Let $G$ be a finite group and $H$ a subgroup.
Let $V_1, \dots , V_r$ be (equivalence class representatives for) the irreducible complex representations of $G$. Let the stabilizer subspace $V_i^H = \{ v \in V_i \ | \ hv=v \ , \forall h \in H \}$
What is a reference for the following equality? $$|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$$
If you know a proof but not a reference, then you can post it, your answer will be a reference.
So expanding on my comment, let $1_H^G$ denote the trivial representation of $H$ induced to $G$ which clearly has dimension $|G:H|$.
On the other hand, denoting by $[V,W]_K$ the usual normalized inner product of representations of the group $K$ the dimension is also (by Frobenius reciprocity) $$\sum_{i=1}\dim(V_i)[V_i,1_H^G]_G = \sum_i\dim(V_i)[V_i,1_H]_H = \sum_i\dim(V_i)\dim(V_i^H)$$