# Rings of invariants for modular representations of elementary abelian *p*-groups

- 173 Downloads
- 3 Citations

## Abstract

We initiate a study of the rings of invariants of modular representations of elementary abelian *p*-groups. With a few notable exceptions, the modular representation theory of an elementary abelian *p*-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.

## Keywords

Complete Intersection Group Homomorphism Polynomial Algebra Left Action Modular Representation## Preview

Unable to display preview. Download preview PDF.

## References

- [AL]W. W. Adams, P. Loustaunau,
*An Introduction to Gröbner Bases*, Graduate Studies in Mathematics, Vol. 3, Amer. Math. Society, Providence, RI, 1994.Google Scholar - [B1]D. J. Benson,
*Representations and Cohomology*I, Cambridge Studies in Advanced Mathematics, Vol. 30, Cambridge University Press, 1991.Google Scholar - [B2]D. J. Benson,
*Polynomial Invariants of Finite Groups*, London Mathematical Society Lecture Note Series, Vol. 190, Cambridge University Press, 1993.Google Scholar - [BCP]W. Bosma, J. J. Cannon, C. Playoust,
*The Magma algebra system I: the user language*, J. Symbolic Comput.**24**(1997), 235–265.MathSciNetMATHCrossRefGoogle Scholar - [CC]H. E. A. Campbell, J. Chuai,
*On the invariant fields and localized invariant rings of p-groups*, Quarterly J. of Math.**58**(2007), 151–157.MathSciNetMATHCrossRefGoogle Scholar - [CW]H. E. A. Campbell, D. L. Wehlau,
*Modular Invariant Theory*, Encyclopaedia of Mathematical Sciences, Vol. 139, Subseries*Invariant Theory and Algebraic Transformation Groups*, Vol. VIII, Springer, Berlin, 2011.CrossRefGoogle Scholar - [DK]H. Derksen, G. Kemper,
*Computational Invariant Theory*, Encyclopaedia of Mathematical Sciences, Vol. 130, Subseries*Invariant Theory and Algebraic Transformation Groups*, Vol. I, Springer, Berlin, 2002.Google Scholar - [D1]L. E. Dickson,
*A fundamental system of invariants of the general modular linear group with a solution of the form problem*, Trans. Amer. Math. Soc.**12**(1911), 75–98.MathSciNetMATHCrossRefGoogle Scholar - [D2]L. E. Dickson,
*On Invariants and the Theory of Numbers*, The Madison Colloquium (1913, Part 1), Amer. Math. Society, reprinted by Dover, New York, 1966.Google Scholar - [HS]A. Hobson, R. J. Shank,
*The invariants of the second symmetric power representation of SL*_{2}(\( \mathbb{F}_q \)), J. Pure Appl. Algebra**215**(2011), no. 10, 2481–2485.MathSciNetMATHCrossRefGoogle Scholar - [Ka]M. Kang,
*Fixed fields of triangular matrix groups*, J. Algebra**302**(2006), 845–847.MathSciNetMATHCrossRefGoogle Scholar - [KM]D. Kapur, K. Madlener,
*A completion procedure for computing a canonical basis of a k-subalgebra*, Proceedings of Computers and Mathematics 89 (1989), MIT, 1–11.Google Scholar - [Ke]G. Kemper,
*A Course in Commutative Algebra*, Graduate Texts in Mathematics, Vol. 256, Springer, Berlin, 2011.CrossRefGoogle Scholar - [LS]P. S. Landweber, R. E. Stong,
*The depth of rings of invariants over finite fields*, in:*Number Theory*(New York, 1984–1985), Lecture Notes in Mathematics, Vol. 1240, Springer, Berlin, 1987, pp. 259–274.Google Scholar - [NS]M. D. Neusel, L. Smith,
*Invariant Theory of Finite Groups*, Mathematical Surveys and Monographs, Vol. 94, Amer. Math. Society, Providence, RI, 2002.MATHGoogle Scholar - [RS]L. Robbianno, M. Sweedler,
*Subalgebra bases*, in:*Commutative Algebra*(Salvador, 1988), Lecture Notes in Mathematics, Vol. 1430, Springer, Berlin, 1990, pp. 61–87.CrossRefGoogle Scholar - [S]B. Sturmfels,
*Gröbner Bases and Convex Polytopes*, University Lecture Series, Vol. 8, Amer. Math. Society, Providence, RI, 1996.MATHGoogle Scholar - [W]C. W. Wilkerson,
*A primer on the Dickson invariants*, in:*Proceedings of the Northwestern Homotopy Theory Conference*(Evanston, Ill., 1982), Contemp. Math., Vol. 19, Amer. Math. Society, Providence, RI, 1983, pp. 421–434.CrossRefGoogle Scholar