Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
invariant ring; transvection; generalized transvection group
invariant ring; transvection; generalized transvection group
Summary:
We investigate the invariant rings of two classes of finite groups $G\leq {\rm GL}(n,F_q)$ 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.
References:
[1] Bass, H.: On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8-28. DOI 10.1007/BF01112819 | MR 0153708 | Zbl 0112.26604
[2] Bertin, M.-J.: Anneaux d'invariants d'anneaux de polynômes, en caractéristique $p$. C. R. Acad. Sci., Paris, Sér. A 264 (1967), 653-656 French. MR 0215826 | Zbl 0147.29503
[3] Braun, A.: On the Gorenstein property for modular invariants. J. Algebra 345 (2011), 81-99. DOI 10.1016/j.jalgebra.2011.07.030 | MR 2842055 | Zbl 1243.13003
[4] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608681 | MR 1251956 | Zbl 0909.13005
[5] Campbell, H. E. A., Geramita, A. V., Hughes, I. P., Shank, R. J., Wehlau, D. L.: Non-Cohen-Macaulay vector invariants and a Noether bound for a Gorenstein ring of invariants. Can. Math. Bull. 42 (1999), 155-161. DOI 10.4153/CMB-1999-018-4 | MR 1692004 | Zbl 0942.13007
[6] Derksen, H., Kemper, G.: Computational Invariant Theory. Encyclopaedia of Mathematical Sciences 130, Invariant Theory and Algebraic Transformation Groups 1, Springer, Berlin (2002). DOI 10.1007/978-3-662-04958-7 | MR 1918599 | Zbl 1011.13003
[7] Dickson, L. E.: Invariants of binary forms under modular transformations. Amer. M. S. Trans. 8 (1907), 205-232 \99999JFM99999 38.0147.02. DOI 10.1090/S0002-9947-1907-1500782-9 | MR 1500782
[8] Han, X., Nan, J., Nam, K.: The invariants of generalized transvection groups in the modular case. Commun. Math. Res. 33 (2017), 160-176. MR 3676528
[9] Hochster, M., Eagon, J. A.: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93 (1971), 1020-1058. DOI 10.2307/2373744 | MR 0302643 | Zbl 0244.13012
[10] Huang, J.: A gluing construction for polynomial invariants. J. Algebra 328 (2011), 432-442. DOI 10.1016/j.jalgebra.2010.09.010 | MR 2745575 | Zbl 1225.14039
[11] Kemper, G., Malle, G.: The finite irreducible linear groups with polynomial ring of invariants. Transform. Groups 2 (1997), 57-89. DOI 10.1007/BF01234631 | MR 1439246 | Zbl 0899.13004
[12] Milnor, J. W.: Introduction to Algebraic $K$-Theory. Annals of Mathematics Studies 72, Princeton University Press and University of Tokyo Press, Princeton (1971). DOI 10.1515/9781400881796 | MR 0349811 | Zbl 0237.18005
[13] Nakajima, H.: Invariants of finite groups generated by pseudo-reflections in positive characteristic. Tsukuba J. Math. 3 (1979), 109-122. DOI 10.21099/tkbjm/1496158618 | MR 0543025 | Zbl 0418.20041
[14] Nakajima, H.: Modular representations of abelian groups with regular rings of invariants. Nagoya Math. J. 86 (1982), 229-248. DOI 10.1017/s0027763000019875 | MR 0661227 | Zbl 0443.14005
[15] Nakajima, H.: Regular rings of invariants of unipotent groups. J. Algebra 85 (1983), 253-286. DOI 10.1016/0021-8693(83)90094-7 | MR 0725082 | Zbl 0536.20028
[16] Neusel, M. D., Smith, L.: Polynomial invariants of groups associated to configurations of hyperplanes over finite fields. J. Pure Appl. Algebra 122 (1997), 87-105. DOI 10.1016/S0022-4049(96)00079-5 | MR 1479349 | Zbl 0901.13006
[17] Neusel, M. D., Smith, L.: Invariant Theory of Finite Groups. Mathematical Surveys and Monographs 94, American Mathematical Society, Providence (2002). DOI 10.1090/surv/094 | MR 1869812 | Zbl 0999.13002
[18] Smith, L.: Some rings of invariants that are Cohen-Macaulay. Can. Math. Bull. 39 (1996), 238-240. DOI 10.4153/CMB-1996-030-2 | MR 1390361 | Zbl 0868.13006
[19] Smith, L., Stong, R. E.: On the invariant theory of finite groups: Orbit polynomials and splitting principles. J. Algebra 110 (1987), 134-157. DOI 10.1016/0021-8693(87)90040-8 | MR 0904185 | Zbl 0652.20046
[20] Stanley, R. P.: Invariants of finite groups and their applications to combinatorics. Bull. Am. Math. Soc., New Ser. 1 (1979), 475-511. DOI 10.1090/S0273-0979-1979-14597-X | MR 0526968 | Zbl 0497.20002
[21] Steinberg, R.: On Dickson's theorem on invariants. J. Fac. Sci., Univ. Tokyo, Sect. I A 34 (1987), 699-707. MR 0927606 | Zbl 0656.20052
[22] You, H., Lan, J.: Decomposition of matrices into 2-involutions. Linear Algebra Appl. 186 (1993), 235-253. DOI 10.1016/0024-3795(93)90294-X | MR 1217208 | Zbl 0773.15005
Search
Partner of
