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On rings of invariants of non-modular Abelian groups

Published online by Cambridge University Press:  17 April 2009

H.E.A. Campbell ,
J.C. Harris  and
D.L. Wehlau
H.E.A. Campbell
Affiliation:
Department of Mathematics and StatisticsQueen's UniversityKingston, Ontario, Canada, K7L 3N6 e-mail: eddy@mast.queensu.ca
J.C. Harris
Affiliation:
Department of MathematicsUniversity of TorontoToronto, OntarioCanadaM5S 1A1 e-mail: harris@math.toronto.edu
D.L. Wehlau
Affiliation:
Department of Mathematics and Computer ScienceRoyal Military CollegeKingston, OntarioCanadaM5S 1A1 and Department of Mathematics and StatisticsQueen's UniversityKingston, OntarioCanada K7L 3N6 e-mail: wehlau@mast.queensu.ca
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We study the ring of invariant Laurent polynomials associated to the action of a finite diagonal group G on the symmetric algebra of a vector space over a field F. Here the characteristic p of the field F necessarily does not divide the order q = |G| of the group, so G is said to be non-modular. For certain representations of such groups, we can characterise generators of the ring of invariant polynomials in the original symmetric algebra, extending results of Campbell, Hughes, Pappalardi and Selick. In particular we obtain a recursive formula for the number of minimal generators for these rings of invariants.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 3 , December 1999 , pp. 509 - 520
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Artin, M., Algebra (Prentice Hall, Englewood Cliffs, N.J., 1991).Google Scholar
[2]Campbell, H.E.A., Harris, J.C. and Wehlau, D.L., ‘Internal duality in resolutions of certain rings of invariants’, J. Algebra 215 (1999), 1–33.CrossRefGoogle Scholar
[3]Campbell, H.E.A., Hughes, I.P., Pappalardi, F. and Selick, P.S., ‘On the ring of invariants of F2’, Comment. Math. Helv. 66 (1991), 322–331.CrossRefGoogle Scholar
[4]Campbell, H.E.A. and Selick, P.S., ‘Polynomial algebras over the Steenrod algebra’, Comment. Math. Helv. 65 (1990), 171–180.CrossRefGoogle Scholar
[5]Cartan, H. and Eilenberg, S., Homological algebra, Princeton Math. Series 19 (Princeton University Press, Princeton, N.J., 1956).Google Scholar
[6]de Bruijn, N.G., ‘On Mahler's partition problem’, Indiag. Math. 10 (1948), 220–230.Google Scholar
[7]Erdös, P., Dixmier, J. and Nicolas, J.-L., ‘Sur le nombre d'invariants fondamentaux des formes binaires’, C.R. Acad Sci. Paris Sér. I Math. 305 (1987), 319–322.Google Scholar
[8]Elashvili, A. and Jibladze, M., ‘Hermite reciprocity for the regular representations of cyclic groups’, (preprint).Google Scholar
[9]Elashvili, A. and Jibladze, M., Untitled, Institute of Mathematics of Georgian Academy of Sciences, (preprint 1996).Google Scholar
[10]Kac, V., ‘Root systems, representations of quivers and invariant theory’, in Invariant theory, Lecture Notes in Math. 996 (Springer-Verlag, Berlin, Heidelberg, New York, 1983), pp. 74–108.CrossRefGoogle Scholar
[11]Schmid, B., ‘Finite groups and invariant theory’, in Topics in invariant theory, Lecture Notes in Math. 1478 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 35–66.CrossRefGoogle Scholar
[12]Wehlau, D.L., ‘Constructive invariant theory for tori’, Ann. Inst. Fourier (Grenoble) 43 (1993), 1055–1066.CrossRefGoogle Scholar
[13]Wehlau, D.L., ‘When is a ring of torus invariants a polynomial ring?’, Manuscripta Math, 82 (1994), 161–170.CrossRefGoogle Scholar