Class function

In mathematics, particularly in the field of group theory, a class function on a finite group G is a function f: G \to K (where K is typically the complex numbers \mathbb{C} or another field) that is constant on each conjugacy class of G, meaning f(g) = f(hgh^{-1}) for all g, h \in G.^[1] These functions are invariant under the conjugation action of the group on itself and thus depend only on the conjugacy class structure of G.^[2] Class functions form a vector space \mathcal{C}_G over the base field, with dimension equal to the number of conjugacy classes in G, and they admit a natural inner product defined by \langle f_1, f_2 \rangle_G = \frac{1}{|G|} \sum_{g \in G} f_1(g) \overline{f_2(g)} when working over \mathbb{C}.^[3] This structure makes them particularly significant in representation theory, where irreducible characters—traces of matrix representations—are class functions, enabling the decomposition of representations via orthogonality relations and the computation of character tables.^[3] Examples include indicator functions on conjugacy classes and functions arising from homomorphisms to other groups, with applications extending to the study of symmetric functions and algebraic combinatorics.^[1]

Definition and preliminaries

Formal definition

In group theory, a class function on a finite group G is a function f: G \to \mathbb{C} such that f(g) = f(hgh^{-1}) for all g, h \in G.^[4] More generally, the codomain may be any field K.^[5] This invariance under conjugation is the defining property, ensuring that f is constant on each conjugacy class of G.^[4] The set of all such class functions is denoted \mathrm{Cl}(G) or \mathrm{cf}(G), forming the space of functions that depend solely on the conjugacy class of their argument.^[6] Class functions are valuable because they classify group elements by their conjugate equivalence, capturing intrinsic structural properties independent of specific embeddings in the group.^[5]

Relation to conjugacy classes

In group theory, the conjugacy class of an element g in a group G, denoted \mathrm{Cl}(g), is the set \{ h g h^{-1} \mid h \in G \} consisting of all elements conjugate to g.^[7] Conjugation defines an equivalence relation on G, where two elements are equivalent if one is a conjugate of the other; thus, the conjugacy classes form a partition of G into disjoint subsets that cover the entire group.^[7] The size of a conjugacy class \mathrm{Cl}(g) is given by the formula |\mathrm{Cl}(g)| = |G| / |C_G(g)|, where C_G(g) = \{ h \in G \mid h g = g h \} is the centralizer of g in G; this follows from the fact that the conjugacy class is the orbit of g under the conjugation action, and its size equals the index of the centralizer subgroup.^[7] Class functions, which are functions f: G \to \mathbb{C} invariant under conjugation (i.e., f(h g h^{-1}) = f(g) for all g, h \in G), are precisely those functions that take constant values on each conjugacy class \mathrm{Cl}(g).^[8]

Algebraic structure

Vector space properties

The space of class functions on a finite group G, denoted \mathrm{Cl}(G), forms a vector space over the complex numbers \mathbb{C} (or more generally over a field K of characteristic not dividing |G|), equipped with pointwise addition and scalar multiplication: for \phi, \psi \in \mathrm{Cl}(G) and c \in \mathbb{C}, the functions (\phi + \psi)(g) = \phi(g) + \psi(g) and (c\phi)(g) = c \cdot \phi(g) are also class functions.^[9]^[10] The dimension of \mathrm{Cl}(G) equals the number of conjugacy classes in G, denoted k(G).^[10]^[9] Since G is finite, \mathrm{Cl}(G) is finite-dimensional with \dim \mathrm{Cl}(G) = k(G).^[10] A basis for \mathrm{Cl}(G) is given by the indicator functions of the conjugacy classes of G; for each conjugacy class \mathrm{Cl}(g) = \{hgh^{-1} \mid h \in G\}, the corresponding indicator function \chi_{\mathrm{Cl}(g)} is defined by

\chi_{\mathrm{Cl}(g)}(x) = \begin{cases} 1 & \text{if } x \in \mathrm{Cl}(g), \\ 0 & \text{otherwise}. \end{cases}

These k(G) functions are linearly independent and span \mathrm{Cl}(G), as any class function is uniquely determined by its values on the conjugacy classes.^[9]

