Character table

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.

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:

	(1)	(u)	(u²)
1	1	1	1
χ₁	1	ω	ω²
χ₂	1	ω²	ω

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 $S_{3}$ :

	(1)	(12)	(123)
χ_triv	1	1	1
χ_sgn	1	$-$ 1	1
χ_stand	2	0	$-$ 1

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. 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:

\left\langle \alpha ,\beta \right\rangle :={\frac {1}{\left|G\right|}}\sum _{g\in G}\alpha (g){\overline {\beta (g)}}

where ${\overline {\beta (g)}}$ means the complex conjugate of the value of $\beta$ on g. 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:

\left\langle \chi _{i},\chi _{j}\right\rangle ={\begin{cases}0&{\mbox{ if }}i\neq j,\\1&{\mbox{ if }}i=j.\end{cases}}

For $g,h\in G$ the orthogonality relation for columns is as follows:

\sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(h)}}={\begin{cases}\left|C_{G}(g)\right|,&{\mbox{ if }}g,h{\mbox{ are conjugate }}\\0&{\mbox{ otherwise.}}\end{cases}}

where the sum is over all of the irreducible characters $\chi _{i}$ of G and the symbol $\left|C_{G}(g)\right|$ denotes the order of the centralizer of $g$ .

For an unknown character $\chi _{i}$ , it is irreducible if and only if $\left\langle \chi _{i},\chi _{i}\right\rangle =1$ .

The orthogonality relations can aid many computations including:

Decomposing an unknown character as a linear combination of irreducible characters.
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.

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 $\chi (e)=\left|G\right|$ and $\chi (g)=0$ for g not the identity. Then given a irreducible representation $V_{i}$ ,

\left\langle \chi _{reg},\chi _{i}\right\rangle ={\frac {1}{\left|G\right|}}\sum _{g\in G}\chi _{i}(g){\overline {\chi _{reg}(g)}}={\frac {1}{\left|G\right|}}\chi _{i}(1){\overline {\chi _{reg}(1)}}=dimV_{i}

Then decomposing the regular representations as a sum of irreducible representations of G, we get $V_{reg}=\oplus V_{i}^{dimV_{i}}$ . From which we conclude

\left|G\right|=dimV_{reg}=\sum (dimV_{i})^{2}

over all irreducible representations $V_{i}$ . 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 $10=1^{2}+1^{2}+2^{2}+2^{2}$ , 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 (the degrees of the irreducible characters). (See Representation theory of finite groups#Applying Schur's lemma.) 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 (and thus whether or not G is simple) can be recognised from its character table. The kernel of a character χ is the set of elements g in G for which χ(g) = χ(1); 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 derived subgroup of G is the intersection of the kernels of the linear characters of G. In particular, G is Abelian if and only if all its irreducible characters are 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 (an observation of Graham Higman).

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 $g\mapsto g^{-1}$ , 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 $C_{3}$ above, this map sends $u\mapsto u^{2},u^{2}\mapsto u,$ and accordingly switches $\chi_1$ and $\chi_2$ (switching their values of $\omega$ and $\omega ^{2}$ ). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.

Formally, if $\phi \colon G\to G$ is an automorphism of G and $\rho \colon G\to \operatorname {GL}$ is a representation, then $\rho ^{\phi }:=g\mapsto \rho (\phi (g))$ is a representation. If $\phi =\phi _{a}$ 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.

References

Isaacs, I. Martin (1976). Character Theory of Finite Groups. Dover.
Rowland, Todd; Weisstein, Eric W. "Character Table". MathWorld.

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