 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2020/1/d/solution

All three vector spaces now have the same dimension

n

. Each of the nine blocks of

P

commutes with the common irreducible representation, so the Schur lemma gives

P = a_{11} I_{n} a_{21} I_{n} a_{31} I_{n} a_{12} I_{n} a_{22} I_{n} a_{32} I_{n} a_{13} I_{n} a_{23} I_{n} a_{33} I_{n} = A \otimes I_{n}, A \in M_{3} (C) .

Conversely

R (g) = I_{3} \otimes R_{1} (g)

, so the Kronecker product identity immediately verifies commutation for every

A

. The commutant of a semisimple representation acts arbitrarily on the three-dimensional multiplicity vector space while acting as a scalar on each irreducible factor. Its complex dimension is nine.

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