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.[1][2][3][4][5][6]

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 G on a 1-dimensional vector space by ${\displaystyle \rho (g)=1}$ for all ${\displaystyle g\in G}$. 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 = <u>, the cyclic group with three elements and generator u:

	(1)	(u)	(u^2)
1	1	1	1
χ1	1	ω	ω^2
χ2	1	ω^2	ω

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 ${\displaystyle S_{3}}$:

	(1)	(12)	(123)
χtriv	1	1	1
χsgn	1	${\displaystyle -}$1	1
χstand	2	0	${\displaystyle -}$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. (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

Main article: Schur orthogonality relations

The space of complex-valued class functions of a finite group G has a natural inner-product:

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

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

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

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

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

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

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

The orthogonality relations can aid many computations including:

• Decomposing an unknown character as a linear combination of irreducible characters, i.e. # of copies of irreducible representation V_i in V = ${\displaystyle \left\langle \chi ,\chi _{i}\right\rangle }$.
• 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, ${\displaystyle \left|G\right|=\left|Cl(g)\right|*\sum _{\chi _{i}}\chi _{i}(g){\overline {\chi _{i}(g)}}}$, for any g in G.

If the irreducible representation V is non-trivial, then ${\displaystyle \sum _{g}\chi (g)=0}$.

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

${\displaystyle \left\langle \chi _{\text{reg}},\chi _{i}\right\rangle ={\frac {1}{\left|G\right|}}\sum _{g\in G}\chi _{i}(g){\overline {\chi _{\text{reg}}(g)}}={\frac {1}{\left|G\right|}}\chi _{i}(1){\overline {\chi _{\text{reg}}(1)}}=\operatorname {dim} V_{i}}$.

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

${\displaystyle \left|G\right|=\operatorname {dim} V_{\text{reg}}=\sum (\operatorname {dim} V_{i})^{2}}$

over all irreducible representations ${\displaystyle V_{i}}$. 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 ${\displaystyle 10=1^{2}+1^{2}+2^{2}+2^{2}}$, 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 number of irreducible representations of G equals to the number of conjugacy classes that G has.
• The commutator subgroup of G is the intersection of the kernels of the linear characters of G.
• If G is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that G is abelian iff each conjugacy class is a singleton iff the character table of G is ${\displaystyle |G|\times |G|}$ 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 (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 representations of G are themselves a group under the tensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, if ${\displaystyle \rho _{1}:G\to V_{1}}$ and ${\displaystyle \rho _{2}:G\to V_{2}}$ are linear representations, then ${\displaystyle \rho _{1}\otimes \rho _{2}(g)=(\rho _{1}(g)\otimes \rho _{2}(g))}$ defines a new linear representation. This gives rise to a group of linear characters, called the character group under the operation ${\displaystyle [\chi _{1}*\chi _{2}](g)=\chi _{1}(g)\chi _{2}(g)}$. 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 ${\displaystyle g\mapsto g^{-1}}$, 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 ${\displaystyle C_{3}}$ above, this map sends ${\displaystyle u\mapsto u^{2},u^{2}\mapsto u,}$ and accordingly switches ${\displaystyle \chi _{1}}$ and ${\displaystyle \chi _{2}}$ (switching their values of ${\displaystyle \omega }$ and ${\displaystyle \omega ^{2}}$). Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation.

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

See also

• Irreducible representation § Applications in theoretical physics and chemistry
• Molecular symmetry
• List of character tables for chemically important 3D point groups
• Character tables of small groups on GroupNames
• Isaacs, I. Martin (1976). Character Theory of Finite Groups. Dover.
• Rowland, Todd; Weisstein, Eric W. "Character Table". MathWorld.CS1 maint: multiple names: authors list (link)

References

1. Quantum Chemistry, 3rd ed. John P. Lowe, Kirk Peterson ISBN 0-12-457551-X
2. Physical Chemistry: A Molecular Approach by Donald A. McQuarrie, John D. Simon ISBN 0-935702-99-7
3. The chemical bond, 2nd ed. J.N. Murrell, S.F.A. Kettle, J.M. Tedder ISBN 0-471-90760-X
4. Physical Chemistry, 8th ed. P.W. Atkins and J. de Paula, W.H. Freeman, 2006 ISBN 0-7167-8759-8, chap.12
5. Molecular Symmetry and Spectroscopy, 2nd ed. Philip R. Bunker and Per Jensen, NRC Research Press, Ottawa, 1998 ISBN 9780660196282
6. G. L. Miessler and D. A. Tarr Inorganic Chemistry, 2nd ed. Pearson, Prentice Hall, 1998 ISBN 0-13-841891-8, chap.4.