nLab
induced character

Induced characters

Definition

Let ϕ:HG\phi\colon H\to G be a group homomorphism, VV a representation of HH, and χ\chi the character of VV. The induced character ϕ !(χ)\phi_!(\chi) of ff is the character of the induced GG-representation

ϕ !(V)=Ind H G(V)=V 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:

ϕ !(χ)(g)=1|H| k 1gk=ϕ(h)χ(h) \phi_!(\chi)(g) = \frac{1}{|H|} \sum_{k^{-1} g k = \phi(h)} \chi(h)

where the sum is over all pairs (kG,hH)(k\in G, h\in H) such that k 1gk=ϕ(h)k^{-1} g k = \phi(h).

This formula is usually given only in the case when ϕ\phi is injective, when it can be re-expressed as a sum over cosets. The case when ϕ\phi 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)