Rings of invariants for modular representations of elementary abelian p-groups
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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.
KeywordsComplete Intersection Group Homomorphism Polynomial Algebra Left Action Modular Representation
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- [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
- [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
- [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
- [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
- [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