Definition of a unity representation

Question

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.

Answer 1

What your book calls the unity representation is also sometimes called the trivial representation, the identity representation, or the scalar representation. The unity/trivial/identity/scalar representation $\rho_\mathbb I$ is the simplest possible representation, mapping every element of the group to the identity operator on the Hilbert space; this trivially satisfies the condition $\rho_\mathbb I(g_1 g_2) = \rho_\mathbb I(g_1)\rho_\mathbb I(g_2)$.

Answer 2

OP's QM textbook apparently calls the trivial representation for the unity representation, cf. the mentioned Schur orthogonality relations.

(In the above context it is implicitly implied that all the involved representations $\alpha$, $\beta$, etc, are irreducible.)