Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 655-698, 2017

Invariants of finite groups generated by generalized transvections in the modular case

Xiang Han, Jizhu Nan, Chander K. Gupta

Received February 2, 2016.   First published July 12, 2017.

Xiang Han, Jizhu Nan, School of Mathematical Sciences, Dalian University of Technology, No. 2 Linggong Road, Dalian, 116024, Ganjingzi, Liaoning, P. R. China, e-mail: xianghan328@yahoo.com, jznan@163.com; Chander K. Gupta, Department of Mathematics, University of Manitoba, Machray Hall 420, 186 Dysart Rd, Winnipeg, MB R3T 2M8, Canada

Abstract: 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.

Keywords: invariant ring; transvection; generalized transvection group

Classification (MSC 2010): 13A50, 20F55, 20F99

DOI: 10.21136/CMJ.2017.0044-16

Full text available as PDF.


References:
  [1] H. Bass: On the ubiquity of Gorenstein rings. Math. Z. 82 (1963), 8-28. DOI 10.1007/BF01112819 | MR 0153708 | Zbl 0112.26604
  [2] M.-J. Bertin: Anneaux d'invariants d'anneaux de polynômes, en caractéristique $p$. C. R. Acad. Sci., Paris, Sér. A 264 (1967), 653-656. (In French.) MR 0215826 | Zbl 0147.29503
  [3] A. Braun: 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] W. Bruns, J. Herzog: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608681 | Zbl 0909.13005
  [5] 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. DOI 10.4153/CMB-1999-018-4 | MR 1692004 | Zbl 0942.13007
  [6] H. Derksen, G. Kemper: 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] L. E. Dickson: Invariants of binary forms under modular transformations. Amer. M. S. Trans. 8 (1907), 205-232. DOI 10.1090/S0002-9947-1907-1500782-9 | MR 1500782 | JFM 38.0147.02
  [8] X. Han, J. Nan, K. Nam: The invariants of generalized transvection groups in the modular case. Commun. Math. Res. 33 (2017), 160-176.
  [9] M. Hochster, J. A. Eagon: 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] J. Huang: 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] G. Kemper, G. Malle: 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] J. W. Milnor: 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] H. Nakajima: Invariants of finite groups generated by pseudo-reflections in positive characteristic. Tsukuba J. Math. 3 (1979), 109-122. MR 0543025 | Zbl 0418.20041
  [14] H. Nakajima: 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] H. Nakajima: 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] 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. DOI 10.1016/S0022-4049(96)00079-5 | MR 1479349 | Zbl 0901.13006
  [17] M. D. Neusel, L. Smith: 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] L. Smith: 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] L. Smith, R. E. Stong: 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] R. P. Stanley: 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] R. Steinberg: On Dickson's theorem on invariants. J. Fac. Sci., Univ. Tokyo, Sect. I A 34 (1987), 699-707. MR 0927606 | Zbl 0656.20052
  [22] H. You, J. Lan: 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


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