Character group

In mathematics, particularly in the fields of abstract algebra and harmonic analysis, the character group (also known as the dual group) of a locally compact abelian group G is defined as the set of all continuous group homomorphisms from G to the circle group \mathbb{T} (the multiplicative group of complex numbers with absolute value 1), equipped with pointwise multiplication as the group operation and the compact-open topology.^[1] This structure plays a fundamental role in Pontryagin duality, a cornerstone theorem stating that every locally compact abelian group G is topologically isomorphic to its double dual \hat{\hat{G}}, where the duality map sends each element g \in G to the evaluation functional g' \in \hat{\hat{G}} defined by g'(\chi) = \chi(g) for \chi \in \hat{G}.^[1] For finite abelian groups, which are discrete and compact in the discrete topology, the character group \hat{G} consists of all group homomorphisms \chi: G \to S^1 (where S^1 denotes the unit circle), and Pontryagin duality implies that |\hat{G}| = |G| and G \cong \hat{G} as groups, though the isomorphism is not canonical.^[2] Key properties include the fact that characters take values in roots of unity whose orders divide the orders of elements in G, and the trivial character (constant 1) serves as the identity element.^[2] The character group enables Fourier analysis on abelian groups, where characters form an orthogonal basis for the group algebra or L^2(G), facilitating decompositions similar to those using exponentials on \mathbb{[R](/page/R)}.^[2] Notable examples include the character group of \mathbb{[Z](/page/Z)}, which is isomorphic to \mathbb{T}; the character group of \mathbb{[R](/page/R)}, isomorphic to itself; and for a cyclic group \mathbb{Z}/n\mathbb{Z}, whose characters are precisely the n-th roots of unity.^[1] In broader contexts, such as representation theory of finite groups, the linear characters (one-dimensional representations) form the character group of the abelianization G/[G,G].^[2]

Background Concepts

Abelian Groups

An abelian group is a group G equipped with a binary operation * that satisfies the group axioms and the commutativity condition: for all a, b \in G, a * b = b * a. This commutative property distinguishes abelian groups from non-abelian groups, where the order of operation may affect the result. The concept forms a foundational structure in abstract algebra, essential for studying symmetries and transformations in mathematics.^[3] Basic examples illustrate the structure clearly. The set of integers \mathbb{Z} under addition forms an infinite abelian group, where the operation is commutative since a + b = b + a for all integers a, b. Finite examples include the cyclic groups \mathbb{Z}/n\mathbb{Z}, consisting of integers modulo n under addition, which are abelian by construction. Direct products, such as \mathbb{Z} \times \mathbb{Z}, also yield abelian groups, combining multiple copies of \mathbb{Z} with componentwise addition. These examples highlight how abelian groups capture both discrete and modular arithmetic structures.^[3] A key result is the classification theorem for finite abelian groups, which states that every such group is isomorphic to a direct sum of cyclic groups of prime power order. This decomposition uniquely determines the group's structure up to isomorphism, providing a complete invariant for finite cases. The theorem was established by Leopold Kronecker in 1870.^[4] Abelian groups are named after the Norwegian mathematician Niels Henrik Abel (1802–1829), who demonstrated that commutativity in the group of a polynomial is necessary for solvability by radicals; the term was coined by Camille Jordan in the late 19th century. Early developments trace to the 19th century, with contributions from Augustin-Louis Cauchy, who introduced group concepts in 1812, and Kronecker, who formalized abelian structures in number theory contexts.^[5]

Homomorphisms to the Circle Group

The circle group, denoted \mathbb{T}, is the multiplicative group of all complex numbers with modulus 1, consisting of elements z \in \mathbb{C} such that |z| = 1.^[1] This group is equipped with the topology induced from the complex plane and is isomorphic to the additive group \mathbb{R}/\mathbb{Z}.^[6] For an abelian group G, a group homomorphism \chi: G \to \mathbb{T} is a function satisfying \chi(gh) = \chi(g) \chi(h) for all g, h \in G, with the group operation in G written multiplicatively.^[2] Such maps preserve the group structure, mapping the identity of G to 1 in \mathbb{T} and respecting inverses via \chi(g^{-1}) = \chi(g)^{-1}.^[2] When G is a topological abelian group, these homomorphisms are typically required to be continuous, ensuring compatibility with the topologies on G and \mathbb{T}.^[1] The kernel of such a \chi, defined as \{ g \in G \mid \chi(g) = 1 \}, forms a subgroup of G.^[2] These homomorphisms are commonly referred to as characters of G.^[1] For any character \chi, the image satisfies |\chi(g)| = 1 for all g \in G, allowing representation as \chi(g) = e^{2\pi i \theta(g)} for some real-valued function \theta: G \to \mathbb{R}.^[2]

