A Periodicity Property of Symmetric Algebras with Actions of Metacyclic Groups in the Modular Case

Abstract:

      In this paper, we consider the decomposition of the symmetric algebra $\mathbb{F}[V]$ into indecomposables with linear actions of a metacyclic group $G=C_p\times H$, where $H$ is a $p^{\prime}$-group, and prove a periodicity property of the symmetric algebra $\mathbb{F}[V]$ if $V$ is a direct sum of indecomposable $G$-module such that the norm polynomial of the simple $H$-module is the power of the product of the basis elements of the dual.