nLab
induced character

Definition

Let $\varphi :H\to G$ be a group homomorphism, $V$ a representation of $H$, and $\chi$ the character of $V$. The induced character ${\varphi }_{!}(\chi )$ of $f$ is the character of the induced $G$-representation

$${\varphi }_{!}(V)={\mathrm{Ind}}_H^G(V)=V{\otimes }_{k[H]}k[G].$$\phi_!(V) = Ind^G_H(V) = V\otimes_{k[H]} k[G].

Formula

There is a formula for the induced character:

$${\varphi }_{!}(\chi )(g)=\frac{1}{\mid H\mid }\sum _{k^{-1}gk=\varphi (h)}\chi (h)$$\phi_!(\chi)(g) = \frac{1}{|H|} \sum_{k^{-1} g k = \phi(h)} \chi(h)

where the sum is over all pairs $(k\in G,h\in H)$ such that $k^{-1}gk=\varphi (h)$.

This formula is usually given only in the case when $\varphi$ is injective, when it can be re-expressed as a sum over cosets. The case when $\varphi$ is surjective is Exercise 7.1 of (Serre) and the general case is easy to put together from these. It can also be derived abstractly using bicategorical trace.

References

• J-P. Serre, Linear Representations of Finite Groups

• MO question about the non-injective case.