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

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

\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 \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.

References

Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, New York: Springer-Verlag, 7.2, Proposition 20, ISBN 0-387-90190-6, MR 0450380. Translated from the second French edition by Leonard L. Scott. \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 \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.
References

Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, New York: Springer-Verlag, 7.2, Proposition 20, ISBN 0-387-90190-6, MR 0450380. Translated from the second French edition by Leonard L. Scott.

Mathematics Encyclopedia