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 that arise 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: 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 that arise 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: 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 f : G → C ∖ { 0 } {displaystyle f:G ightarrow mathbb {C} ackslash {0}} mapping the group to the non-zero complex numbers is called a character of G if it is a group homomorphism from G {displaystyle G} to C × {displaystyle mathbb {C} ^{ imes }} —that is, if ∀ g 1 , g 2 ∈ G f ( g 1 g 2 ) = f ( g 1 ) f ( g 2 ) {displaystyle forall g_{1},g_{2}in G;;f(g_{1}g_{2})=f(g_{1})f(g_{2})} . If f is a character of a finite group G, then each function value f(g) is a root of unity (since ∀ g ∈ G ∃ k ∈ N {displaystyle forall gin G;;exists kin mathbb {N} } such that g k = e {displaystyle g^{k}=e} , f ( g ) k = f ( g k ) = f ( e ) = 1 {displaystyle f(g)^{k}=f(g^{k})=f(e)=1} ). 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, ∀ g ∈ G f 1 ( g ) = 1 {displaystyle forall gin G;;f_{1}(g)=1} . It is called the principal character of G; the others are called the non-principal characters. The non-principal characters have the property that f i ( g ) ≠ 1 {displaystyle f_{i}(g) eq 1} for some g ∈ G {displaystyle gin G} . If G is an abelian group of order n, then the set of characters fk forms an abelian group under multiplication ( f j f k ) ( g ) = f j ( g ) f k ( g ) {displaystyle (f_{j}f_{k})(g)=f_{j}(g)f_{k}(g)} for each element g ∈ G {displaystyle gin G} . This group is the character group of G and is sometimes denoted as G ^ {displaystyle {hat {G}}} . It is of order n. The identity element of G ^ {displaystyle {hat {G}}} is the principal character f1. The inverse of fk is the reciprocal 1/fk. Note that since ∀ g ∈ G | f k ( g ) | = 1 {displaystyle forall gin G;;|f_{k}(g)|=1} , the inverse is equal to the complex conjugate. Consider the n × n {displaystyle n imes n} matrix A=A(G) whose matrix elements are A j k = f j ( g k ) {displaystyle A_{jk}=f_{j}(g_{k})} where g k {displaystyle g_{k}} is the kth element of G. The sum of the entries in the jth row of A is given by