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. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.
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 any representative 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 character of the trivial representation, which is the trivial action of on a 1-dimensional vector space by formula_1 for all formula_2. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by 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 "C"3 = "", 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 (a scalar multiple of) the DFT matrix.
Another example is the character table of formula_3:
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 .
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. (A class function is one that is constant on conjugacy classes.) 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 formula_5 means the complex conjugate of the value of formula_6 on formula_7. 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 formula_9 the orthogonality relation for columns is as follows:
where the sum is over all of the irreducible characters formula_11 of "G" and the symbol formula_12 denotes the order of the centralizer of formula_7.
For an arbitrary character formula_11, it is irreducible if and only if formula_15.
The orthogonality relations can aid many computations including:
If the irreducible representation "V" is non-trivial, then formula_18.
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 formula_19 and formula_20 for formula_7 not the identity. Then given an irreducible representation formula_22,
Then decomposing the regular representations as a sum of irreducible representations of "G", we get formula_24. From which we conclude
over all irreducible representations formula_22. This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, the dihedral group of order 10) then the only way to express the order of the group as a sum of four squares is formula_27, so we know the dimensions of all the 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 ("D"4) 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 representations of are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if formula_29 and formula_30 are linear representations, then formula_31 defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation formula_32. This group is connected to Dirichlet characters and Fourier analysis.
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 formula_33, which is non-trivial except for elementary abelian 2-groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example of formula_34 above, this map sends formula_35 and accordingly switches formula_36 and formula_37 (switching their values of formula_38 and formula_39). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.
Formally, if formula_40 is an automorphism of "G" and formula_41 is a representation, then formula_42 is a representation. If formula_43 is an inner automorphism (conjugation by some element "a"), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group Aut descends to the quotient Out.
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.