Character theory
Character theory is a central component of representation theory in mathematics, focusing on the study of finite groups through their characters—complex-valued functions defined on the group elements as the traces of the matrices representing those elements in a given linear representation over the complex numbers.[1] These characters are class functions, constant on conjugacy classes, and they encode essential structural information about the group, allowing representations to be classified up to isomorphism and facilitating the decomposition of representations into irreducible components.[2] Developed primarily by Georg Frobenius in 1896 as an extension of earlier work on abelian groups by Carl Friedrich Gauss and Richard Dedekind, character theory originated from efforts to factor the group determinant, a polynomial constructed from the group's multiplication table.[3] Frobenius's groundbreaking contributions established that irreducible characters form an orthonormal basis for the space of class functions under a specific inner product, with the orthogonality relations providing a linear algebraic framework to compute character tables and determine the number of irreducible representations, which equals the number of conjugacy classes in the group.[4] A key theorem, due to Frobenius, states that the sum of the squares of the dimensions of the irreducible representations equals the order of the group, underscoring the completeness of this decomposition.[2] Beyond its foundational role in abstract algebra, character theory simplifies the analysis of group actions by reducing problems to computations involving traces and inner products, making it indispensable for applications in number theory, combinatorics, and Lie theory.[1] For finite groups over fields of characteristic zero, such as the complex numbers, characters are particularly well-behaved due to the algebraic closure, enabling the full machinery of Schur's lemma and the Artin-Wedderburn theorem to classify semisimple algebras associated with the group ring.[5] Modern extensions, including modular character theory in positive characteristic, build on these classical results to handle broader contexts like p-groups and symmetric groups.[6]Foundations
Definitions
In the context of representation theory for finite groups, a representation of a finite group G is defined as a group homomorphism \rho: G \to \mathrm{[GL](/page/GL)}(V), where V is a finite-dimensional vector space over the complex numbers \mathbb{C} and \mathrm{[GL](/page/GL)}(V) denotes the general linear group of invertible linear transformations on V.[7] This setup encodes the action of G on V via linear transformations, preserving the group structure. The dimension of V, denoted \dim V, is called the degree of the representation. The character \chi associated to a representation \rho is the function \chi: G \to \mathbb{C} given by \chi(g) = \operatorname{Tr}(\rho(g)) for each g \in G, where \operatorname{Tr} is the trace of the matrix representing \rho(g) with respect to any basis of V.[8] Characters are class functions, meaning \chi(g) = \chi(hgh^{-1}) for all g, h \in G, as the trace is invariant under simultaneous conjugation of the matrix. The space of all class functions on G, denoted \mathrm{CF}(G), consists of complex-valued functions constant on the conjugacy classes \mathrm{Cl}(G) of G.[4] An irreducible representation of G is one that admits no proper nontrivial invariant subspace under the action of \rho. The characters of the irreducible representations form an orthonormal basis for the vector space \mathrm{CF}(G) with respect to the inner product \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi(g)} \psi(g), though the orthogonality details are explored elsewhere.[9] The set of irreducible characters is denoted \mathrm{Irr}(G), and its cardinality equals the number of conjugacy classes |\mathrm{Cl}(G)|. A fundamental example is the trivial representation, where \rho(g) is the identity transformation on V for all g \in G, yielding the trivial character \chi_{\mathrm{triv}}(g) = 1 for every g \in G. This one-dimensional representation is always irreducible.[1]Representations and linear characters
