 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/10/a/solution

W \subseteq C^{n}

D

D (g) W \subseteq W for every g \in G .

Because

g^{- 1}

is also in the group and

D (g)

is invertible, the inclusion is equivalently

D (g) W = W

for every

g

For

n > 0

, the representation is an irreducible representation when its only invariant vector subspaces are

W = {0} and W = C^{n} .

Thus no nonzero proper subspace is preserved by all representing matrices.

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