The Invariant Rings of the Generalized Transvection Groups in the Modular Case (Communications in Mathematical Research, 2019)
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 invariantsubspaces $H$ over a finite field $F_q$ in the modular case. Then we are concerned withgeneral 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 calculatetheir 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