Linear characters are the characters of one-dimensional representations of a finite group G, which are group homomorphisms \chi: G \to \mathbb{C}^\times satisfying \chi(gh) = \chi(g)\chi(h) for all g, h \in G.[10] These representations map elements of G to the multiplicative group of nonzero complex numbers, preserving the group operation multiplicatively.[10] For a finite abelian group G, every irreducible representation is one-dimensional, meaning all irreducible characters are linear.[10] The set of all linear characters of G forms a group under pointwise multiplication, known as the dual group \hat{G}, which is isomorphic to G itself.[10] The trivial character, which sends every element to 1, serves as the identity element in this dual group. The kernel of a linear character \chi, defined as \ker(\chi) = \{g \in G \mid \chi(g) = 1\}, is a normal subgroup of G.[10] By the first isomorphism theorem for groups, the quotient G / \ker(\chi) is isomorphic to the image \operatorname{im}(\chi) \subseteq \mathbb{C}^\times, which is a finite cyclic subgroup of the unit circle.[4] A concrete example arises with the cyclic group \mathbb{Z}/n\mathbb{Z}, generated by 1 modulo n. Its linear characters are given by \chi_k(m) = e^{2\pi i k m / n} for k = 0, 1, \dots, n-1 and m \in \mathbb{Z}/n\mathbb{Z}, corresponding to the nth roots of unity.[10] In general, the number of linear characters of a finite abelian group G equals |G|, matching the order of the group since the dual group is isomorphic to G.[10]Properties
Arithmetic properties
In the theory of representations of finite groups over the complex numbers, the character \chi associated to a representation \rho: G \to \mathrm{GL}(V) is defined by \chi(g) = \mathrm{tr}(\rho(g)) for g \in G. These characters are constant on conjugacy classes of G, meaning \chi(hgh^{-1}) = \chi(g) for all h, g \in G, and thus belong to the space of class functions on G.[11] The degree of a character \chi, denoted \chi(1), is the value at the identity element and equals the dimension of the representation space V. This degree is a positive integer. For an irreducible character, the degree \chi(1) divides the order of the group |G|.[11] A fundamental inequality states that for any character \chi and element g \in G, |\chi(g)| \leq \chi(1), with equality holding if and only if \rho(g) is a scalar multiple of the identity matrix on V. This bound reflects the unitary nature of representations of finite groups up to equivalence.[11] The arithmetic structure of characters is further illuminated by the inner product on the space of class functions, defined as \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \psi(g^{-1}). For irreducible characters \chi, \psi, this inner product takes integer values: it equals 1 if \chi = \psi and 0 otherwise.[11] In particular, the norm \langle \chi, \chi \rangle = 1 if and only if \chi is the character of an irreducible representation. By Schur's lemma, which asserts that the endomorphism algebra of an irreducible representation over \mathbb{C} is isomorphic to \mathbb{C}, distinct irreducible representations cannot have the same character, ensuring that irreducible characters are pairwise distinct.Multiplicativity and orthogonality
One key property of characters in representation theory is their multiplicativity under direct products of groups. For finite groups G and H, if \chi \in \operatorname{Irr}(G) and \psi \in \operatorname{Irr}(H), then the irreducible characters of the direct product G \times H are precisely the products \chi \times \psi, defined by (\chi \times \psi)(g, h) = \chi(g) \psi(h) for g \in G and h \in H.[12] This reflects how representations of G \times H arise as external tensor products of representations of G and H. A related multiplicativity holds for tensor products of representations of the same group. If \rho: G \to \operatorname{GL}(V) and \sigma: G \to \operatorname{GL}(W) are representations with characters \chi_\rho and \chi_\sigma, the tensor product representation \rho \otimes \sigma on V \otimes W has character \chi_{\rho \otimes \sigma}(g) = \chi_\rho(g) \chi_\sigma(g) for all g \in G.[12] This product structure facilitates the analysis of how representations combine under tensoring, preserving the trace via the multiplicativity of traces on tensor products of matrices. In contrast, characters exhibit additivity under direct sums of representations. For representations \rho on V and \sigma on W, the direct sum \rho \oplus \sigma on V \oplus W has character \chi_{\rho \oplus \sigma}(g) = \chi_\rho(g) + \chi_\sigma(g) for all g \in G.[12] This linearity allows any representation to be expressed as a direct sum of irreducible ones, with the character serving as an additive invariant. Irreducible characters also display orthogonality properties when viewed as functions constant on conjugacy classes. Specifically, the set of irreducible characters \operatorname{Irr}(G) forms an orthogonal basis for the space of class functions on G with respect to the inner product \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, where the sum can be grouped over conjugacy classes due to class constancy.