- Character group
In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters which arises in the related context of character theory. Whenever a group is represented by matrices, the function defined by the trace of the matrices is called a character; however, these traces do not in general form a group. Some important properties of these one-dimensional characters apply to characters in general:
- Characters are invariant on conjugacy classes.
- The characters of irreducible representations are orthogonal.
The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier transform. For locally compact abelian groups, the character group (with an assumption of continuity) is central to Fourier analysis.
Let G be an abelian group. A function mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism—that is, if and f(e) = 1 where e is the identity of the group.
If f is a character of a finite group G, then each function value f(g) is a root of unity (since all elements of a finite group have finite order).
Each character f is a constant on conjugacy classes of G, that is, f(h g h−1) = f(g). For this reason, the character is sometimes called the class function.
A finite abelian group of order n has exactly n distinct characters. These are denoted by f1, ..., fn. The function f1 is the trivial representation; that is, . It is called the principal character of G; the others are called the non-principal characters. The non-principal characters have the property that for some .
If G is an abelian group of order n, then the set of characters fk forms an abelian group under multiplication (fjfk)(g) = fj(g)fk(g) for each element . This group is the character group of G and is sometimes denoted as . It is of order n. The identity element of is the principal character f1. The inverse of fk is the reciprocal 1/fk. Note that since , the inverse is equal to the complex conjugate.
Orthogonality of characters
Consider the matrix A=A(G) whose matrix elements are Ajk = fj(gk) where gk is the kth element of G.
The sum of the entries in the jth row of A is given by
- if , and
The sum of the entries in the kth column of A is given by
- if , and
Let denote the conjugate transpose of A. Then
This implies the desired orthogonality relationship for the characters: i.e.,
where δij is the Kronecker delta and is the complex conjugate of fk(gi).
Wikimedia Foundation. 2010.
Look at other dictionaries:
Character theory — This article refers to the use of the term character theory in mathematics. For the media studies definition, see Character theory (Media). In mathematics, more specifically in group theory, the character of a group representation is a function… … Wikipedia
Character table — Main article: Character theory In group theory, a character table is a two dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to classes of group elements. The entries consist of characters … Wikipedia
Character (mathematics) — In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word character are almost always… … Wikipedia
Character education — is an umbrella term loosely used to describe the teaching of children in a manner that will help them develop variously as moral, civic, good, mannered, behaved, non bullying, healthy, critical, successful, traditional, compliant and/ or socially … Wikipedia
Group marriage — Group marriage, also known as multi lateral marriage, is a form of polyamory in which more than two persons form a family unit, with all the members of the group marriage being considered to be married to all the other members of the group… … Wikipedia
Character Map — A component of Microsoft Windows Character Map is a utility included with Microsoft Windows operating systems and is used to view the characters in any installed … Wikipedia
Group psychotherapy — is a form of psychotherapy in which one or more therapists treat a small group of clients together as a group. The term can legitimately refer to any form of psychotherapy when delivered in a group format, including Cognitive behavioural therapy… … Wikipedia
Gröûp X — (also known as Gröûp X: Arabian Rap Sensations) is a faux Arabian rock band, claiming to be from the fictional village of Cramshananteen, Saudi Arabia. They play a mix of rap and rock. They are known for using comically accented and mangled… … Wikipedia
Group Captain — (Gp Capt in the RAF and Indian Air Force, GPCAPT in the RNZAF and RAAF, G/C in the former RCAF) is a senior commissioned rank in the Royal Air Force [cite web |url=http://www.raf.mod.uk/structure/commissionedranks.cfm |title=Ranks and Badges of… … Wikipedia
Group-dynamic game — Group dynamic games are experiential education exercises which help people to learn about themselves, interpersonal relationships, and how groups function from a group dynamics or social psychological point of view.Group dynamics can be… … Wikipedia