## The module structure of a group action on a polynomial ring: A finiteness theorem

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- by Dikran B. Karagueuzian and Peter Symonds;
- J. Amer. Math. Soc.
**20**(2007), 931-967 - DOI: https://doi.org/10.1090/S0894-0347-07-00563-2
- Published electronically: April 11, 2007
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## 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.## References

- Gert Almkvist and Robert Fossum,
*Decomposition of exterior and symmetric powers of indecomposable $\textbf {Z}/p\textbf {Z}$-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-New York, 1978, pp. 1–111. MR**499459** - J. Alperin and L. G. Kovacs,
*Periodicity of Weyl modules for $\textrm {SL}(2,\,q)$*, J. Algebra**74**(1982), no. 1, 52–54. MR**644217**, DOI 10.1016/0021-8693(82)90004-7
BC Bleher, F.M. and Chinburg, T., - 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 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 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 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**, DOI 10.1007/978-3-662-04958-7 - 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 10.1016/0021-8693(85)90109-7 - 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 10.1007/BF01206635 - Roger Howe,
*Asymptotics of dimensions of invariants for finite groups*, J. Algebra**122**(1989), no. 2, 374–379. MR**999080**, DOI 10.1016/0021-8693(89)90223-8 - 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 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 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 10.1016/0021-8693(78)90116-3 - 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 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 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 10.1016/S0022-4049(98)00036-X - Peter Symonds,
*Group action on polynomial and power series rings*, Pacific J. Math.**195**(2000), no. 1, 225–230. MR**1781621**, DOI 10.2140/pjm.2000.195.225
Sy2 Symonds, P.,

*Galois structure of homogeneous coordinate rings*, Trans. Amer. Math. Soc. (to appear).

*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.

## Bibliographic 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
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2328711