Abstract
We investigate the invariant rings of two classes of finite groups G ≤ GL(n, F q) which are generated by a number of generalized transvections with an invariant subspace H over a finite field F q in the modular case. We name these groups generalized transvection groups. One class is concerned with a given invariant subspace which involves roots of unity. Constructing quotient groups and tensors, we deduce the invariant rings and study their Cohen-Macaulay and Gorenstein properties. The other is concerned with different invariant subspaces which have the same dimension. We provide a explicit classification of these groups and calculate their invariant rings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bass: On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8–28.
M.-J. Bertin: Anneaux d’invariants d’anneaux de polynomes, en caractéristique p. C. R. Acad. Sci., Paris, Sér. A 264 (1967), 653–656. (In French.)
A. Braun: On the Gorenstein property for modular invariants. J. Algebra 345 (2011), 81–99.
W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1998.
H. E. A. Campbell, A. V. Geramita, I. P. Hughes, R. J. Shank, D. L. Wehlau: Non-Cohen-Macaulay vector invariants and a Noether bound for a Gorenstein ring of invariants. Can. Math. Bull. 42 (1999), 155–161.
H. Derksen, G. Kemper: Computational Invariant Theory. Encyclopaedia of Mathematical Sciences 130, Invariant Theory and Algebraic Transformation Groups 1, Springer, Berlin, 2002.
L. E. Dickson: Invariants of binary forms under modular transformations. Amer. M. S. Trans. 8 (1907), 205–232.
X. Han, J. Nan, K. Nam: The invariants of generalized transvection groups in the modular case. Commun. Math. Res. 33 (2017), 160–176.
M. Hochster, J. A. Eagon: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93 (1971), 1020–1058.
J. Huang: A gluing construction for polynomial invariants. J. Algebra 328 (2011), 432–442.
G. Kemper, G. Malle: The finite irreducible linear groups with polynomial ring of invariants. Transform. Groups 2 (1997), 57–89.
J. W. Milnor: Introduction to Algebraic K-Theory. Annals of Mathematics Studies 72, Princeton University Press and University of Tokyo Press, Princeton, 1971.
H. Nakajima: Invariants of finite groups generated by pseudo-reflections in positive characteristic. Tsukuba J. Math. 3 (1979), 109–122.
H. Nakajima: Modular representations of abelian groups with regular rings of invariants. Nagoya Math. J. 86 (1982), 229–248.
H. Nakajima: Regular rings of invariants of unipotent groups. J. Algebra 85 (1983), 253–286.
M. D. Neusel, L. Smith: Polynomial invariants of groups associated to configurations of hyperplanes over finite fields. J. Pure Appl. Algebra 122 (1997), 87–105.
M. D. Neusel, L. Smith: Invariant Theory of Finite Groups. Mathematical Surveys and Monographs 94, American Mathematical Society, Providence, 2002.
L. Smith: Some rings of invariants that are Cohen-Macaulay. Can. Math. Bull. 39 (1996), 238–240.
L. Smith, R. E. Stong: On the invariant theory of finite groups: Orbit polynomials and splitting principles. J. Algebra 110 (1987), 134–157.
R. P. Stanley: Invariants of finite groups and their applications to combinatorics. Bull. Am. Math. Soc., New Ser. 1 (1979), 475–511.
R. Steinberg: On Dickson’s theorem on invariants. J. Fac. Sci., Univ. Tokyo, Sect. I A 34 (1987), 699–707.
H. You, J. Lan: Decomposition of matrices into 2-involutions. Linear Algebra Appl. 186 (1993), 235–253.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Grant No. 11371343).
Rights and permissions
About this article
Cite this article
Han, X., Nan, J. & Gupta, C.K. Invariants of finite groups generated by generalized transvections in the modular case. Czech Math J 67, 655–698 (2017). https://doi.org/10.21136/CMJ.2017.0044-16
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2017.0044-16