This seems like the kind of question that could be answered with Google, but I have done exhaustive searching and come across nothing so far:

My quantum mechanics textbook is discussing group theory in the context of quantum mechanics, and it just introduced the theorem$$\sum_GG_{ik}^{(\alpha)} G_{lm}^{(\beta)}=\frac g{f_\alpha}\delta_{\alpha\beta}\delta_{il}\delta_{km},$$where $G_{ik}$ and $G_{lm}$ are matrix elements corresponding to the symmetry operation $G$ in representations $\alpha$ and $\beta$. My book claims this theorem implies that $\sum_GG_{ik}^{\alpha}=0$ for any nonunity representation $\alpha$. My question is: what is a unity representation? My book, strangely, never seems to define it, and I can't find information online.

This seems like the kind of question that could be answered with Google, but I have done exhaustive searching and come across nothing so far:

My quantum mechanics textbook is discussing group theory in the context of quantum mechanics, and it just introduced the theorem $$\sum_GG_{ik}^{(\alpha)} G_{lm}^{(\beta)}=\frac g{f_\alpha}\delta_{\alpha\beta}\delta_{il}\delta_{km},\tag{1}$$ where $G_{ik}$ and $G_{lm}$ are matrix elements corresponding to the symmetry operation $G$ in representations $\alpha$ and $\beta$. My book claims this theorem implies that $$\sum_GG_{ik}^{\alpha}=0\tag{2}$$ for any non-unity representation $\alpha$. My question is: what is a unity representation? My book, strangely, never seems to define it, and I can't find information online.