The subclass algebra associated with a finite group and subgroup

Abstract:

Let $G$ be a finite group and let $H$ be a subgroup of $G$. If $g \in G$, then the set ${E_g} = \{ hg{h^{ - 1}}|h \in H\}$ is the subclass of $G$ containing $g$ and ${\Sigma _{x \in {E_g}}}x$ is the subclass sum containing $g$. The algebra over the field of complex numbers generated by these subclass sums is called the subclass algebra (denoted by $S$) associated with $G$ and $H$. The irreducible modules of $S$ are demonstrated, and results about Schur algebras are used to develop formulas relating the irreducible characters of $S$ to the irreducible characters of $G$ and $H$.