 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-342/1/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

W \in S L (n, Z)

, then both

W

and

W^{- 1} = adj (W)

have integer entries. Thus

q^{'} = W q

Z^{n}

into itself, and the inverse map shows that it maps onto all of

Z^{n}

Writing the source contraction as

j^{T} a

, with

j^{'} = Wj

, invariance requires

j^{' T} a^{'} = j^{T} a .

Hence

j^{T} W^{T} a^{'} = j^{T} a

for every current, so

a^{'} = W^{- T} a, a = W^{T} a^{'} .

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