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ORBITS IN THE LEECH LATTICE DANIEL ALLCOCK
 

Summary: ORBITS IN THE LEECH LATTICE
DANIEL ALLCOCK
Abstract. We provide an algorithm for determining whether two
vectors in the Leech lattice are equivalent under its isometry group,
the Conway group Co0 of order 8 × 1018
. Our methods rely on
and develop the work of R. T. Curtis, and we describe our in-
tended applications to the symmetry groups of Lorentzian lattices
and the enumeration of lattices of dimension 24 with good prop-
erties such as having small determinant. Our algorithm reduces
the test of equivalence to 4 tests under the subgroup 212
:M24,
and a test under this subgroup to 12 tests under M24. We also
give algorithms for testing equivalence under these two subgroups.
Finally, we analyze the performance of the algorithm.
1. Introduction
The Leech lattice is a lattice in 24-dimensional Euclidean space
with many remarkable properties, for us the most important of which
is that its isometry group (modulo {ħI}) is one of the sporadic finite
simple groups. The isometry group is called the Conway group Co0,

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics