Induced character for non-injective homomorphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:19:24Z http://mathoverflow.net/feeds/question/89928 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89928/induced-character-for-non-injective-homomorphisms Induced character for non-injective homomorphisms Mike Shulman 2012-03-01T05:43:59Z 2012-03-01T08:58:34Z <p>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?</p> http://mathoverflow.net/questions/89928/induced-character-for-non-injective-homomorphisms/89932#89932 Answer by Evan Jenkins for Induced character for non-injective homomorphisms Evan Jenkins 2012-03-01T07:06:16Z 2012-03-01T07:06:16Z <p><a href="http://books.google.com/books?id=NCfZgr54TJ4C&amp;pg=PA57&amp;lpg=PA57" rel="nofollow">Exercise 7.1 in Serre's <i>Linear Representations of Finite Groups</i></a> 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.</p>