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 concise 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:

where ω is a primitive cube 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 ${\displaystyle S_{3}}$:

where (12) represents the conjugacy class consisting of (12), (13), (23), while (123) represents the 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:

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

where ${\displaystyle {\overline {\beta (g)}}}$ denotes 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 ${\displaystyle V=\left\langle \chi ,\chi _{i}\right\rangle }$.
• 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 (the free vector space spanned by) 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 ${\displaystyle V_{i}}$,

${\displaystyle ⟨{\mathrm{\chi <!--\; \chi \; -->}}_{\text{reg}},{\mathrm{\chi <!--\; \chi \; -->}}_i⟩=\frac{1}{\left|G\right|}\underset{g\in <!--\; \in \; -->G}{\sum <!--\; \sum \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Character_table
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