Induced character

In mathematics, an induced character is the character of the representation V of a finite group G induced from a representation W of a subgroup H ≤ G. More generally, there is also a notion of induction $\mathrm{Ind}(f)$ of a class function f on H given by the formula

$\mathrm{Ind}(f)(s)=\frac{1}{|H|}\sum _{t\in G,t^{-1}st\in H}f(t^{-1}st).$

If f is a character of the representation W of H, then this formula for $\mathrm{Ind}(f)$ calculates the character of the induced representation V of G.

The basic result on induced characters is Brauer's theorem on induced characters. It states that every irreducible character on G is a linear combination with integer coefficients of characters induced from elementary subgroups.