Dihedral Group D_3

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The dihedral group D_3 is a particular instance of one of the two distinct abstract groups of group order 6. Unlike the cyclic group C_6 (which is Abelian), D_3 is non-Abelian. In fact, D_3 is the non-Abelian group having smallest group order.

Examples of D_3 include the point groups known as C_(3h), C_(3v), S_3, D_3, the symmetry group of the equilateral triangle (Arfken 1985, p. 246), and the permutation group of three objects (Arfken 1985, p. 249).

CycleGraphD3

The cycle graph of D_3 is shown above. D_3 has cycle index given by

Z(D_3)=1/6x_1^3+1/2x_2x_1+1/3x_3.

(1)

DihedralGroupD3Table

Its multiplication table is illustrated above and enumerated below, where 1 denotes the identity element. Equivalent but slightly different forms are given by (Arfken 1985, p. 247) and Cotton (1990, p. 12), the latter of which denotes the abstract group of D_3 by G_6^((2)).

D_3 1 A B C D E

1 1 A B C D E

A A B 1 D E C

B B 1 A E C D

C C E D 1 B A

D D C E A 1 B

E E D C B A 1

Like all dihedral groups, a reducible two-dimensional representation using real matrices has generators given by S and R, where S is a rotation by pi radians about an axis passing through the center of a regular n-gon and one if its vertices and R is a rotation by 2pi/n about the center of the n-gon. The multiplication table above corresponds to the following matrices:

1 = [画像:S^0R^0=[1 0; 0 1]]

(2)

A = [画像:S^0R^1=[-1/2 -1/2sqrt(3); 1/2sqrt(3) -1/2]]

(3)

B = [画像:S^0R^2=[-1/2 1/2sqrt(3); -1/2sqrt(3) -1/2]]

(4)

C = [画像:S^1R^1=[1/2 1/2sqrt(3); 1/2sqrt(3) -1/2]]

(5)

D = [画像:S^1R^0=[-1 0; 0 1]]

(6)

E = [画像:S^1R^2=[1/2 -1/2sqrt(3); -1/2sqrt(3) -1/2].]

(7)

The elements X=1, C, D, and E of D_3 satisfy X^2=1, the elements X=1, A, and B satisfy X^3=1, the elements X=1, C, D, and E satisfy X^4=1, and all elements satisfy X^6=1.

The conjugacy classes are {1}, {A,B}, and {C,D,E}. There are 6 subgroups of D_3: {1}, {1,C}, {1,D}, {1,E}, {1,A,B}, and {1,A,B,C,D,E}. Of these, the subgroups {1}, {1,A,B}, and {1,A,B,C,D,E} are normal

To find the irreducible representation, note that there are three conjugacy classes. The fifth rule of irreducible representations requires that there be three irreducible representations, and the second rule requires that

h=l_1^2+l_2^2+l_3^2=6,

(8)

so it must be true that

l_1=l_2=1,l_3=2.

(9)

By rule 6, we can let the first representation have all 1s.

D_3 1 A B C D E

Gamma_1 1 1 1 1 1 1

To find a representation orthogonal to the totally symmetric representation, we must have three +1 and three -1 group characters. We can also add the constraint that the components of the identity element 1 be positive. The three conjugacy classes have 1, 2, and 3 elements. Since we need a total of three +1s and we have required that a +1 occur for the conjugacy class of order 1, the remaining +1s must be used for the elements of the conjugacy class of order 2, i.e., D and E.

D_3 1 A B C D E

Gamma_1 1 1 1 1 1 1

Gamma_2 1 -1 -1 -1 1 1

Using group rule 1, we see that

1^2+1^2+chi_3^2(1)=6,

(10)

so the final representation for 1 has group character 2. Orthogonality with the first two representations (group rule 3) then yields the following constraints:

1·1·2+1·2·chi_2+1·3·chi_3=2+2chi_2+3chi_3 =

(11)

1·1·2+1·2·chi_2+(-1)·3·chi_3=2+2chi_2-3chi_3 = 0.

(12)

Solving these simultaneous equations by adding and subtracting (12) from (11), we obtain chi_2=-1, chi_3=0. The full character table is then

D_3 1 A B C D E

Gamma_1 1 1 1 1 1 1

Gamma_2 1 -1 -1 -1 1 1

Gamma_3 2 0 0 0 -1 -1

Since there are only three conjugacy classes, this table is conventionally written simply as

D_3 1 A=B=C D=E

Gamma_1 1 1 1

Gamma_2 1 -1 1

Gamma_3 2 0 -1

Writing the irreducible representations in matrix form then yields

1 = [画像:[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1]]

(13)

A = [画像:[1 0 0 0; 0 1 0 0; 0 0 -1/2 -1/2sqrt(3); 0 0 1/2sqrt(3) -1/2]]

(14)

B = [画像:[1 0 0 0; 0 1 0 0; 0 0 -1/2 1/2sqrt(3); 0 0 -1/2sqrt(3) -1/2]]

(15)

C = [画像:[1 0 0 0; 0 -1 0 0; 0 0 -1 0; 0 0 0 1]]

(16)

D = [画像:[1 0 0 0; 0 -1 0 0; 0 0 1/2 -1/2sqrt(3); 0 0 -1/2sqrt(3) -1/2]]

(17)

E = [画像:[1 0 0 0; 0 -1 0 0; 0 0 1/2 1/2sqrt(3); 0 0 1/2sqrt(3) -1/2].]

(18)

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 246-248, 1985.Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

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Dihedral Group D_3

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Weisstein, Eric W. "Dihedral Group D_3." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DihedralGroupD3.html

Dihedral Group D_3

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