Connection to the group algebra

The group algebra K[G] of a finite group G over a field K (typically the complex numbers \mathbb{C}) consists of all formal linear combinations \sum_{g \in G} a_g g where a_g \in K, equipped with addition componentwise and multiplication extended linearly from the group operation: \left( \sum a_g g \right) \left( \sum b_h h \right) = \sum_{g,h \in G} a_g b_h (g h).^[11] This structure makes K[G] an associative algebra with unit e, the identity element of G.^[11] The center Z(K[G]) comprises those elements z = \sum_{g \in G} a_g g that commute with every element of K[G], equivalently, with every group element: z h = h z for all h \in G. This condition implies a_{h g h^{-1}} = a_g for all g, h \in G, so the coefficients a_g are constant on each conjugacy class of G. Thus, Z(K[G]) has a basis given by the class sums s_C = \sum_{g \in C} g for each conjugacy class C of G, and its dimension equals the number of conjugacy classes. The vector space of class functions \mathrm{Cl}(G, K), consisting of functions f: G \to K constant on conjugacy classes, is isomorphic as a K-vector space to Z(K[G]) via the linear map \phi: f \mapsto \sum_{g \in G} f(g) g. This map is well-defined because if f is constant on classes, \phi(f) lies in the center.^[11] For finite G, this embedding identifies class functions with central elements, facilitating techniques like averaging operators over conjugacy classes to project onto the center.

Role in representation theory

Characters as class functions

In representation theory, a representation of a finite group G is a group homomorphism \rho: G \to \mathrm{GL}(V), where V is a finite-dimensional vector space over a field K of characteristic zero, typically the complex numbers \mathbb{C}. The character associated to this representation, denoted \chi_\rho, is the function \chi_\rho: G \to K defined by \chi_\rho(g) = \mathrm{tr}(\rho(g)) for each g \in G, where \mathrm{tr} denotes the trace of the linear operator \rho(g). This trace is independent of the choice of basis for V, making \chi_\rho well-defined as a function on G.^[12] Characters are class functions, meaning \chi_\rho is constant on conjugacy classes of G. To see this, consider conjugate elements hgh^{-1} for h \in G:

\chi_\rho(hgh^{-1}) = \mathrm{tr}(\rho(hgh^{-1})) = \mathrm{tr}(\rho(h)\rho(g)\rho(h)^{-1}).

By the cyclic property of the trace, \mathrm{tr}(AB) = \mathrm{tr}(BA) for matrices A, B, it follows that \mathrm{tr}(\rho(h)\rho(g)\rho(h)^{-1}) = \mathrm{tr}(\rho(g)\rho(h)\rho(h)^{-1}) = \mathrm{tr}(\rho(g)), so \chi_\rho(hgh^{-1}) = \chi_\rho(g). Thus, \chi_\rho depends only on the conjugacy class of its argument.^[13] The irreducible characters of G, which arise from irreducible representations, play a central role: they form an orthonormal basis for the vector space \mathcal{C}_G of class functions on G with respect to the standard inner product on class functions. For any representations \rho and \sigma of the finite group G, the multiplicity of the irreducible representation \rho in a direct sum decomposition of \sigma is given by the inner product \langle \chi_\sigma, \chi_\rho \rangle.^[12]

Inner product formula

In the context of representation theory for a finite group G, the space of class functions is endowed with a Hermitian inner product defined by

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

where \overline{\phi(g)} denotes the complex conjugate.^[14] This inner product is Hermitian, positive definite, and non-degenerate on the space of class functions.^[15] Because class functions are constant on conjugacy classes, the sum simplifies to

\langle \phi, \psi \rangle = \frac{1}{|G|} \sum_{C} |C| \overline{\phi(c)} \psi(c),

where the sum runs over the conjugacy classes C of G and c \in C is a representative element.^[14] The irreducible characters \{\chi_i\} of G are orthonormal with respect to this inner product, satisfying \langle \chi_i, \chi_j \rangle = \delta_{ij}, where \delta_{ij} is the Kronecker delta.^[16] This orthogonality relation implies that the dimension of the space of class functions equals the number of conjugacy classes k(G), and thus the number of irreducible representations of G (up to isomorphism) is also k(G).^[16]

