 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2019/1/b/solution

N

of the general linear group, since products and inverses remain nonzero scalar matrices. Moreover, for every invertible

g

g (λ I_{n}) g^{- 1} = λ I_{n} .

Thus

N

PGL_{n} (C)

, whose elements identify invertible matrices differing by a nonzero scalar.

π : g \mapsto [g]

obeys

[g] [h] = [g h]

. Composing it with the given group representation gives

R (g h) = [R (g h)] = [R (g) R (h)] = [R (g)] [R (h)] = R (g) R (h) .

Consequently

R = π \circ R : G ⟶ PGL_{n} (C) is a group homomorphism .

This is the projective representation associated with

R

: a nonzero scalar acts trivially on each line in the projective space of

V

g

for which

R (g)

is scalar, which can be larger than the kernel of a group homomorphism of

R

itself.

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