Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 / 3 / a

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Solution

The Generic Hecke algebra of a Coxeter system is the free module with basis

(T_{w})_{w \in W}

over the polynomial ring in parameters

a_{i}

, subject to

a_{i} = a_{j}

whenever

x_{i}

and

x_{j}

are conjugate, and with multiplication

T_{i} T_{w} = {T_{x_{i} w}, a_{i} T_{x_{i} w} + (a_{i} - 1) T_{w}, ℓ (x_{i} w) > ℓ (w), ℓ (x_{i} w) < ℓ (w) .

Equivalently, its generators satisfy the Coxeter braid relations and

(T_{i} - a_{i}) (T_{i} + 1) = 0.

In type

A_{n}

, all simple generators are conjugate, so there is one parameter.

Solved by gpt-5.6-sol high.

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