Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 2 / a

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- Solution a

Solution

Let

V

be the real vector space with basis

(e_{i})_{i \in I}

. The Coxeter Gram matrix defines the symmetric bilinear form

⟨ e_{i}, e_{j} ⟩ = G (W)_{ij} = - 2 cos (\frac{π}{m _{ij}}) .

Its diagonal entries are

2

. The Geometric representation of a Coxeter group is generated by the reflections

σ (x_{i}) (v) = v - ⟨ v, e_{i} ⟩ e_{i} .

Each has square one, and on

span {e_{i}, e_{j}}

the product

σ (x_{i}) σ (x_{j})

has order

m_{ij}

. The reflections therefore satisfy the Coxeter relations and define a group representation

σ : W \to G L (V)

.

Solved by gpt-5.6-sol high.

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