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

Xiang Han; Jizhu Nan; Chander K. Gupta

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 655-698
  • ISSN: 0011-4642

Abstract

top
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.

How to cite

top

Han, Xiang, Nan, Jizhu, and Gupta, Chander K.. "Invariants of finite groups generated by generalized transvections in the modular case." Czechoslovak Mathematical Journal 67.3 (2017): 655-698. <http://eudml.org/doc/294101>.

@article{Han2017,
abstract = {We investigate the invariant rings of two classes of finite groups $G\le \{\rm GL\}(n,F_q)$ 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.},
author = {Han, Xiang, Nan, Jizhu, Gupta, Chander K.},
journal = {Czechoslovak Mathematical Journal},
keywords = {invariant ring; transvection; generalized transvection group},
language = {eng},
number = {3},
pages = {655-698},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Invariants of finite groups generated by generalized transvections in the modular case},
url = {http://eudml.org/doc/294101},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Han, Xiang
AU - Nan, Jizhu
AU - Gupta, Chander K.
TI - Invariants of finite groups generated by generalized transvections in the modular case
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 655
EP - 698
AB - We investigate the invariant rings of two classes of finite groups $G\le {\rm GL}(n,F_q)$ 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.
LA - eng
KW - invariant ring; transvection; generalized transvection group
UR - http://eudml.org/doc/294101
ER -

References

top
  1. Bass, H., 10.1007/BF01112819, Math. Z. 82 (1963), 8-28. (1963) Zbl0112.26604MR0153708DOI10.1007/BF01112819
  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. (1967) Zbl0147.29503MR0215826
  3. Braun, A., 10.1016/j.jalgebra.2011.07.030, J. Algebra 345 (2011), 81-99. (2011) Zbl1243.13003MR2842055DOI10.1016/j.jalgebra.2011.07.030
  4. Bruns, W., Herzog, J., 10.1017/CBO9780511608681, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). (1998) Zbl0909.13005MR1251956DOI10.1017/CBO9780511608681
  5. Campbell, H. E. A., Geramita, A. V., Hughes, I. P., Shank, R. J., Wehlau, D. L., 10.4153/CMB-1999-018-4, Can. Math. Bull. 42 (1999), 155-161. (1999) Zbl0942.13007MR1692004DOI10.4153/CMB-1999-018-4
  6. Derksen, H., Kemper, G., 10.1007/978-3-662-04958-7, Encyclopaedia of Mathematical Sciences 130, Invariant Theory and Algebraic Transformation Groups 1, Springer, Berlin (2002). (2002) Zbl1011.13003MR1918599DOI10.1007/978-3-662-04958-7
  7. Dickson, L. E., 10.1090/S0002-9947-1907-1500782-9, Amer. M. S. Trans. 8 (1907), 205-232 9999JFM99999 38.0147.02. (1907) MR1500782DOI10.1090/S0002-9947-1907-1500782-9
  8. Han, X., Nan, J., Nam, K., The invariants of generalized transvection groups in the modular case, Commun. Math. Res. 33 (2017), 160-176. (2017) MR3676528
  9. Hochster, M., Eagon, J. A., 10.2307/2373744, Am. J. Math. 93 (1971), 1020-1058. (1971) Zbl0244.13012MR0302643DOI10.2307/2373744
  10. Huang, J., 10.1016/j.jalgebra.2010.09.010, J. Algebra 328 (2011), 432-442. (2011) Zbl1225.14039MR2745575DOI10.1016/j.jalgebra.2010.09.010
  11. Kemper, G., Malle, G., 10.1007/BF01234631, Transform. Groups 2 (1997), 57-89. (1997) Zbl0899.13004MR1439246DOI10.1007/BF01234631
  12. Milnor, J. W., 10.1515/9781400881796, Annals of Mathematics Studies 72, Princeton University Press and University of Tokyo Press, Princeton (1971). (1971) Zbl0237.18005MR0349811DOI10.1515/9781400881796
  13. Nakajima, H., 10.21099/tkbjm/1496158618, Tsukuba J. Math. 3 (1979), 109-122. (1979) Zbl0418.20041MR0543025DOI10.21099/tkbjm/1496158618
  14. Nakajima, H., 10.1017/s0027763000019875, Nagoya Math. J. 86 (1982), 229-248. (1982) Zbl0443.14005MR0661227DOI10.1017/s0027763000019875
  15. Nakajima, H., 10.1016/0021-8693(83)90094-7, J. Algebra 85 (1983), 253-286. (1983) Zbl0536.20028MR0725082DOI10.1016/0021-8693(83)90094-7
  16. Neusel, M. D., Smith, L., 10.1016/S0022-4049(96)00079-5, J. Pure Appl. Algebra 122 (1997), 87-105. (1997) Zbl0901.13006MR1479349DOI10.1016/S0022-4049(96)00079-5
  17. Neusel, M. D., Smith, L., 10.1090/surv/094, Mathematical Surveys and Monographs 94, American Mathematical Society, Providence (2002). (2002) Zbl0999.13002MR1869812DOI10.1090/surv/094
  18. Smith, L., 10.4153/CMB-1996-030-2, Can. Math. Bull. 39 (1996), 238-240. (1996) Zbl0868.13006MR1390361DOI10.4153/CMB-1996-030-2
  19. Smith, L., Stong, R. E., 10.1016/0021-8693(87)90040-8, J. Algebra 110 (1987), 134-157. (1987) Zbl0652.20046MR0904185DOI10.1016/0021-8693(87)90040-8
  20. Stanley, R. P., 10.1090/S0273-0979-1979-14597-X, Bull. Am. Math. Soc., New Ser. 1 (1979), 475-511. (1979) Zbl0497.20002MR0526968DOI10.1090/S0273-0979-1979-14597-X
  21. Steinberg, R., On Dickson's theorem on invariants, J. Fac. Sci., Univ. Tokyo, Sect. I A 34 (1987), 699-707. (1987) Zbl0656.20052MR0927606
  22. You, H., Lan, J., 10.1016/0024-3795(93)90294-X, Linear Algebra Appl. 186 (1993), 235-253. (1993) Zbl0773.15005MR1217208DOI10.1016/0024-3795(93)90294-X

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.