Characters and Class Functions

Character theory provides a powerful computational tool for studying representations through trace functions, reducing infinite-dimensional representation data to finite combinatorial information.

DefinitionCharacter of a Representation

Let $(\rho, V)$ be a finite-dimensional representation of a group $G$ over a field $\mathbb{F}$ . The character of $\rho$ is the function $\chi_V: G \to \mathbb{F}$ defined by: $\chi_V(g) = \text{tr}(\rho(g))$ where $\text{tr}$ denotes the trace of the linear transformation $\rho(g): V \to V$ .

The degree of the character is $\chi_V(e) = \dim V$ , where $e$ is the identity element of $G$ .

ExampleBasic Examples

Trivial representation: $\chi_{\text{triv}}(g) = 1$ for all $g \in G$

Regular representation: For finite $G$ , $\chi_{\text{reg}}(g) = \begin{cases} |G| & g = e \\ 0 & g \neq e \end{cases}$

Standard representation of $S_3$ : Acting on $\mathbb{C}^3$ by permuting coordinates, $\chi(e) = 3$ , $\chi(\text{transposition}) = 1$ , $\chi(\text{3-cycle}) = 0$

DefinitionClass Function

A function $f: G \to \mathbb{F}$ is a class function if it is constant on conjugacy classes: $f(hgh^{-1}) = f(g) \text{ for all } g, h \in G$

The set of class functions forms a vector space, denoted $\mathcal{C}(G, \mathbb{F})$ .

TheoremCharacters are Class Functions

For any representation $\rho$ , the character $\chi_V$ is a class function.

Proof: Using the cyclic property of trace and $\rho$ being a homomorphism: $\chi_V(hgh^{-1}) = \text{tr}(\rho(hgh^{-1})) = \text{tr}(\rho(h)\rho(g)\rho(h)^{-1}) = \text{tr}(\rho(g)) = \chi_V(g)$

Remark

Class functions on a finite group form a vector space of dimension equal to the number of conjugacy classes. For $G$ with $k$ conjugacy classes, $\dim \mathcal{C}(G, \mathbb{F}) = k$ .

The key insight: characters depend only on conjugacy class structure, dramatically reducing computational complexity. Instead of tracking $|G|$ values, we need only track $k$ values where $k$ is often much smaller (e.g., for $S_n$ , the conjugacy classes correspond to cycle types, giving far fewer than $n!$ classes).

DefinitionInner Product on Class Functions

For finite groups over $\mathbb{C}$ , define the Hermitian inner product on $\mathcal{C}(G, \mathbb{C})$ : $\langle f, h \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{f(g)} h(g)$

This can also be written as a sum over conjugacy classes: $\langle f, h \rangle = \frac{1}{|G|} \sum_{i=1}^k |C_i| \overline{f(g_i)} h(g_i)$ where $C_1, \ldots, C_k$ are the conjugacy classes and $g_i \in C_i$ .

Characters encode essential information about representations while being much simpler to compute and manipulate than the full representation data. They are the bridge between abstract representation theory and concrete computation.