模情况下对称代数在亚循环群作用下的周期性质

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

中文关键词:  不可分解模  对称代数  周期性质

英文关键词:indecomposable module  symmetric algebra  periodicity property

中文摘要:

      在本文中,我们考虑在亚循环群$G=C_p \times H$作用下将对称代数$\mathbb{F}[V]$分解为不可分解模的直和,其中$H$是一个$p^{\prime}$-模.当向量空间$V$作为$G$-模的不可分解直和部分对应的单$H$-模的规范多项式是它的对偶模的基底元素乘积的幂时, 我们证明了对称代数 $\mathbb{F}[V]$的周期性质.

英文摘要:

      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.