Character Operations and Properties

Characters interact with representation-theoretic constructions in predictable and computable ways, providing powerful computational tools.

TheoremProperties of Characters

Let $\chi_V$ and $\chi_W$ be characters of representations $V$ and $W$ of a group $G$ over $\mathbb{C}$ . Then:

Direct sum: $\chi_{V \oplus W}(g) = \chi_V(g) + \chi_W(g)$
Tensor product: $\chi_{V \otimes W}(g) = \chi_V(g) \cdot \chi_W(g)$
Dual: $\chi_{V^*}(g) = \overline{\chi_V(g^{-1})} = \overline{\chi_V(g)}$ (for finite groups)
Complex conjugate: $\overline{\chi_V(g)}$ is the character of the complex conjugate representation $\overline{V}$
Identity value: $\chi_V(e) = \dim V$
Inversion: $\chi_V(g^{-1}) = \overline{\chi_V(g)}$ for unitary representations

Proof of (2): For the tensor product, if $\{e_i\}$ is a basis for $V$ and $\{f_j\}$ for $W$ , then $\{e_i \otimes f_j\}$ is a basis for $V \otimes W$ . The matrix of $\rho_{V \otimes W}(g)$ in this basis is the Kronecker product of $\rho_V(g)$ and $\rho_W(g)$ , whose trace is the product of traces.

ExampleComputing with Characters

For $S_3$ : The three irreducible characters are:

Trivial: $\chi_1 = (1, 1, 1)$ on classes $(e)$ , (transpositions), (3-cycles)
Sign: $\chi_{\text{sgn}} = (1, -1, 1)$
Standard: $\chi_{\text{std}} = (2, 0, -1)$

The tensor product $\chi_{\text{std}} \otimes \chi_{\text{std}}$ has character $(4, 0, 1)$ , which decomposes as: $\chi_{\text{std}}^2 = \chi_1 + \chi_{\text{sgn}} + \chi_{\text{std}}$ using the inner product to find multiplicities.

DefinitionIrreducible Character

A character $\chi$ is irreducible if it is the character of an irreducible representation. Equivalently, $\chi$ is irreducible if and only if: $\langle \chi, \chi \rangle = \frac{1}{|G|} \sum_{g \in G} |\chi(g)|^2 = 1$

TheoremCharacter Table

For a finite group $G$ with $k$ conjugacy classes and irreducible characters $\chi_1, \ldots, \chi_k$ , the character table is the $k \times k$ matrix: $[\chi_i(g_j)]$ where $g_1, \ldots, g_k$ are representatives of the conjugacy classes.

This matrix is invertible, and the irreducible characters form an orthonormal basis for the space of class functions.

Remark

The character table encodes complete information about the representation theory of $G$ in characteristic zero. From the character table, we can:

Determine all irreducible representations (up to isomorphism)
Decompose any representation into irreducibles
Compute tensor products
Answer group-theoretic questions (e.g., find normal subgroups, determine solvability)

The character table is a finite object that captures the entire representation theory, making it an exceptionally powerful computational tool.