A note on the degree bounds of the invariant ring

Abstract

<abstract><p>Let $ G = C_p\times H $ be a finite group, where $ C_p $ is a cyclic group of prime order $ p $ and $ H $ is a $ p^{\prime} $-group. Let $ \mathbb{F} $ be an algebraically closed field in characteristic $ p $. Let $ V $ be a direct sum of $ m $ non-trivial indecomposable $ G $-modules such that the norm polynomials of the simple $ H $-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring $ \mathbb{F}[V] $ with actions of $ G $. In this paper, by the periodicity property, we showed that $ \mathbb{F}[V]^G $ is generated by $ m $ norm polynomials together with homogeneous invariants of degree at most $ m|G|-{\rm dim}(V) $ and transfer invariants, which yields the well-known degree bound $ {\rm dim} $$ (V)\cdot(|G|-1) $. More precisely, we found that this bound gets less sharp as the dimensions of simple $ H $-modules increase.</p></abstract>