group theory

# Induced characters

## Definition

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

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

## Formula

There is a formula for the induced character:

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

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

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

Revised on March 1, 2012 22:10:25 by Urs Schreiber (82.169.65.155)