“Finite Group D3”的概念、定义、习题解题方法及翻译-数学辞典

单词

Finite Group D3

释义

Finite Group D3

The Dihedral Group is one of the two groups of Order 6. It is the non-Abelian group ofsmallest Order. Examples of include the Point Groups known as , , ,, the symmetry group of the Equilateral Triangle, and the group of permutation of three objects. Its elements satisfy , and four of its elements satisfy , where 1 is the Identity Element. The CycleGraph is shown above, and the Multiplication Table is given below.

	1
1	1
		1
			1
				1
						1
					1

The Conjugacy Classes are ,

			(1)
			(2)
			(3)
			(4)
			(5)

and

			(6)
			(7)

A reducible 2-D representation using Real Matrices can be found by performing thespatial rotations corresponding to the symmetry elements of . Take the z-Axis along the axis.

			(8)

			(9)
			(10)
			(11)
			(12)
			(13)

To find the irreducible representation, note that there are three Conjugacy Classes. Rule 5requires that there be three irreducible representations satisfying

(14)

so it must be true that

(15)

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

	1
	1	1	1	1	1	1

To find a representation orthogonal to the totally symmetric representation, we must have three and three Characters. We can also add the constraint that the components of the Identity Element 1 bepositive. The three Conjugacy Classes have 1, 2, and 3 elements. Since we need a total of three sand we have required that a occur for the Conjugacy Class of Order 1, theremaining +1s must be used for the elements of the Conjugacy Class of Order 2, i.e., and .

1
1	1	1	1	1	1
1	1	1

Using Group rule 1, we see that

(16)

so the final representation for 1 has Character 2. Orthogonality with the first tworepresentations (rule 3) then yields the following constraints:

			(17)
			(18)

Solving these simultaneous equations by adding and subtracting (18) from (17), we obtain

. The full Character Table is then

1
1	1	1	1	1	1
1	1	1
2			0	0	0

Since there are only three Conjugacy Classes, this table is conventionally written simply as

1
1	1	1
1	1
2		0

Writing the irreducible representations in matrix form then yields

			(19)
			(20)
			(21)
			(22)
			(23)
			(24)

See also Dihedral Group, Finite Group D4, Finite Group Z6

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