Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations: $$ \hom_G(\phi_! M, N) \cong \hom_H(M, \phi^* N) $$ However, it seems that most textbooks and web pages about representation theory inexplicably consider only the case when $\phi$ is injective, i.e. exhibits $H$ as a subgroup of $G$. In this case, there are formulas for the character of $\phi_! M$ in terms of the character of $M$: $$ \chi_{\phi_!(M)}(g) = \frac{1}{|H|} \sum_{k\in G \atop k^{-1} g k \in H} \chi_M(k^{-1} g k) = \sum_{\text{cosets } k H \atop k^{-1} g k \in H} \chi_M(k^{-1} g k) . $$ Can someone give a reference for versions of these formulas when $\phi$ is not injective?
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asked Mar 1, 2012 at 5:43
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$\begingroup$ Isn't it enough to replace $H$ bu $\phi (H)$ to reduce to the injective case ? $\endgroup$– Paul BroussousCommented Mar 1, 2012 at 11:47
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Exercise 7.1 in Serre's Linear Representations of Finite Groups gives a formula (without proof) in the case where $\phi$ is surjective. It is probably straightforward to compose this formula with your formula for the injective case to get the general formula.
answered Mar 1, 2012 at 7:06
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$\begingroup$ This should yield the general formula by induction by steps. $\endgroup$ Commented Mar 1, 2012 at 8:52
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$\begingroup$ Thanks! I've put them together into the general formula at nlab.mathforge.org/nlab/show/induced+character . I'd still be interested to hear any other references for the general (or just the surjective) case. $\endgroup$ Commented Mar 1, 2012 at 18:11