Condensed Matter II
Problem set #3
Spring 2023
1 Td group representation
1.1 Background
The group of symmetry operations of the regular tetrahedron Td is isomorphic to the
group of permutations of four objects P(4).
Figure 1: Regular tetrahedron with vertices abcd.
The elements of the group Td are (Schoenflies notation):
• E (identity)
• 8 rotations C3 about the diagonals of a cube.
• 3 rotations C2 about axes x, y, z.
• 6 improper rotations S4 about axis x, y, z (rotations of angle π/2 followed by a
reflection in a plane perpendicular to the axis of rotation).
• 6 reflections σd in planes containing one edge and the center of the tetrahedron.
The elements of the group P(4) are:
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• E =(abcd) A =(acbd) B =(cbad) C =(bacd) D =(cabd) F =(bcad) (perm. abc;d)
• G =(abdc) H =(adbc) J =(dbac) K =(badc) L =(dabc) M =(bdac) (perm. abd;c)
• N =(adcb) O =(acdb) P =(cdab) Q =(dacb) R =(cadb) S =(dcab) (perm. acd;b)
• T =(dbca) U =(dcba) V =(cbda) W =(bdca) X =(cdba) Y =(bcda) (perm. bcd;a)
1.2 Questions
(i) Show that P(4) and Td are isomorphic.
(ii) Partition the elements of P(4) so that the elements of each subset have the same
order (the order n of element X is the smallest n ∈ N such that Xn = E).
(iii) Determine at least 10 subgroups. Among those, determine the subgroups isomorphic
to P(3).
(iv) Explain why {E, G, N, T} is or is not a subgroup.
(v) Determine the classes (sets of equivalent elements, through the relation A ∼ B ⇔
∃X ∈ G, B = XAX−1)
(vi) Find several representations.
(vii) Determine the irreducible representations, and the character table of the group.
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