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While reading Fulton and Harris's book 'Representation Theory', I came across Proposition 5.1, which basically states the following:

Let $V$ be an irreducible representation of $G$ and let $W$ be the representation when restricted to $H$, a normal subgroup of $G$. Let |G / H| = 2. Then, either $W$ is still irreducible or $W$ splits up into the direct sum of two irreducible representations of the same dimension that are not isomorphic.

I am confused on the second case. I believe that this will happen when $V$ is an irreducible representation for $G$, but not for $H$. How do we know $W$ will split up into two irreducible representations, that they are of the same dimension and are not isomorphic?

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Recall the inner product of characters

$$ \langle \chi_V,\chi_W\rangle_G=\frac{1}{|G|}\sum_{g\in G} \chi_V(g)\overline{\chi_W(g)} = \sum_{i=1}^r m_in_i $$

where $V_1,\cdots,V_r$ are the irreps of $G$ (up to iso), which appear in $V$ and $W$ with multiplicities $m_1,\cdots,m_r$ and $n_1,\cdots,n_r$ respectively. So here's a hint on your problem:

$$ \frac{1}{|H|}\sum_{h\in H}|\chi_V(h)|^2\le \frac{2}{|G|}\sum_{g\in G}|\chi_V(g)|^2. $$

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