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. The columns are labelled by 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 for all. 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 C3 = , the cyclic group with three elements and generator u:
1111
χ11ωω2
χ21ω2ω

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 :
χtriv111
χsgn111
χstand201

where represents conjugacy class consisting of,,, and represents conjugacy class consisting of,. 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. Further, the character table is always square because irreducible characters are pairwise orthogonal, and 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 means the complex conjugate of the value of on. 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 of G and the symbol denotes the order of the centralizer of.
For an arbitrary character, it is irreducible if and only if.
The orthogonality relations can aid many computations including:
  • Decomposing an unknown character as a linear combination of irreducible characters, i.e. # of copies of irreducible representation Vi in V =.
  • Constructing the complete character table when only some of the irreducible characters are known.
  • Finding the orders of the centralizers of representatives of the conjugacy classes of a group.
  • Finding the order of the group,, for any g in G.
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 itself. The characters of this representation are and for not the identity. Then given an irreducible representation,
Then decomposing the regular representations as a sum of irreducible representations of G, we get. From which we conclude
over all irreducible representations. 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 then the only way to express the order of the group as a sum of four squares is, 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 order of G is given by the sum of the squares of the entries of the first column. More generally, the sum of the squares of the absolute values of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
  • All normal subgroups of G can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ = χ; this is a normal subgroup of G. Each normal subgroup of G is the intersection of the kernels of some of the irreducible characters of G.
  • The number of irreducible representations of G equals to the number of conjugacy classes that G has.
  • The commutator subgroup of is the intersection of the kernels of the linear characters of.
  • If is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that is abelian iff each conjugacy class is a singleton iff the character table of is iff each irreducible character is linear.
  • It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group 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 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 and are linear representations, then defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation. This group is connected to Dirichlet characters and Fourier analysis.

Outer automorphisms

The outer automorphism group acts on the character table by permuting columns and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism, which is non-trivial except for elementary abelian 2-groups, and outer because abelian groups are precisely those for which conjugation acts trivially. In the example of above, this map sends and accordingly switches and . Note that this particular automorphism agrees with complex conjugation.
Formally, if is an automorphism of G and is a representation, then is a representation. If is an inner automorphism, then it acts trivially on representations, because representations are class functions. 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, and conversely, one can restrict possible outer automorphisms based on the character table.