Character group
   HOME

TheInfoList



OR:

In
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 ...
, a character group is the group of representations of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
by
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-valued functions. These functions can be thought of as one-dimensional
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
representations and so are special cases of the group
characters 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 ...
that arise in the related context of
character theory In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information ab ...
. 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 class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es. * The characters of irreducible representations are orthogonal. The primary importance of the character group for
finite 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 marke ...
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s is in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, where it is used to construct Dirichlet characters. The character group of the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
also appears in the theory of the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
. For
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
abelian groups, the character group (with an assumption of continuity) is central to
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
.


Preliminaries

Let G be an abelian group. A function f: G \to \mathbb\setminus\ mapping the group to the non-zero complex numbers is called a character of G if it is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
from G to \mathbb C^\times—that is, if f(g_1 g_2) = f(g_1)f(g_2) for all g_1, g_2 \in G. If f is a character of a finite group G, then each function value f(g) is a
root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
, since for each g \in G there exists k \in \mathbb such that g^ = e, and hence f(g)^ = f(g^) = f(e) = 1. Each character ''f'' is a constant on conjugacy classes of ''G'', that is, ''f''(''hgh''−1) = ''f''(''g''). For this reason, a character is sometimes called a class function. A finite abelian group of order ''n'' has exactly ''n'' distinct characters. These are denoted by ''f''1, ..., ''f''''n''. The function ''f''1 is the trivial representation, which is given by f_1(g) = 1 for all g \in G. It is called the principal character of G; the others are called the non-principal characters.


Definition

If ''G'' is an abelian group, then the set of characters ''fk'' forms an abelian group under pointwise multiplication. That is, the product of characters f_j and f_k is defined by (f_j f_k)(g)= f_j(g) f_k(g) for all g \in G. This group is the character group of G and is sometimes denoted as \hat. The identity element of \hat is the principal character ''f''1, and the inverse of a character ''fk'' is its reciprocal 1/''fk''. If G is finite of order ''n'', then \hat is also of order ''n''. In this case, since , f_k(g), = 1 for all g \in G, the inverse of a character is equal to the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.


Alternative definition

There is another definition of character grouppg 29 which uses U(1) = \ as the target instead of just \mathbb^*. This is useful while studying
complex tori In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
because the character group of the lattice in a complex torus V/\Lambda is canonically isomorphic to the dual torus via the Appell-Humbert theorem. That is,
\text(\Lambda, U(1)) \cong V^\vee/\Lambda^\vee = X^\vee
We can express explicit elements in the character group as follows: recall that elements in U(1) can be expressed as
e^
for x \in \mathbb. If we consider the lattice as a subgroup of the underlying real vector space of V, then a homomorphism
\phi: \Lambda \to U(1)
can be factored as a map
\phi : \Lambda \to \mathbb \xrightarrow U(1)
This follows from elementary properties of homomorphisms. Note that
\begin \phi(x+y) &= \exp(f(x+y)) \\ &= \phi(x) + \phi(y) \\ &= \exp(2\pi i f(x))\exp(2\pi i f(y)) \end
giving us the desired factorization. As the group
\text(\Lambda,\mathbb) \cong \text(\mathbb^,\mathbb)
we have the isomorphism of the character group, as a group, with the group of homomorphisms of \mathbb^ to \mathbb. Since \text(\mathbb,G)\cong G for any abelian group G, we have
\text(\mathbb^, \mathbb) \cong \mathbb^
after composing with the complex exponential, we find that
\text(\mathbb^, U(1)) \cong \mathbb^/\mathbb^
which is the expected result.


Examples


Finitely generated abelian groups

Since every finitely generated abelian group is isomorphic to
G \cong \mathbb^\oplus \bigoplus_^m \mathbb/a_i
the character group can be easily computed in all finitely generated cases. From universal properties, and the isomorphism between finite products and coproducts, we have the character groups of G is isomorphic to
\text(\mathbb,\mathbb^*)^\oplus\bigoplus_^k\text(\mathbb/n_i,\mathbb^*)
for the first case, this is isomorphic to (\mathbb^*)^, the second is computed by looking at the maps which send the generator 1 \in \mathbb/n_i to the various powers of the n_i-th roots of unity \zeta_ = \exp(2\pi i/n_i).


Orthogonality of characters

Consider the n \times n matrix ''A'' = ''A''(''G'') whose matrix elements are A_ = f_j(g_k) where g_k is the ''k''th element of ''G''. The sum of the entries in the ''j''th row of ''A'' is given by :\sum_^n A_ = \sum_^n f_j(g_k) = 0 if j \neq 1, and :\sum_^n A_ = n. The sum of the entries in the ''k''th column of ''A'' is given by :\sum_^n A_ = \sum_^n f_j(g_k) = 0 if k \neq 1, and :\sum_^n A_ = \sum_^n f_j(e) = n. Let A^\ast denote the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of ''A''. Then :AA^\ast = A^\ast A = nI. This implies the desired orthogonality relationship for the characters: i.e., :\sum_^n ^* (g_i) f_k (g_j) = n \delta_ , where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
and f^*_k (g_i) is the complex conjugate of f_k (g_i).


See also

*
Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...


References

* See chapter 6 of {{Apostol IANT Number theory Group theory Representation theory of groups