Semigroups and weights for group representations

Let $G$ be a finite group. Consider a pair $\chi=(\chi_+,\chi_-)$ of linear characters of subgroups $P,P^-$ of $G$ with $\chi_+$ and $\chi_-$ agreeing on $P\cap P^-$. Naturally associated with $\chi$ is a finite monoid $M_\chi$. Semigroup representation theory then yields a representation $\theta$ of $G$. If $\theta$ is irreducible, we say that $\chi$ is a weight for $\theta$. When the underlying field is the field of complex numbers, we obtain a formula for the character of $\theta$ in terms of $\chi_+$ and $\chi_-$. We go on to construct weights for some familiar group representations.