Commun. Math. Res., 33 (2017), pp. 160-176.
Published online: 2019-11
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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} }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.