 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3/49/b/i/solution

For representation spaces

V, W

D_{V \otimes W} (g) = D_{V} (g) \otimes D_{W} (g) .

Its character is the trace. Summing diagonal elements explicitly,

χ_{V \otimes W} (g) = \sum_{i, j} [D_{V} (g)]_{ii} [D_{W} (g)]_{jj} = χ_{V} (g) χ_{W} (g) .

No step assumes one-dimensional matrices, so the result applies to both degenerate and nondegenerate representations.

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