– (1979)
[20] (1960)
[21] J. Fac. Sci. Univ. Tokyo, Sect. IA Math 30 pp 379– (1983)
[22] Lecture Notes in Math (1977)
[23] DOI: 10.4153/CJM-1953-042-7 · Zbl 0052.16403 · doi:10.4153/CJM-1953-042-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.