在两个群的直积上对有限群特征标余次数的推广

Generalization of the Co-Degree of Characters for Finite Groups on the Direct Product of Two Groups

摘要: 本文旨在推广钱国华老师的特征标余次数。设$G_1$ 和$G_2$ 是两个有限群，${\chi }_{\xi },{\chi }_{\zeta }$ 分别是$G_1$ 和$G_2$ 的任意不可约特征标，复合得到的特征标${\chi }_{\xi }{{}_{\#}}_{\zeta }$ 是$G_1\times G_2$ 的不可约特征标。我们探讨了$G_1\times G_2$ 的特征标余次数$cod\left({\chi }_{\xi }{{}_{\#}}_{\zeta }\right)$ 与$G_1$ 的特征标余次数$cod\left({\chi }_{\xi }\right)$ 及$G_2$ 的特征标余次数$cod\left({\chi }_{\zeta }\right)$ 之间的数量关系，并在此基础上对原有特征标余次数的相关性质进行了推广和拓展。

Abstract: This paper aims to extend the definition of character codegrees as presented by Professor Qian Guohua. Let $G_1$ and $G_2$ be two finite groups, with ${\chi }_{\xi }$ and ${\chi }_{\zeta }$ being arbitrary irreducible characters of $G_1$ and $G_2$ , respectively. The combined character ${\chi }_{\xi }{{}_{\#}}_{\zeta }$ is an irreducible character of $G_1\times G_2$ . We explore the quantitative relationship between the character codegree $cod\left({\chi }_{\xi }{{}_{\#}}_{\zeta }\right)$ of$G_1\times G_2$ , the character codegrees $cod\left({\chi }_{\xi }\right)$ of $G_1$ , and $cod\left({\chi }_{\zeta }\right)$ of $G_2$ . Based on this relationship, we extend and expand the properties of the original character codegrees.