Introduction

This work presents progress on two relevant problems within the framework of algebraic geometry and commutative algebra that thrive unsolved. First, the Gröbner’s longstanding problem regarding the arithmetic Cohen–Macaulayness of projections of Veronese varieties and, second, the fundamental problem of determining the internal structure of the algebra of invariants of finite groups. We work to evince the symbiosis between these two subjects and to understand their connection, a priori unexpected, with the Lefschetz properties of artinian ideals.