The module structure of a group action on a polynomial ring: A finiteness theorem
Authors:
Dikran B. Karagueuzian and Peter Symonds
Journal:
J. Amer. Math. Soc. 20 (2007), 931-967
MSC (2000):
Primary 16W22; Secondary 20C20
DOI:
https://doi.org/10.1090/S0894-0347-07-00563-2
Published electronically:
April 11, 2007
MathSciNet review:
2328711
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Consider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example, any indecomposable module that appears as a summand must also appear in low degree, and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators.
- Gert Almkvist and Robert Fossum, Decomposition of exterior and symmetric powers of indecomposable ${\bf Z}/p{\bf Z}$-modules in characteristic $p$ and relations to invariants, Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Math., vol. 641, Springer, Berlin, 1978, pp. 1–111. MR 499459
- J. Alperin and L. G. Kovacs, Periodicity of Weyl modules for ${\rm SL}(2,\,q)$, J. Algebra 74 (1982), no. 1, 52–54. MR 644217, DOI https://doi.org/10.1016/0021-8693%2882%2990004-7 BC Bleher, F.M. and Chinburg, T., Galois structure of homogeneous coordinate rings, Trans. Amer. Math. Soc. (to appear).
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI https://doi.org/10.1006/jsco.1996.0125
- Roger M. Bryant, Symmetric powers of representations of finite groups, J. Algebra 154 (1993), no. 2, 416–436. MR 1206130, DOI https://doi.org/10.1006/jabr.1993.1023
- H. E. A. Campbell and I. P. Hughes, The ring of upper triangular invariants as a module over the Dickson invariants, Math. Ann. 306 (1996), no. 3, 429–443. MR 1415072, DOI https://doi.org/10.1007/BF01445259
- Harm Derksen and Gregor Kemper, Computational invariant theory, Invariant Theory and Algebraic Transformation Groups, I, Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical Sciences, 130. MR 1918599
- Stephen R. Doty, The submodule structure of certain Weyl modules for groups of type $A_n$, J. Algebra 95 (1985), no. 2, 373–383. MR 801273, DOI https://doi.org/10.1016/0021-8693%2885%2990109-7
- Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, DOI https://doi.org/10.1007/BF01206635
- Roger Howe, Asymptotics of dimensions of invariants for finite groups, J. Algebra 122 (1989), no. 2, 374–379. MR 999080, DOI https://doi.org/10.1016/0021-8693%2889%2990223-8
- Dikran B. Karagueuzian and Peter Symonds, The module structure of a group action on a polynomial ring, J. Algebra 218 (1999), no. 2, 672–692. MR 1705758, DOI https://doi.org/10.1006/jabr.1999.7867
- D. B. Karagueuzian and P. Symonds, The module structure of a group action on a polynomial ring: examples, generalizations, and applications, Invariant theory in all characteristics, CRM Proc. Lecture Notes, vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 139–158. MR 2066462, DOI https://doi.org/10.1090/crmp/035/08
- D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425–475. MR 476841, DOI https://doi.org/10.1016/0021-8693%2878%2990116-3
- Ian Hughes and Gregor Kemper, Symmetric powers of modular representations, Hilbert series and degree bounds, Comm. Algebra 28 (2000), no. 4, 2059–2088. MR 1747371, DOI https://doi.org/10.1080/00927870008826944
- Ian Hughes and Gregor Kemper, Symmetric powers of modular representations for groups with a Sylow subgroup of prime order, J. Algebra 241 (2001), no. 2, 759–788. MR 1843324, DOI https://doi.org/10.1006/jabr.2000.8710
- Huỳnh Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 422451
- R. James Shank and David L. Wehlau, The transfer in modular invariant theory, J. Pure Appl. Algebra 142 (1999), no. 1, 63–77. MR 1716047, DOI https://doi.org/10.1016/S0022-4049%2898%2900036-X
- Peter Symonds, Group action on polynomial and power series rings, Pacific J. Math. 195 (2000), no. 1, 225–230. MR 1781621, DOI https://doi.org/10.2140/pjm.2000.195.225 Sy2 Symonds, P., Cyclic group actions on polynomial rings, to appear in Bull. London Math. Soc. Sy3 Symonds, P., Structure theorems over polynomial rings, Advances in Math. 208 (2007), pp. 408–421.
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Additional Information
Dikran B. Karagueuzian
Affiliation:
Mathematics Department, Binghamton University, P.O. Box 6000, Binghamton, New York 13902-6000
Email:
dikran@math.binghamton.edu
Peter Symonds
Affiliation:
School of Mathematics, University of Manchester, P.O. Box 88, Manchester M60 1QD, United Kingdom
Email:
Peter.Symonds@manchester.ac.uk
Received by editor(s):
March 17, 2005
Published electronically:
April 11, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.