An LLL algorithm with symmetries

TL;DR

This work generalises LLL lattice reduction to arbitrary split semisimple groups by introducing a notion of reduction on the symmetric space $X_G=K\backslash G$ and an algorithm that, given a point $x$, produces a group element $\gamma\in\mathbf{G}(\mathbb{Z})$ such that $x\gamma$ is reduced. In the classical SL$_g$ case the method recovers Lenstra–Lenstra–Lovász, and it is made explicit for the symplectic, even orthogonal, and exceptional $G_2$ groups, with detailed constructions of reduction data, fundamental sets, and reflection steps. Termination is established by a descent argument using exterior powers, and the paper provides concrete implementations in Sp$_{2g}$, SO$_{2g}$, and $G_2$, including numerical examples. The approach connects to reduction covariants and applications to 2-descent and stability questions, offering a broad, computable framework for symmetry-aware lattice reduction with potential arithmetic and geometric applications.

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 $G = G_2$.

Theorems & Definitions (38)

• Definition 1.1
• Proposition 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Lemma 2.6
• proof