# A reference for the equality $|G:H| = \sum_i \dim(V_i)\dim(V_i^H)$

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.

• Hmm, this seems like it should follow just by inducing the trivial representation of $H$ to $G$ and describing the dimension of this in two ways, but I will need a pen to write out the details. – Tobias Kildetoft Apr 13 '16 at 9:46

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) $$\sum_{i=1}\dim(V_i)[V_i,1_H^G]_G = \sum_i\dim(V_i)[V_i,1_H]_H = \sum_i\dim(V_i)\dim(V_i^H)$$
• Can you give more details about the Frobenius reciprocity and the equality $[V_i,1_H^G]_G = [V_i,1_H]_H$? – Sebastien Palcoux Apr 16 '16 at 19:26
• The same argument, but without the use of characters: Let $A = K\left[G\right]$. The Artin-Wedderburn theorem yields that $A \cong \bigoplus_{i=1}^r \left(V_i\right)^{\oplus \dim\left(V_i\right)}$ as left $A$-modules. Hence, $A^H \cong \left(\bigoplus_{i=1}^r \left(V_i\right)^{\oplus \dim\left(V_i\right)}\right)^H \cong \bigoplus_{i=1}^r \left(V_i^H\right)^{\oplus \dim\left(V_i\right)}$ as vector spaces (since the operation of taking $H$-fixed points commutes with direct sums). Taking dimensions on both sides of this isomorphism, we obtain ... – darij grinberg Mar 12 '17 at 0:38
• ... the equality $\dim \left(A^H\right) = \sum_{i=1}^r \dim\left(V_i\right) \dim\left(V_i^H\right)$. Now, it remains to prove that $\dim \left(A^H\right) = \left[G : H\right]$ (where $\left[G : H\right]$ is my notation for what you call $\left|G : H\right|$). But this is easy: Just observe that $\left(\sum_{g \in K} g\right)_{K \in H \backslash G}$ is a basis of the vector space $A^H$, where we regard $H \backslash G$ as the set of all right cosets of $H$ in $G$ (that is, subsets of $G$ having the form $Hk$ for some $k \in G$). – darij grinberg Mar 12 '17 at 0:41