Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-14T23:22:10.418Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1] Bourbaki, N., Groupes et algebres de Lie, Chapitres 4, 5 et 6. Hermann, Paris, 1968.Google Scholar
[1] Coxeter, H. S. M., The Symmetry Groups of the Regular Complex Polygons, Arch. Math., 13 (1962), 86–97.CrossRefGoogle Scholar
[1] Coxeter, H. S. M., Finite groups generated by unitary reflections, Abh. Math. Sem. Univ. Hamburg, 31 (1967), 127–135.Google Scholar
[1] Coxeter, H. S. M., Regular Complex Polytopes, Cambridge Univ. Press, 1974.Google Scholar
[1] Gordan, P. and Noether, M., Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann., 10 (1876), 547–568.CrossRefGoogle Scholar
[1] Grace, J. H. and Young, A., The Algebra of Invariants, Cambridge Univ. Press, 1903.Google Scholar
[1] Hesse, L. O., Ueber die Bedingung, unter welcher eine homogene ganze Function von n unabhängigen Variabeln durch lineare Substitutionen von n anderen unabhängigen Variabeln auf eine homogene Function sich zurückführen lässt, die eine Variable weniger enthält, J. reine angew. Math., 42 (1852), 117–124, reprinted in Gesammelte Werke, Chelsea, 1972.Google Scholar
[1] Hesse, L. O., Zur Thorie der ganzen homogenen Functionen, J. reine angew. Math., 56 (1859), 481–496, reprinted in Gesammelte Werke, Chelsea, 1972.Google Scholar
[1] Klein, F., Lectures on the Icosahedron, Kegan Paul, London 1913, reprinted Dover 1956.Google Scholar
[10] Koster, D. W., Complex Reflection Groups, Thesis, Univ. of Wisconsin, Madison, 1975.Google Scholar
[11] Maschke, H., Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen, Math. Ann., 33 (1888), 317–344.Google Scholar
[12] Noether, M., Anmerkungen zu den Abhandlungen L. O. Hesse, Note 30, p. 704 in L. O. Hesse, Gesammelte Werke, Chelsea, 1972.Google Scholar
[13] Orlik, P. and Solomon, L., Unitary reflection groups and cohomology, Invent, math., 59 (1980), 77–94.CrossRefGoogle Scholar
[14] Orlik, P. and Solomon, L., Discriminants in the invariant theory of reflection groups, Nagoya Math. J., 109 (1988), 23–45.CrossRefGoogle Scholar
[15] Saito, K., On a linear structure of a quotient variety by a finite reflection group, RIMS Report 288, Kyoto, 1979.Google Scholar
[16] Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo, Sect. IA, 27 (1980), 265–291.Google Scholar
[17] Shephard, G. C., Regular Complex Polytopes, Proc. London Math. Soc, (3) 2 (1952), 82–97.Google Scholar
[18] Shephard, G. C., Unitary Groups Generated by Reflections, Canad. J. Math., 5 (1953), 364–383.Google Scholar
[19] Shephard, G. C.and Todd, J. A., Finite Unitary Reflection Groups, Canad. J. Math., 6 (1954), 274–304.CrossRefGoogle Scholar
[20] Solomon, L., Invariants of Euclidean reflection groups, Trans. Amer. Math. Soc, 113 (1964), 274–286.CrossRefGoogle Scholar
[21] Springer, T. A., Invariant Theory, Lecture Notes in Math. vol. 585, Springer Verlag, 1977.Google Scholar
[22] Springer, T. A., Regular elements of finite reflection groups, Invent, math., 25 (1974), 159–198.CrossRefGoogle Scholar
[23] Sylvester, J. J., On the conditions necessary and sufficient to be satisfied in order that a function of any number of variables may be linearly equivalent to a function of any less number of variables, Phil. Mag., 5 (1853), 119–126, reprinted in Collected Works I, Chelsea 1973.Google Scholar
[24] Terao, H., Free Arrangements of Hyperplanes and Unitary Reflection Groups, Proc. Japan. Acad., 56, Ser. A (1980), 389–392.Google Scholar
[25] Wedekind, L., Studien im Binaren Werthgebiet, Habilitationsschrift Polytechnikum zu Carlsruhe, Carlsruhe, 1876.Google Scholar