Character table

In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements.

The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation.

In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons.

[1][2][3][4][5][6] The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form.

The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the character of the trivial representation, which is the trivial action of G on a 1-dimensional vector space by

Here is the character table of C3 = , the cyclic group with three elements and generator u: where ω is a primitive cube root of unity.

The character table for general cyclic groups is (a scalar multiple of) the DFT matrix.

This is tied to the important fact that the irreducible representations of a finite group G are in bijection with its conjugacy classes.

This bijection also follows by showing that the class sums form a basis for the center of the group algebra of G, which has dimension equal to the number of irreducible representations of G. The space of complex-valued class functions of a finite group G has a natural inner product: where

With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table: For

the orthogonality relation for columns is as follows: where the sum is over all of the irreducible characters

The orthogonality relations can aid many computations including: If the irreducible representation V is non-trivial, then

More specifically, consider the regular representation which is the permutation obtained from a finite group G acting on (the free vector space spanned by) itself.

This sum can help narrow down the dimensions of the irreducible representations in a character table.

Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism.

This group is connected to Dirichlet characters and Fourier analysis.

The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table.

Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.

This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.

To find the total number of vibrational modes of a water molecule, the irreducible representation Γirreducible needs to calculate from the character table of a water molecule first.

point group, which is also the character table for a water molecule.

In here, the first row describes the possible symmetry operations of this point group and the first column represents the Mulliken symbols.

A simple way to determine the characters for the reducible representation

An easiest way to calculate "contribution per unshifted atom" for

A simplified version of above statements is summarized in the table below per unshifted atom Character of

Contribution per unshifted atom along each of three axis From the above discussion, a new character table for a water molecule (

point group) can be written as Using the new character table including

Translational motion will corresponds with the reducible representations in the character table, which have

Rotational motion will corresponds with the reducible representations in the character table, which have

Similar rules will apply for rest of the irreducible representations