Self-dual rational lattice

A rational lattice is self-dual iff it is its own dual lattice, or equivalently it is ^integral and unimodular. geo

Proof

Note for a unimodular integral matrix $𝐺$ we have $𝐺 ℤ 𝑛 = ℤ 𝑛$ .

Let $𝐴$ be a basis matrix for $𝐿$ so that $𝐴 𝖳 𝐴 = 𝐺$ , so the basis matrix $𝐴 \circ$ of $𝐿 \circ$ is $𝐴 𝐺 - 1$ . Now assuming $𝐺$ is unimodular, so is $𝐺 - 1$ and we have
$𝐿 \circ = 𝐴 \circ ℤ 𝑛 = 𝐴 𝐺 - 1 ℤ 𝑛 = 𝐴 ℤ 𝑛 = 𝐿$
as required.

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Self-dual rational lattice

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