mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an induced character is the

character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

of the representation ''V'' of a

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

''G'' induced from a representation ''W'' of a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

''H'' ≤ ''G''. More generally, there is also a notion of

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

\operatorname(f)

of a

class function In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjugat ...

''f'' on ''H'' given by the formula :

\operatorname(f)(s) = \frac \sum_ f(t^ st).

If ''f'' is a character of the representation ''W'' of ''H'', then this formula for

\operatorname(f)

calculates the character of the induced representation ''V'' of ''G''.. Translated from the second French edition by Leonard L. Scott. The basic result on induced characters is

Brauer's theorem on induced characters Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group. Backgrou ...

. It states that every irreducible character on ''G'' is a linear combination with integer coefficients of characters induced from elementary subgroups.

References

Group theory {{algebra-stub