Degree Homogeneous Subgroups

[摘要] Let $G$ be a finite group and $H$ be a subgroup. We say that $H$is emph{degree homogeneous }if, for each $chiin Irr(G)$, allthe irreducible constituents of the restriction $chi_{H}$ havethe same degree. Subgroups which are either normal or abelian areobvious examples of degree homogeneous subgroups. Following aquestion by E.~M. Zhmud', we investigate general properties ofsuch subgroups. It appears unlikely that degree homogeneoussubgroups can be characterized entirely by abstract groupproperties, but we provide mixed criteria (involving both groupstructure and character properties) which are both necessary andsufficient. For example, $H$ is degree homogeneous in $G$ if andonly if the derived subgroup $H^{prime}$ is normal in $G$ and,for every pair $alpha,eta$ of irreducible $G$-conjugatecharacters of $H^{prime}$, all irreducible constituents of$alpha^{H}$ and $eta^{H}$ have the same degree.

[学科分类] 数学（综合）

[关键词] Yang-Mills connection;vector bundle;gauge transformation