On the Top Degree of Coinvariants

TL;DR

This paper investigates how the maximum degree of coinvariants behaves under group actions, showing it grows unbounded in the modular case and can remain constant in the non-modular case, with a new proof of Steinberg's theorem.

Contribution

It establishes the unbounded growth of top degree in the modular case and provides a simpler proof of Steinberg's theorem relating group order and coinvariant dimension.

Findings

Top degree grows unboundedly in modular case

Top degree remains constant in certain non-modular cases

Elementary proof of Steinberg's theorem provided

Abstract

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: $\lim_{m\to\infty} \topdeg F[V^{m}]_{G}=\infty$. In contrast, in the non-modular 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.