Condensed Matter II
Problem #1
Spring 2023
1 C3v group representation
1.1 Background
The group of symmetry operations of the equilateral triangle, C3v, is isomorphic to the group
of permutations of three objects P(3). The elements of the group P(3) are:
E =(123) A =(132) B =(321) C =(213) D =(312), F =(231), in which each parenthesis
indicates the final order of the initial elements (123).
The elements of the group C3v are (Schoenflies notation):
• E (identity)
• rotations C3(1) about the center of the triangle, by angle 2π/3.
• rotations C3(2) about the center of the triangle, by angle 4π/3.
• reflection σv(1) with respect to the vertical plane containing vertex 1, and the center of the
triangle.
• reflection σv(2) with respect to the vertical plane containing vertex 2, and the center of the
triangle.
• reflection σv(3) with respect to the vertical plane containing vertex 3, and the center of the
triangle.
1.2 Questions
(i) Prove that P(3) and C3v are isomorphic.
(ii) Find the periods of the group (the Abelian subgroups {E, A, A2
, . . . , An−1
} where n is the
period of element A).
(iii) Find the subgroups of the group.
(iv) Determine the classes (the set of all elements associated to the others in the set through the
relation: B ∼ A ⇔ ∃X ∈ G, B = XAX−1
)
(v) Find several representations (groups isomorphic to the group of square matrices).
(vi) Find the irreducible representations and determine the character table.
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