Volume 33, Issue 2
The Invariant Rings of the Generalized Transvection Groups in the Modular Case

Xiang Han, Jizhu Nan & Ki-Bong Nam

Commun. Math. Res., 33 (2017), pp. 160-176.

Published online: 2019-11

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  • Abstract

In this paper, first we investigate the invariant rings of the finite groups $G ≤ GL(n, F_q)$ generated by $i$-transvections and $i$-reflections with given invariant subspaces $H$ over a finite field $F_q$ in the modular case. Then we are concerned with general groups $G_i(ω)$ and $G_i(ω)^t$ named generalized transvection groups where $ω$ is a $k$-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay, Gorenstein, complete intersection, polynomial and Poincare series of these rings.

  • Keywords

invariant, $i$-transvection, $i$-reflection, generalized transvection group

  • AMS Subject Headings

13A50, 20F55, 20F99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xianghan328@yahoo.com (Xiang Han)

  • BibTex
  • RIS
  • TXT
@Article{CMR-33-160, author = {Han , Xiang and Nan , Jizhu and Nam , Ki-Bong}, title = {The Invariant Rings of the Generalized Transvection Groups in the Modular Case}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {33}, number = {2}, pages = {160--176}, abstract = {

In this paper, first we investigate the invariant rings of the finite groups $G ≤ GL(n, F_q)$ generated by $i$-transvections and $i$-reflections with given invariant subspaces $H$ over a finite field $F_q$ in the modular case. Then we are concerned with general groups $G_i(ω)$ and $G_i(ω)^t$ named generalized transvection groups where $ω$ is a $k$-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay, Gorenstein, complete intersection, polynomial and Poincare series of these rings.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2017.02.08}, url = {http://global-sci.org/intro/article_detail/cmr/13396.html} }
TY - JOUR T1 - The Invariant Rings of the Generalized Transvection Groups in the Modular Case AU - Han , Xiang AU - Nan , Jizhu AU - Nam , Ki-Bong JO - Communications in Mathematical Research VL - 2 SP - 160 EP - 176 PY - 2019 DA - 2019/11 SN - 33 DO - http://doi.org/10.13447/j.1674-5647.2017.02.08 UR - https://global-sci.org/intro/article_detail/cmr/13396.html KW - invariant, $i$-transvection, $i$-reflection, generalized transvection group AB -

In this paper, first we investigate the invariant rings of the finite groups $G ≤ GL(n, F_q)$ generated by $i$-transvections and $i$-reflections with given invariant subspaces $H$ over a finite field $F_q$ in the modular case. Then we are concerned with general groups $G_i(ω)$ and $G_i(ω)^t$ named generalized transvection groups where $ω$ is a $k$-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay, Gorenstein, complete intersection, polynomial and Poincare series of these rings.

Xiang Han, Ji-zhu Nan & Ki-bong Nam. (2019). The Invariant Rings of the Generalized Transvection Groups in the Modular Case. Communications in Mathematical Research . 33 (2). 160-176. doi:10.13447/j.1674-5647.2017.02.08
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