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In Magma, a G-lattice L is a lattice upon which a finite integral
matrix group G acts by right multiplication. Magma allows various
computations with lattices associated with finite integral matrix groups
by use of G-lattices.
The computation of the automorphism group of a lattice (i.e. the largest
matrix group that acts on the lattice) and the testing
of lattices for isometry is performed within Magma by a
search designed by Bill Unger, which is based on the Plesken-Souvignier
backtrack algorithm [PS97], together with ordered partition methods.
Optionally, this may be combined with orthogonal decomposition code of
Gabi Nebe.
If G is a finite integral matrix group, then Magma uses Plesken's
centering algorithm ([Ple74]) to construct all G-invariant sublattices of a
given G-lattice L. The lattice of G-invariant sublattices of L can be
explored much like the lattice of submodules over finite fields.
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