Character table
In group theory, a character table is a square array that displays the values of the irreducible characters of a finite group G, arranged with rows corresponding to the distinct irreducible characters and columns corresponding to the conjugacy classes of G.[1] The entry in the row for an irreducible character \chi and the column for a conjugacy class C is the value \chi(g) for any g \in C, which equals the trace of the matrix representing the action of g in the corresponding irreducible representation.[2] The number of rows and columns in the table equals the number of conjugacy classes in G, which also equals the number of irreducible representations up to isomorphism.[1] Character tables are constructed using the fact that characters are class functions—constant on conjugacy classes—and satisfy orthogonality relations that form the basis for much of representation theory.[2] The column orthogonality relation states that for conjugacy classes C_i and C_j, the sum over all irreducible characters \chi_k of \overline{\chi_k(g_i)} \chi_k(g_j) equals |G| / |C_i| if i = j and 0 otherwise, where g_i \in C_i.[1] Similarly, the row orthogonality relation implies that the inner product \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{\chi(g)} \psi(g) is 1 if \chi = \psi and 0 otherwise for distinct irreducible characters.[2] These relations enable the decomposition of any representation into irreducibles via character projections and provide a complete encoding of the representation theory of G.[1] The development of character theory, including character tables, traces back to the late 19th century, building on earlier work with characters of abelian groups by Gauss in the early 1800s; the modern framework for non-abelian groups was established by Frobenius in 1896 and further advanced by Schur.[3] Character tables play a central role in reducing complex problems about finite groups—such as counting subgroups or determining normal subgroups—to linear algebra over the complex numbers.[2] They also facilitate applications in related fields, including the study of symmetries in physics and chemistry through point groups, though their foundational importance lies in abstract algebra.[4]Basics of Character Tables
Definition
A character table in group theory is a square array that organizes the irreducible representations of a finite group, with rows corresponding to the irreducible representations and columns corresponding to the conjugacy classes of the group elements, where each entry is the character, defined as the trace of the matrix representing the group element in that representation.[2][5] Characters are class functions, meaning they remain invariant under conjugation, so χ(x^{-1}gx) = χ(g) for any group elements x and g, which allows the table to be structured by conjugacy classes rather than individual elements.[2][5] In standard notation, the character of a group element g with respect to a representation is denoted χ(g), and for the identity element e, χ(e) gives the dimension of the representation.[2] The orthogonality relations among these characters ensure the uniqueness and completeness of the irreducible representations listed in the table.[2] In chemistry, character tables are applied to point groups, which describe the symmetry operations of molecules, enabling the classification of molecular orbitals, vibrations, and other symmetry-adapted functions to predict properties such as spectroscopic transitions./04:_Symmetry_and_Group_Theory/4.03:_Properties_and_Representations_of_Groups/4.3.03:_Character_Tables)[6] For instance, by assigning molecular features to irreducible representations via the table, chemists can determine which modes are infrared- or Raman-active based on their transformation properties under the group's operations./04:_Symmetry_and_Group_Theory/4.03:_Properties_and_Representations_of_Groups/4.3.03:_Character_Tables)[6]Construction
The construction of a character table for a finite group or point group involves a systematic process to determine the characters of its irreducible representations organized by conjugacy classes. The first step is to identify all conjugacy classes of the group, which are the equivalence classes of elements under conjugation: two elements h and k are conjugate if there exists g \in G such that k = g^{-1} h g. These classes form the columns of the character table (aside from the row indicating group order and class sizes), and the number of classes equals the number of irreducible representations.[7][8] The second step is to find the irreducible representations by constructing explicit matrix representations \rho(g) for each group element g that satisfy the group's multiplication relations, ensuring the matrices are unitary or orthogonal as appropriate for the field (typically complex numbers for characters). The characters are then the traces \chi(g) = \operatorname{tr}(\rho(g)), which are constant on conjugacy classes. Irreducibility of each representation is verified using Schur's lemma, which asserts that the representation is irreducible if the only matrices A commuting with all \rho(g) (i.e., A \rho(g) = \rho(g) A for all g \in G) are scalar multiples of the identity matrix.[7][5] The third step is to compute the characters for each conjugacy class using the representation matrices or projection operators to isolate irreducible components from a known reducible representation, such as the regular representation.[8] For the cyclic group C_3 = \langle r \mid r^3 = e \rangle, the conjugacy classes are \{e\}, \{r\}, and \{r^2\} since the group is abelian. The irreducible representations are one-dimensional, parameterized by k = 0, 1, 2, with \rho_k(r) = \omega^k where \omega = e^{2\pi i / 3} is a primitive cube root of unity satisfying \omega^3 = 1 and $1 + \omega + \omega^2 = 0. The characters are thus \chi_k(r^s) = \omega^{k s} for s = 0, 1, 2:- For k=0 (trivial representation): \chi_0(e) = 1, \chi_0(r) = 1, \chi_0(r^2) = 1.
- For k=1: \chi_1(e) = 1, \chi_1(r) = \omega, \chi_1(r^2) = \omega^2.
- For k=2: \chi_2(e) = 1, \chi_2(r) = \omega^2, \chi_2(r^2) = \omega.
Example
A representative example of a character table is that for the C_{2v} point group, which is commonly encountered in molecular symmetry analysis. The C_{2v} group consists of four symmetry operations: the identity E, a twofold rotation C_2 about the principal axis, and two vertical reflection planes \sigma_v (typically the xz-plane) and \sigma_v' (typically the yz-plane). All irreducible representations (irreps) of C_{2v} are one-dimensional, as the group is abelian. The complete character table is as follows:| C_{2v} | E | C_2 | \sigma_v | \sigma_v' |
|---|---|---|---|---|
| A_1 | 1 | 1 | 1 | 1 |
| A_2 | 1 | 1 | -1 | -1 |
| B_1 | 1 | -1 | 1 | -1 |
| B_2 | 1 | -1 | -1 | 1 |