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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.

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  • $\begingroup$ 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. $\endgroup$ – Tobias Kildetoft Apr 13 '16 at 9:46
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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) $$\sum_{i=1}\dim(V_i)[V_i,1_H^G]_G = \sum_i\dim(V_i)[V_i,1_H]_H = \sum_i\dim(V_i)\dim(V_i^H)$$

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  • $\begingroup$ Can you give more details about the Frobenius reciprocity and the equality $[V_i,1_H^G]_G = [V_i,1_H]_H$? $\endgroup$ – Sebastien Palcoux Apr 16 '16 at 19:26
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    $\begingroup$ @SébastienPalcoux That equality is precisely Frobenius reciprocity as it is usually stated for characters, since the inner product counts multiplicities due to the irreducibles forming an orthonormal basis, and the multiplies are also given as the dimensions of homomorphism spaces since everything is semisimple. $\endgroup$ – Tobias Kildetoft Apr 16 '16 at 21:09
  • $\begingroup$ Do you know a reference where the proof you suggested appear? $\endgroup$ – Sebastien Palcoux Nov 10 '16 at 8:49
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    $\begingroup$ 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 ... $\endgroup$ – darij grinberg Mar 12 '17 at 0:38
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    $\begingroup$ ... 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 $k \in G$). $\endgroup$ – darij grinberg Mar 12 '17 at 0:41

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