On The Top Degree of Coinvariants

For a finite group G acting faithfully on a finite-dimensional F-vector space V, we show that in the modular case, the top degree of the vector coinvariants grows unboundedly: ${\mathrm lim}_{m\rightarrow \infty}\ {\mathrm topdeg}\ F[V^m]_G=\infty$. In contrast, in the nonmodular case we identify a situation where the top degree of the vector coinvariants remains constant. Furthermore, we present a more elementary proof of Steinberg’s theorem which says that the group order is a lower bound for the dimension of the coinvariants which is sharp if and only if the invariant ring is polynomial.