[12] This column orthogonality underpins the uniqueness of character tables and decomposition into irreducibles. To illustrate these properties, consider the symmetric group S_3, which has three conjugacy classes: the identity \{e\}, transpositions (order 2), and 3-cycles (order 3). Its irreducible characters are the trivial character \chi_1 \equiv 1, the sign character \chi_{\operatorname{sgn}} with values $1, -1, 1, and the 2-dimensional character \chi_2 with values $2, 0, -1.[4] The additivity is evident in the permutation representation, whose character $3, 1, 0 decomposes as \chi_1 + \chi_2. For multiplicativity, the tensor product of \chi_2 with itself yields the character $4, 0, 1, which further decomposes but demonstrates the pointwise product rule. Restricting to the subgroup A_3 \cong C_3, the characters of S_3 multiply consistently with those of the subgroup's irreducibles (the trivial and two complex 1-dimensional characters), aligning with the direct product structure for abelian factors.[12]Character Tables
Construction and examples
The character table of a finite group G is constructed by indexing the columns with the conjugacy classes of G and the rows with its irreducible characters. The number of irreducible characters equals the number of conjugacy classes.[13] To determine the table, first compute the conjugacy classes explicitly for small groups. The degrees (values at the identity) of the irreducible characters \chi(1) must satisfy \sum \chi(1)^2 = |G|, and each degree divides |G|. One standard approach uses the decomposition of known representations, such as the regular representation or permutation representations on cosets. The regular representation of G acts on the vector space of functions on G by left translation, with character \chi_{\mathrm{reg}}(g) = |G| if g = e (the identity) and $0 otherwise.[14] This character decomposes as \chi_{\mathrm{reg}} = \sum_{\chi} \chi(1) \cdot \chi, where the sum is over all irreducible characters \chi, so the multiplicity of each irreducible is its degree \chi(1).[14] For small groups, the remaining character values can be found by decomposing the permutation representation (e.g., the action on cosets of subgroups) into irreducibles or by solving systems based on known values and verification via orthogonality relations.Example: Symmetric Group S_3
The symmetric group S_3 has order 6 and three conjugacy classes: the identity \{e\} (size 1), the 3-cycles \{(123), (132)\} (size 2), and the transpositions \{(12), (13), (23)\} (size 3).[15] There are thus three irreducible characters, with degrees satisfying d_1^2 + d_2^2 + d_3^2 = 6; the possible degrees are 1, 1, and 2 (as S_3 has two 1-dimensional representations from its abelianization S_3 / A_3 \cong C_2).[15] The trivial representation gives the first row: \chi_1 = (1, 1, 1). The sign representation (det of the permutation representation) gives \chi_2 = (1, 1, -1). The remaining 2-dimensional irreducible is the standard representation on \mathbb{C}^3 modulo the trivial subspace, with character \chi_3 = (2, -1, 0), obtained by subtracting the trivial and sign characters from the permutation character (3, 0, 1).[15] The full table is:| Character / Class | e (size 1) | 3-cycles (size 2) | Transpositions (size 3) |
|---|---|---|---|
| Trivial (\chi_1) | 1 | 1 | 1 |
| Sign (\chi_2) | 1 | 1 | -1 |
| Standard (\chi_3) | 2 | -1 | 0 |
Example: Quaternion Group Q_8
The quaternion group Q_8 = \{\pm 1, \pm i, \pm j, \pm k\} has order 8 and five conjugacy classes: \{1\} (size 1), \{-1\} (size 1), \{i, -i\} (size 2), \{j, -j\} (size 2), and \{k, -k\} (size 2).[16] Thus, there are five irreducible characters, with degrees 1, 1, 1, 1, and 2 (as \sum d_i^2 = 8). The four 1-dimensional characters arise from the quotient Q_8 / \langle -1 \rangle \cong C_2 \times C_2, which is abelian. The trivial character is \chi_1 = (1, 1, 1, 1, 1). The other three 1-dimensional characters are the sign-like representations with kernels \langle i \rangle, \langle j \rangle, and \langle k \rangle: \chi_2 = (1, 1, 1, -1, -1), \chi_3 = (1, 1, -1, 1, -1), and \chi_4 = (1, 1, -1, -1, 1).[16] The 2-dimensional irreducible is faithful, realized over \mathbb{C} using quaternionic units (e.g., via matrices with i and j satisfying i^2 = j^2 = -1, ij = -ji = k), with character \chi_5 = (2, -2, 0, 0, 0). This representation is not realizable over \mathbb{R} without extension, highlighting the need for complex coefficients despite real-valued characters.[16] The full table is:| Character / Class | \{1\} (size 1) | \{-1\} (size 1) | \{i, -i\} (size 2) | \{j, -j\} (size 2) | \{k, -k\} (size 2) |
|---|---|---|---|---|---|
| Trivial (\chi_1) | 1 | 1 | 1 | 1 | 1 |
| i-kernel (\chi_2) | 1 | 1 | 1 | -1 | -1 |
| j-kernel (\chi_3) | 1 | 1 | -1 | 1 | -1 |
| k-kernel (\chi_4) | 1 | 1 | -1 | -1 | 1 |
| Faithful 2D (\chi_5) | 2 | -2 | 0 | 0 | 0 |