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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/10/i/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 10 / i / Solution by

Codex  0 2026-10-18

For a group representation

D : G \to GL (V)

, a faithful representation is injective, equivalently

D (g) = I

only for

g = e

. Thus no distinct group elements are represented by the same linear map.

Two equivalent representations are related by one fixed invertible change of basis:

E (g) = S^{- 1} D (g) S

for every

g

. The same

S

must work simultaneously for all group elements.

An irreducible representation acts on a nonzero vector space and has no invariant subspace other than

{0}

and the whole space. An invariant subspace is one carried into itself by every representing matrix.

The character of a representation is

χ_{D} (g) = tr D (g) .

The cyclic property of the trace makes it invariant under equivalence and constant on conjugacy classes. Its value at the identity is the dimension of the representation.

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