Coinvariants and the regular representation of a cyclic \(P\)-group
Let \(p > 0\) be a prime number and let \(F\) be a field of characteristic \(p\). Let \(V\) be a finite dimensional representation of a finite group \(G\) over \(F\). \(F[V]\) denotes the ring of polynomial functions on \(V\), and the ring of invariant polynomials \(F\left[V \right]^G = \left\{f\in F\left[V \right] |g\left(f\right) = f \forall g \in G\right\}\), which is a graded, finitely generated subalgebra of \(F\left[V \right]\). The author of this article considered an indecomposable representation of the cyclic p-group \(G = \mathbb Z_{p^r}\) over a field \(F\) of characteristic \(p\). He showed that the top degree of the corresponding ring of coinvariants is less than \(\frac{(r^2+3r)p^r}{2}\). This bound also applies to the degrees of the generators for the invariant ring of the regular representation.
scientific article; zbMATH DE number 6134693
scientific article