9
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ Isn't it enough to replace $H$ bu $\phi (H)$ to reduce to the injective case ? $\endgroup$ Mar 1, 2012 at 11:47

1 Answer 1

5
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ This should yield the general formula by induction by steps. $\endgroup$
    – Marc Palm
    Mar 1, 2012 at 8:52
  • $\begingroup$ 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. $\endgroup$ Mar 1, 2012 at 18:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.