A chiral rigid rotor carries four identical point masses at the vertices of a tetrahedron

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

A chiral rigid rotor carries four identical point masses at the vertices of a tetrahedron, labelled

1, 2, 3, 4

, with no mirror planes present. Its symmetry is the chiral tetrahedral rotation group

T

, of order

12

, whose elements fall into the four classes

E

(size

1

4C_3

(size

4

4C_3^2

(size

4

) and

3C_2

(size

3

), as established from the proper rotations of a tetrahedron. ((a)) Take the

C_3

rotation about the axis through vertex

1

and the centroid of the opposite face, which fixes vertex

1

and cyclically permutes

2\to3\to4\to2

. Write the

4\times4

permutation matrix

D(C_3)

acting on the column of labels

(1,2,3,4)

, and verify

D(C_3)^3 = D(E)

. ((b)) Using the four classes and the order

12

, together with the equality

\sum_i d_i^2 = |G|

and the fact that one irreducible representation is always the one-dimensional totally symmetric representation, find the only possible list of dimensions of the irreducible representations of

T

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