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Character Table


A finite group G has a finite number of conjugacy classes and a finite number of distinct irreducible representations. The group character of a group representation is constant on a conjugacy class. Hence, the values of the characters can be written as an array, known as a character table. Typically, the rows are given by the irreducible representations and the columns are given the conjugacy classes.

A character table often contains enough information to identify a given abstract group and distinguish it from others. However, there exist nonisomorphic groups which nevertheless have the same character table, for example D_4 (the symmetry group of the square) and Q_8 (the quaternion group).

For example, the symmetric group on three letters S_3 has three conjugacy classes, represented by the permutations {1,2,3}, {2,1,3}, and {2,3,1}. It also has three irreducible representations; two are one-dimensional and the third is two-dimensional:

1. The trivial representation phi_1(g)(alpha)=alpha.

2. The alternating representation, given by the signature of the permutation, phi_2(g)(alpha)=sgn(g)alpha.

3. The standard representation on V={(z_1,z_2,z_3):sumz_i=0} with

 phi_3({a,b,c})(z_1,z_2,z_3)=(z_a,z_b,z_c).
(1)

The standard representation can be described on C^2 via the matrices

phi^~_3({2,1,3})=[0 1; 1 0]
(2)
phi^~_3({2,3,1})=[0 -1; 1 -1],
(3)

and hence the group character of the first matrix is 0 and that of the second is -1. The group character of the identity is always the dimension of the vector space. The trace of the alternating representation is just the permutation symbol of the permutation. Consequently, the character table for S_3 is shown below.

123
S_3e(12)(123)
trivial111
alternating1-11
standard20-1
CharacterTable

Chemists and physicists use a special convention for representing character tables which is applied especially to the so-called point groups, which are the 32 finite symmetry groups possible in a lattice. In the example above, the numbered regions contain the following contents (Cotton 1990 pp. 90-92).

1. The symbol used to represent the group in question (in this case C_(3v)).

2. The conjugacy classes, indicated by number and symbol, where the sum of the coefficients gives the group order of the group.

3. Mulliken symbols, one for each irreducible representation.

4. An array of the group characters of the irreducible representation of the group, with one column for each conjugacy class, and one row for each irreducible representation.

5. Combinations of the symbols x, y, z, R_x, R_y, and R_z, the first three of which represent the coordinates x, y, and z, and the last three of which stand for rotations about these axes. These are related to transformation properties and basis representations of the group.

6. All square and binary products of coordinates according to their transformation properties.

The character tables for many of the point groups are reproduced below using this notation.

