Lossless-Join Decomposition: Difference between revisions
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In [[mathematics]], an '''induced character''' is the [[character (mathematics)|character]] of the [[group representation|representation]] ''V'' of a [[finite group]] ''G'' [[induced representation|induced]] from a representation ''W'' of a [[subgroup]] ''H'' ≤ ''G''.<!-- Someone should generalize this article. Taku --> More generally, there is also a notion of [[induction of a class function|induction]] <math>\operatorname{Ind}(f)</math> of a [[class function]] ''f'' on ''H'' given by the formula | |||
:<math>\operatorname{Ind}(f)(s) = \frac{1}{|H|} \sum_{t \in G,\ t^{-1} st \in H} f(t^{-1} st).</math> | |||
If ''f'' is a character of the representation ''W'' of ''H'', then this formula for <math>\operatorname{Ind}(f)</math> calculates the character of the induced representation ''V'' of ''G''.<ref>{{citation | |||
| last = Serre | first = Jean-Pierre | author-link = Jean-Pierre Serre | |||
| isbn = 0-387-90190-6 | |||
| location = New York | |||
| mr = 0450380 | |||
| at = 7.2, Proposition 20 | |||
| publisher = Springer-Verlag | |||
| title = Linear Representations of Finite Groups | |||
| year = 1977}}. Translated from the second French edition by Leonard L. Scott.</ref> | |||
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 group|elementary subgroup]]s. | |||
== References == | |||
{{reflist}} | |||
[[Category:Group theory]] | |||
{{algebra-stub}} | |||
Revision as of 13:47, 11 November 2013
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 of a class function f on H given by the formula
If f is a character of the representation W of H, then this formula for 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
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