Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 19H / Solution

Maschke's theorem says that every invariant subspace of a finite-dimensional complex representation of a finite group has an invariant complement. Average any projection over the group to make it equivariant. Induction on dimension therefore decomposes every representation into irreducibles.

The character of a representation is

χ_{V} (g) = tr (g ∣_{V})

. For a

G

X

, the permutation representation

C X

X

, and its character is the number of fixed points:

χ_{C X} (g) = ∣ X^{g} ∣.

For the regular action,

χ_{C G} (1) = ∣ G ∣, χ_{C G} (g) = 0 (g \neq = 1) .

The multiplicity of an irreducible

V_{i}

is the character inner product

⟨ χ_{reg}, χ_{i} ⟩ = χ_{i} (1) = dim V_{i},

so

C G ≅ i ⨁ (dim V_{i}) V_{i} .

If

V ≅ ⨁_{i} n_{i} V_{i}

, Schur lemma gives

dim Hom_{G} (V, V) = i \sum n_{i}^{2} .

For

V = C G

, this is

\sum_{i} (dim V_{i})^{2} = ∣ G ∣

.

Now suppose

χ (g) = 0

off the identity. The multiplicity of

V_{i}

in

V

is

⟨ χ, χ_{i} ⟩ = \frac{d i m V}{∣ G ∣} dim V_{i} .

Thus

dim V /∣ G ∣

is a nonnegative integer and

V

is that many copies of the regular representation. Finally,

χ_{W \otimes C G} (g) = χ_{W} (g) χ_{reg} (g) = (dim W) χ_{reg} (g),

so

W \otimes C G ≅ (dim W) C G .

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