In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible group representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the trace of the matrices representing group elements of the column's class in the given row's group representation.
Contents
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.
Definition and example
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on the representatives of the respective conjugacy class of G (because characters are class functions). The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the trivial character, and the first column by (the conjugacy class of) the identity. The entries of the first column are the values of the irreducible characters at the identity, the degrees of the irreducible characters. Characters of degree 1 are known as linear characters.
Here is the character table of C3 = <u>, the cyclic group with three elements and generator u:
where ω is a primitive third root of unity. The character table for general cyclic groups is the DFT matrix.
Another example is the character table of
where (12) represents conjugacy class consisting of (12),(13),(23), and (123) represents conjugacy class consisting of (123),(132). To learn more about character table of symmetric groups,see [1].
The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. 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.
Orthogonality relations
The space of complex-valued class functions of a finite group G has a natural inner-product:
where
For
where the sum is over all of the irreducible characters
For an unknown character
The orthogonality relations can aid many computations including:
More specifically, consider the regular representation which is the permutation obtained from a finite group G acting on itself. The characters of this representation are
Then decomposing the regular representations as a sum of irreducible representations of G, we get
over all irreducible representations
Properties
Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-trivial complex values has a conjugate character.
Certain properties of the group G can be deduced from its character table:
The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements (D4) have the same 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. In 1964, this was answered in the negative by E. C. Dade.
The linear characters form a character group, which has important number theoretic connections.
Outer automorphisms
The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism
Formally, if
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.