Formal Definition

Primary Definition

In group theory, the character group of an abelian group G, often denoted \hat{G} or \mathrm{Hom}(G, \mathbb{T}), is defined as the set of all group homomorphisms \chi: G \to \mathbb{T} from G to the circle group \mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \}, equipped with the structure of an abelian group under pointwise multiplication.^[7]^[8] Specifically, for characters \chi_1, \chi_2 \in \hat{G}, their product is given by (\chi_1 \chi_2)(g) = \chi_1(g) \chi_2(g) for all g \in G, and the inverse of \chi is \overline{\chi}, the complex conjugate, since \overline{\chi(g)} = \chi(g)^{-1} as elements of \mathbb{T} have modulus 1.^[7] This construction ensures that \hat{G} is itself an abelian group, with the operation reflecting the multiplicative structure of \mathbb{T}.^[8] The identity element of \hat{G} is the trivial character \varepsilon: G \to \mathbb{T} defined by \varepsilon(g) = 1 for all g \in G, which satisfies \chi \cdot \varepsilon = \chi for any \chi \in \hat{G}.^[2] Alternative notations for the character group include \Gamma(G) in some contexts, particularly in harmonic analysis.^[9] For finite abelian groups G, characters are homomorphisms to the multiplicative group \mathbb{C}^\times of nonzero complex numbers, but since elements of G have finite order, the image lies in the roots of unity, a subgroup of \mathbb{T}, so the definitions coincide up to restriction.^[2]^[10] This setup assumes G is abelian, as the commutativity ensures that characters compose well under pointwise multiplication; for non-abelian groups, the analogous concept involves representation theory rather than a simple character group.^[8]

Alternative Characterizations

In the context of locally compact abelian groups, the character group \hat{G} of an abelian group G is characterized as its Pontryagin dual, consisting of all continuous homomorphisms from G to the circle group \mathbb{T}, equipped with the compact-open topology.^[11] This duality theorem asserts that \hat{G} is also a locally compact abelian group, and furthermore, G is canonically topologically isomorphic to the double dual \hat{\hat{G}}.^[11] Pontryagin duality thus provides an equivalence between the category of locally compact abelian groups and their character groups, enabling generalizations of Fourier analysis to arbitrary such groups. An equivalent formulation of the character group replaces the target circle group \mathbb{T} with the additive group \mathbb{R}/\mathbb{Z}, since \mathbb{T} \cong \mathbb{R}/\mathbb{Z} as topological groups.^[12] In this view, characters are elements of \mathrm{Hom}(G, \mathbb{R}/\mathbb{Z}), where each \chi \in \hat{G} satisfies \chi(g) = \{\theta(g)\} for some \theta: G \to \mathbb{R}, with \{\cdot\} denoting the fractional part modulo 1.^[12] This additive perspective is particularly useful in harmonic analysis, as it aligns characters with exponential functions \chi(g) = e^{2\pi i \theta(g)}.^[12] For discrete abelian groups G, the character group \hat{G} comprises all group homomorphisms from G to \mathbb{T} (without requiring continuity, as the discrete topology makes all maps continuous), and it inherits the compact-open topology, rendering \hat{G} compact.^[13] Conversely, if G is a compact abelian group, then \hat{G} is discrete, consisting of all continuous characters, which separate points in G by the properties of the compact-open topology.^[13] Pontryagin duality was formalized in the 1930s by Lev Pontryagin, building on earlier developments in Fourier analysis for specific groups like the reals and integers, and providing a unified framework for harmonic analysis on general locally compact abelian groups.^[14]

Key Properties

Orthogonality Relations

The orthogonality relations for characters of an abelian group G arise from an inner product structure that highlights their role in decomposing functions on G. For a finite abelian group G, the inner product between two characters \chi, \psi \in \hat{G} is defined as

\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)},

where the bar denotes complex conjugation, since characters map to the unit circle.

\] This inner product satisfies $\langle \chi, \psi \rangle = 1$ if $\chi = \psi$ and $0$ otherwise, demonstrating that the characters are orthonormal.\[

Since every finite abelian group is a direct sum of cyclic groups, all its irreducible representations are one-dimensional, so the characters coincide with the elements of the dual group \hat{G} and form a complete orthonormal basis for the space of all complex-valued functions on G equipped with this inner product.

\] The orthogonality theorem further asserts that distinct irreducible characters are orthogonal under this inner product, enabling the decomposition of the [regular representation](/page/Regular_representation) into a direct sum of one-dimensional representations.\[

A key summation formula derived from these relations is

\sum_{\chi \in \hat{G}} \chi(g) = \begin{cases} |G| & \text{if } g = e, \\ 0 & \text{otherwise}, \end{cases}

which reflects the trace of the regular representation and underscores the isolating property of the identity element.$$] For infinite abelian groups, particularly locally compact abelian (LCA) groups, the orthogonality extends to an L^2 framework using the Haar measure \mu on G. Distinct continuous characters \chi, \psi \in \hat{G} satisfy [ \int_G \chi(g) \overline{\psi(g)} , d\mu(g) = 0