循环群群代数的胞腔性

Cellularity of the Group Algebra of a Cyclic Group

摘要: 文章主要构建了置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基，从而说明置换群群元素对应的置换矩阵在复数域上全矩阵代数的中心化子代数是胞腔代数。文章的证明根据章节分成三部分内容，第一部分给出了胞腔代数的定义，中心化子代数的定义，以及一些符号的意义。然后第二部分根据循环群在复数域上的群代数和某些置换矩阵的中心化子代数的联系，证明了循环群在复数域上的群代数是胞腔的。文章的第三部分是根据置换群的群元素可以表示成不相交轮换的乘积，通过其置换矩阵的分块乘积，发现该中心化子的结构和循环群在复数域上的基的联系，最后根据循环群在复数域上的胞腔基构造了一般的置换矩阵在复数域上全矩阵代数的中心化子代数的胞腔基。

Abstract: The article mainly constructs the cellular basis of the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field, thus showing that the centralizer of the permutation matrix corresponding to the elements of the permutation group in the full matrix algebra over the complex field is a cellular algebra. The proof of the article is divided into three parts according to the sections. The first part gives the definition of cellular algebra, the definition of centralizer algebra, and the meaning of some symbols. Then, the second part proves that the group algebra of the cyclic group over the complex field is cellular based on the connection between the group algebra of the cyclic group over the complex field and the centralizer algebra of some permutation matrices. The third part of the article discovers the connection between the structure of the centralizer and the basis of the cyclic group over the complex field through the block product of the permutation matrix according to the fact that the group elements of the permutation group can be expressed as the product of disjoint cycles, and then constructs the cellular basis of the centralizer of the general permutation matrix in the full matrix algebra over the complex field based on the cellular basis of the cyclic group over the complex field.