By G. Viennot
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D Thus, we first form the vectors V ij whose components are the (i, j)th elements taken from each matrix in the representation in some fixed order. The Great Orthogonality Theorem can then be expressed more concisely as V ij · V ∗i j = |G| δi,i δj,j . d For the given representation of S3 , e= c= 1 0 , 0 1 −1 0 0 1 a= , d= 1 2 1 2 √ 1 − 3 , √ − 3 −1 −1 √ − 3 √ 3 −1 , b= f= 1 2 1 2 1 √ 3 3 −1 √ −1 − 3 √ 3 −1 these vectors are: V 11 = 1, 12 , 12 , −1, − 12 , − 12 , √ √ √ √ V 12 = 0, − 12 3, 12 3, 0, 12 3, − 12 3 , √ √ √ √ V 21 = 0, − 12 3, 12 3, 0, − 12 3, 12 3 , V 22 = 1, − 12 , − 12 , 1, − 12 , − 12 .
1 Inverse. Finally, the inverse of each element ai bj is a−1 i bj because −1 −1 −1 (ai bj )(a−1 i bj ) = (ai ai )(bj bj ) = ea eb and −1 −1 −1 (a−1 i bj )(ai bj ) = (ai ai )(bj bj ) = ea eb . Thus, we have shown that the direct product of two groups is itself a group. Since the elements of this group are obtained by taking all products of elements from Ga and Gb , the order of this group is |Ga ||Gb |. 4. Suppose we have an irreducible representation for each of two groups Ga and Gb . We denote these representations, which may be of different dimensions, by A(ai ) and A(bj ), and their matrix elements by A(ai )ij and A(bj )ij .
The notation above means that cos(kp x) is taken if p is odd, sin(kp x) is taken if p is even, and similarly for the other factor. The corresponding eigenvalues are Ep,q = ¯ 2 π2 2 h (p + q 2 ) 8m (a) Determine the eight planar symmetry operations of a square. These operations form the group of the Hamiltonian for this problem. Assemble the symmetry operations into equivalence classes. (b) Determine the number of irreducible representations and their dimensions for this group. Do these dimensions appear to be broadly consistent with the degeneracies of the energy eigenvalues?