REPRESENTATION THEORY
(because of orthogonality) have characters 1, 1, 1,−1. The summation rules
and orthogonality require the other two character sets to be 2, (−1+√
5)/2,
(− 1 −
√
5)/2, 0 and 2, (− 1 −√
5)/2, (−1+
√
5)/2, 0. Note thatRhas order 5 and
that, e.g., (−1+√
5)/2=exp(2πi/5) + exp(8πi/5).
29.13 The doublet irrep E (characters 2,−1, 0) appears in both 432 and 32 and so
is unsplit. The triplet T 1 (characters 3, 0, 1) splits under 32 into doublet E
(characters 2,−1, 0) and singlet A 1 (characters 1, 1, 1).