Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 19H (Representation Theory)

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- Solution 19H

Solution

The character of

(ρ, V)

is

χ_{V} (g) = tr ρ (g) .

The stated action is a representation because successive action by

(g_{2}, h_{2})

and

(g_{1}, h_{1})

gives

σ (h_{1} h_{2}) α ρ (g_{2}^{- 1} g_{1}^{- 1}) = σ (h_{1} h_{2}) α ρ ((g_{1} g_{2})^{- 1}) .

Identifying

Hom_{C} (V, W)

with

V^{*} \otimes W

gives the character of a Hom representation

χ_{Hom (V, W)} (g, h) = χ_{V} (g^{- 1}) χ_{W} (h) = \overline{χ_{V} (g)} χ_{W} (h) .

Here the final equality follows after unitarizing the finite-group representation.

The permutation character of

C G

counts fixed points:

χ_{C G} (g, h) = # {x \in G : gx h^{- 1} = x} = # {x : x^{- 1} gx = h} .

Thus

χ_{C G} (g, h) = {∣ C_{G} (g) ∣, 0, g and h are conjugate, otherwise .

On the other hand, the character of

⨁_{i} Hom (V_{i}, V_{i})

is

\sum_{i} \overline{χ_{i} (g)} χ_{i} (h) .

The column form of character orthogonality says that this has exactly the same two values above. Complex representations of a finite group are semisimple, and semisimple representations with the same character are isomorphic. Hence the two-sided regular representation decomposition is

C G ≅ i = 1 ⨁ r Hom_{C} (V_{i}, V_{i})

G \times G

representation.

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