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 ${\displaystyle \operatorname {Ind} (f)}$ of a class function f on H given by the formula

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

If f is a character of the representation W of H, then this formula for ${\displaystyle \operatorname {Ind} (f)}$ calculates the character of the induced representation V of G.^[1]

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.