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ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2025/1/a/solution

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Past exam of the groups and representations course of the University of Oxford / 2025 / 1 / a / Solution by

Codex  0 2026-10-25

A group representation on a vector space

V

is a group homomorphism

R : G \to GL (V)

, so

R (g h) = R (g) R (h)

and

R (e) = I

. A nonzero group representation is an irreducible representation if its only invariant subspaces are

0

and

V

. Here the representations and characters are finite-dimensional over the complex numbers.

The character of a representation is

χ_{R} (g) = tr R (g)

. Three useful properties are:

It is a class function: $χ_{R} (h g h^{- 1}) = χ_{R} (g)$ , by the cyclic property of the trace.
It is additive on a direct sum and multiplicative on a tensor product of group representations: $χ_{R \oplus S} = χ_{R} + χ_{S}$ and $χ_{R \otimes S} = χ_{R} χ_{S}$ .
The irreducible characters obey character orthogonality:
$⟨ χ_{r}, χ_{s} ⟩ = \frac{1}{∣ G ∣} \sum_{g \in G} χ_{r} (g)^{*} χ_{s} (g) = δ_{rs} .$
(1)

In particular,

χ_{R} (e) = dim V

and

χ_{R} (g^{- 1}) = χ_{R} (g)^{*}

, the latter following from unitarization of a finite-group representation. The character inner product

⟨ χ_{r}, χ_{R} ⟩

counts the multiplicity of an irreducible representation

r

in

R

.

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