C_1E
A1
C_sEsigma_h
A11x,y,R_zx^2,y^2,z^2,xy
B1-1z,R_x,R_yyz,xz
C_iEi
A_g11R_x,R_y,R_zx^2,y^2,z^2,xy,xz,yz
A_u1-1x,y,z
C_2EC_2
A11z,R_zx^2,y^2,z^2,xy
B1-1x,y,R_x,R_yyz,xz
C_3 E C_3 C_3^2 epsilon=exp(2pii/3)
A111 z,R_zx^2,y^2,z^2,xy
E{1; 1epsilon^ ; epsilon^*epsilon^*; epsilon^ }(x,y)(R_x,R_y)(x^2-y^2,xy)(yz,xz)
C_4 E C_3 C_2 C_4^3
A1 1 11z,R_zx^2+y^2,z^2
B1 -1 1-1x^2-y^2,xy
E{ 1;  1 i;  -i -1;  1 -i;  i}(x,y)(R_x,R_y)(yz,xz)
C_5 E C_5 C_5^2 C_5^3 C_5^4 epsilon=exp(2pii/5)
A11 1 1 1 z,R_zx^2+y^2,z^2
E_1{ 1;  1epsilon^ ; epsilon^(* )epsilon^(2 ); epsilon^(2*)epsilon^(2*); epsilon^(2 )epsilon^(* ); epsilon^ }(x,y)(R_x,R_y)(yz,xz)
E_2{1; 1epsilon^(2 ); epsilon^(2*)epsilon^(* ); epsilon^ epsilon^ ; epsilon^(* )epsilon^(2*); epsilon^(2 )}(x^2-y^2,xy)
C_6 E C_6 C_3 C_2 C_3^2 C_6^5 epsilon=exp(2pii/6)
A1 1 1 1 1 1 z,R_zx^2+y^2,z^2
B1-1 1 -1 1 -1
E_1{ 1;  1 epsilon^ ;  epsilon^*-epsilon^*; -epsilon^ -1; -1-epsilon^ ; -epsilon^* epsilon^*;  epsilon^ }(x,y); (R_x,R_y)(yz,xz)
E_2{ 1;  1-epsilon^ ; -epsilon^*-epsilon^ ; -epsilon^*1; 1-epsilon^*; -epsilon^ -epsilon^ ;  epsilon^*}(x^2-y^2,xy)
D_2 E C_2(z) C_2(y) C_2(x)
A_11111x^2+y^2,z^2
B_111-1-1z,R_zxy
B_21-11-1y,R_yxz
B_31-1-11z,R_zyz
D_3 E 2C_3 3C_2
A_1111x^2+y^2,z^2
A_211-1z,R_zxy
E2-10(x,y)(R_x,R_y)(x^2-y^2,xy)(xz,yz)
D_4 E 2C_4 C_2 2C_2^' 2C_2^('')
A_111111x^2+y^2,z^2
A_2111-1-1z,R_z
B_11-111-1x^2-y^2
B_21-11-11xy
E20-200(x,y)(R_x,R_y)(xz,yz)
D_5 E 2C_5 2C_5^2 5C_2
A_11 1 1 1x^2+y^2,z^2
B_11 1 1 -1z,R_z
B_222cos 72 degrees2cos144 degrees0(x,y)(R_x,R_y)(xz,yz)
B_322cos144 degrees2cos 72 degrees0(x^2-y^2,xy)
D_6 E 2C_6 2C_3 C_2 3C_2^' 3C_2^('')
A_1111111x^2+y^2,z^2
A_21111-1-1z,R_z
B_11-11-11-1
B_21-11-1-11(x,y)(R_x,R_y)
E_121-1-200(xz,yz)
E_22-1-1200(x^2-y^2,xy)
C_(2v) E C_2 sigma_v(xz) sigma_v^'(yz)
A_11111zx^2,y^2,z^2
A_211-1-1R_zxy
B_11-11-1x,R_yxz
B_21-1-11y,R_xyz
C_(3v) E 2C_3 3sigma_v
A_1111zx^2+y^2,z^2
A_211-1R_z
E2-10(x,y)(R_x,R_y)(x^2-y^2,xy)(xz,yz)
C_(4v) E 2C_4 C_2 2sigma_v 2sigma_d
A_111111zx^2+y^2,z^2
A_2111-1-1R_z
B_11-111-1x^2-y^2
B_21-11-11xy
E20-200(x,y)(R_x,R_y)(xz,yz)
C_(5v) E 2C_5 2C_5^2 5sigma_v
A_11 1 1 1zx^2+y^2,z^2
B_11 1 1 -1R_z
B_222cos 72 degrees2cos144 degrees0(x,y)(R_x,R_y) (xz,yz)
B_322cos144 degrees2cos 72 degrees0(x^2-y^2,xy)
C_(6v) E 2C_6 2C_3 C_2 3sigma_v 3sigma_d
A_1111111zx^2+y^2,z^2
A_21111-1-1R_z
B_11-11-11-1
B_21-11-1-11
E_121-1-200(x,y)(R_x,R_y)(xz,yz)
E_22-1-1200(x^2-y^2,xy)
C_(inftyv) E C_infty^Phi ... inftysigma_v
A_1=Sigma^+1 1 ...1zx^2+y^2,z^2
A_2=Sigma^-1 1 ...-1R_z
E_1=Pi22cos Phi...0(x,y);(R_x,R_y)(xz,yz)
E_2=Delta22cos2Phi...0(x^2-y^2,xy)
E_3=Phi22cos3Phi...0
| ||...|

See also

Conjugacy Class, Group, Group Character, Group Representation, Irreducible Representation, Point Groups

Portions of this entry contributed by Todd Rowland

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Referenced on Wolfram|Alpha

Character Table

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Character Table." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharacterTable.html

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