Self-dual rational lattice

A rational lattice is self-dual iff it is its own dual lattice, or equivalently it is integral and unimodular. #m/def/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.

#state/tidy | #lang/en | #SemBr