Editing Character theory (section)

==Induced characters and Frobenius reciprocity==
{{main|Induced character|Frobenius reciprocity}}
The characters discussed in this section are assumed to be complex-valued. Let {{mvar|H}} be a subgroup of the finite group {{mvar|G}}. Given a character {{mvar|χ}} of {{mvar|G}}, let {{math|''χ<sub>H</sub>''}} denote its restriction to {{mvar|H}}. Let {{mvar|θ}} be a character of {{mvar|H}}. [[Ferdinand Georg Frobenius]] showed how to construct a character of {{mvar|G}} from {{mvar|θ}}, using what is now known as ''[[Frobenius reciprocity]]''. Since the irreducible characters of {{mvar|G}} form an orthonormal basis for the space of complex-valued class functions of {{mvar|G}}, there is a unique class function {{math|''θ<sup>G</sup>''}} of {{mvar|G}} with the property that

:<math> \langle \theta^{G}, \chi \rangle_G = \langle \theta,\chi_H \rangle_H </math>

for each irreducible character {{mvar|χ}} of {{mvar|G}} (the leftmost inner product is for class functions of {{mvar|G}} and the rightmost inner product is for class functions of {{mvar|H}}). Since the restriction of a character of {{mvar|G}} to the subgroup {{mvar|H}} is again a character of {{mvar|H}}, this definition makes it clear that {{math|''θ<sup>G</sup>''}} is a non-negative [[integer]] combination of irreducible characters of {{mvar|G}}, so is indeed a character of {{mvar|G}}. It is known as ''the character of'' {{mvar|G}} ''induced from'' {{mvar|θ}}. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.

Given a matrix representation {{mvar|ρ}} of {{mvar|H}}, Frobenius later gave an explicit way to construct a matrix representation of {{mvar|G}}, known as the representation [[induced representation|induced from]] {{mvar|ρ}}, and written analogously as {{math|''ρ<sup>G</sup>''}}. This led to an alternative description of the induced character {{math|''θ<sup>G</sup>''}}. This induced character vanishes on all elements of {{mvar|G}} which are not conjugate to any element of {{mvar|H}}. Since the induced character is a class function of {{mvar|G}}, it is only now necessary to describe its values on elements of {{mvar|H}}. If one writes {{mvar|G}} as a [[disjoint union]] of right [[coset]]s of {{mvar|H}}, say

:<math>G = Ht_1 \cup \ldots \cup Ht_n,</math>

then, given an element {{mvar|h}} of {{mvar|H}}, we have:

:<math> \theta^G(h) = \sum_{i \ : \ t_iht_i^{-1} \in H} \theta \left (t_iht_i^{-1} \right ).</math>

Because {{mvar|θ}} is a class function of {{mvar|H}}, this value does not depend on the particular choice of coset representatives.

This alternative description of the induced character sometimes allows explicit computation from relatively little information about the embedding of {{mvar|H}} in {{mvar|G}}, and is often useful for calculation of particular character tables. When {{mvar|θ}} is the trivial character of {{mvar|H}}, the induced character obtained is known as the '''permutation character''' of {{mvar|G}} (on the cosets of {{mvar|H}}).

The general technique of character induction and later refinements found numerous applications in [[Group_theory#Finite_group_theory|finite group theory]] and elsewhere in mathematics, in the hands of mathematicians such as [[Emil Artin]], [[Richard Brauer]], [[Walter Feit]] and [[Michio Suzuki (mathematician)|Michio Suzuki]], as well as Frobenius himself.