Examples and applications

Abelian groups

In abelian groups, the commutativity implies that every element commutes with all others, so each conjugacy class consists of a single element, or singleton.^[17] Consequently, the space of class functions on a finite abelian group G has dimension |G|, and every complex-valued function on G qualifies as a class function, as there are no nontrivial constraints from conjugation invariance.^[17] For finite abelian groups, all irreducible representations over \mathbb{C} are one-dimensional, with each such representation \rho: G \to \mathbb{C}^\times given by a group homomorphism.^[18] The corresponding character \chi = \chi_\rho is thus \chi(g) = \rho(g) for all g \in G, forming a homomorphism \chi: G \to S^1, where S^1 is the unit circle in \mathbb{C}.^[19] These characters constitute the dual group \hat{G}, which is isomorphic to G itself.^[19] The inner product of two characters \chi, \psi \in \hat{G}, defined as \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g) \overline{\psi(g)}, simplifies to the Kronecker delta: \langle \chi, \psi \rangle = [1](/page/1) if \chi = \psi and $0 otherwise, reflecting the orthogonality of the characters as an orthonormal basis for the space of class functions.^[19] A representative example occurs with the cyclic group \mathbb{Z}/n\mathbb{Z}, generated by $1 \mod n. Its n irreducible characters are \chi_k(m) = \exp(2\pi i k m / n) for k = 0, 1, \dots, n-1 and m \in \{0, 1, \dots, n-1\}, each a homomorphism to S^1.^[19] These satisfy the orthogonality relation, ensuring they form a complete set for decomposing representations of \mathbb{Z}/n\mathbb{Z}.^[19]

Non-abelian finite groups

In non-abelian finite groups, conjugacy classes often partition the group into fewer subsets than in the abelian case, resulting in a class function space whose dimension equals the number of such classes. The symmetric group S_3 provides a simple illustration: it consists of the identity element \{e\} (size 1), the three transpositions \{(1\,2), (1\,3), (2\,3)\} (size 3), and the two 3-cycles \{(1\,2\,3), (1\,3\,2)\} (size 2), yielding three conjugacy classes and thus \dim \mathrm{Cl}(S_3) = 3.^[20] The irreducible characters of S_3, which form a basis for \mathrm{Cl}(S_3), are given by the following character table, with rows corresponding to the trivial representation, the sign representation, and the 2-dimensional standard representation, and columns to the conjugacy classes ordered as identity, transpositions, 3-cycles:

Representation	e	Transpositions	3-cycles
Trivial	1	1	1
Sign	1	-1	1
Standard	2	0	-1

These characters satisfy orthogonality relations via the standard inner product on class functions. For instance, the inner product between the trivial and sign characters is \frac{1}{6}(1 \cdot 1 + 3 \cdot 1 \cdot (-1) + 2 \cdot 1 \cdot 1) = 0, and similarly for the other pairs, confirming their linear independence.^[21] A key application of these characters is the decomposition of the regular representation of S_3, which acts on the group algebra \mathbb{C}[S_3] by left multiplication and has character value 6 at the identity and 0 elsewhere. The multiplicity of each irreducible representation in this decomposition equals its dimension, computed via inner products: the trivial representation appears once, the sign representation once, and the standard representation twice, yielding \mathrm{Reg}(S_3) \cong 1 \oplus \mathrm{sgn} \oplus 2 \cdot \mathrm{std}.^[22] Another illustrative non-abelian example is the quaternion group Q_8 = \{\pm 1, \pm i, \pm j, \pm k\} of order 8, with presentation \langle i, j \mid i^4 = j^4 = 1, i^2 = j^2, ji = i^3 j \rangle. Its five conjugacy classes are \{1\}, \{-1\}, \{i, -i\}, \{j, -j\}, and \{k, -k\}, so \dim \mathrm{Cl}(Q_8) = 5. This group has five irreducible representations over \mathbb{C}: four 1-dimensional and one 2-dimensional. The 1-dimensional characters factor through the abelianization Q_8 / \langle -1 \rangle \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} and distinguish the classes appropriately, while the 2-dimensional representation is faithful with character values 2 at 1, -2 at -1, and 0 on the remaining classes of size 2.