We give a generalisation of the Lenstra-Lenstra-Lov\'asz (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 = Sp2g, SO2g, and for the exceptional group G = G₂.
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Romano et al. (2024) studied this question.
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