An LLL algorithm with symmetries

Abstract

We give a generalisation of the Lenstra-Lenstra-Lovász (LLL) lattice-reduction algorithm that is valid for an arbitrary (split, semisimple) reductive group $G$. This can be regarded as `lattice reduction with symmetries'. We make this algorithm explicit for the classical groups $G={\mathrm{Sp}}_{2g}$, ${\mathrm{SO}}_{2g}$, and for the exceptional group $G=G_2$.

Accepted version