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

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

13A50, 20F55, 20F99

xianghan328@yahoo.com (Xiang Han)

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@Article{CMR-33-160, author = {Xiang and Han and xianghan328@yahoo.com and 5473 and School of Mathematical Sciences, Dalian University of Technology, Liaoning, 116024 and Xiang Han and Jizhu and Nan and and 5474 and School of Mathematical Sciences, Dalian University of Technology, Liaoning, 116024 and Jizhu Nan and Ki-Bong and Nam and and 5475 and Department of Mathematics and Computer Science, University of Wisconsin-Whitewater, Whitewater, WI 53190, United States and Ki-Bong Nam}, 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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