Method of orbit sums in the theory of modular vector invariants

Let $F$ be a field, $V$ a finite-dimensional $F$-vector space, $G\leqslant\operatorname{GL}_F(V)$ a finite group, and $V^m=V\oplus\dots\oplus V$ the $m$-fold direct sum with the diagonal action of $G$. The group $G$ acts naturally on the symmetric graded algebra $A_m=F[V^m]$ as a group of non-degenerate linear changes of the variables. Let $A_m^G$ be the subalgebra of invariants of the polynomial algebra $A_m$ with respect to $G$. A classical result of Noether [1] says that if $\operatorname{char}F=0$, then $A_m^G$ is generated as an $F$-algebra by homogeneous polynomials of degree at most $|G|$, no matter how large $m$ can be. On the other hand, it was proved by Richman [2], [3] that this result does not hold when the characteristic of $F$ is positive and divides the order $|G|$ of $G$. Let $p$, $p>2$, be a prime number, $F=F_p$ a finite field of $p$ elements, $V$ a linear $F_p$-vector space of dimension $n$, and $H\leqslant\operatorname{GL}_{F_p}(V)$ a cyclic group of order $p$ generated by a matrix $\gamma$ of a certain special form. In this paper we describe explicitly (Theorem 1) one complete set of generators of $A_m^H$. After that, for an arbitrary complete set of generators of this algebra we find a lower bound for the highest degree of the generating elements of this algebra. This is a significant extension of the corresponding result of Campbell and Hughes [4] for the particular case of $n=2$. As a consequence we show (Theorem 3) that if $m>n$ and $G\geqslant H$ is an arbitrary finite group, then each complete set of generators of $A_m^G$ contains an element of degree at least $2(m-n+2r)(p-1)/r$, where $r=r(H)$ is a positive integer independent of the structure of the generating matrix $\gamma$ of the group $H$. This results refines considerably the earlier lower bound obtained by Richman [3].
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