I have a set of matrices in GLn(Z) and I want to reduce them by conjugacy. The problem is that the existing tools in GAP seem too slow.
Two matrices (but there are many others) that pose me problem now are: g1:= [ [ -1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1, 0, -1, 0 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0, 1 ], [ 0, 0, -1, 0, 0, 0, 1, 0 ], [ 0, 0, 0, -1, 0, 1, -1, 0 ] ] ; g2:= [ [ 1, 0, 0, 0, 0, 0, 0, 0 ], [ 1, -1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 1, -1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, -1, 0, 0 ], [ 0, 0, 0, 0, 1, 1, 1, 1 ], [ 0, 0, 0, 0, -1, 0, -1, 0 ] ]; Two general strategy are available in GAP 1> Use RepresentativeAction(GL(8, Integers), g1, g2) 2> Use carat by computing the Bravais groups of Gi = BravaisGroup( Group( [gi] ) ), testing for conjugacy, computing the normalizer and then concluding. Both approaches fail for g1 and g2, i.e. take too much time. Of course it is possible to use the eigenvalue -1 of multiplicity 4 to do a reduction, but I have been designing a lot of such special cases algorithm and now I am wondering if there is possibility of improving the general algorithm. In Newman, "Integral matrices" it is mentioned for example that there is a one-to-one correspondence between the conjugacy classes of matrices A in M_n(Z) such that f(A)=0 with f irreducible over Q and the ideal classes of the ring Z[theta] with theta a root of f. Are such methods used in GAP ? But this is of course only one aspect of the problem. For example the matrices [ [0, 1], [1, 0] ] and [ [1, 0], [0,-1]] are not conjugate because the sum of integral eigenspaces form a basis of determinant 2 in the first case and 1 in the second. The enumeration of all sum V1 + V2 of integral spaces with determinant k is equivalent to the enumeration of all double cosets H A K with H = GL(n1,Z) x GL(n2,Z), K = GL(n,Z) and det A = k. Is there a way for achieving this in GAP ? Thanks in advance for any help. Mathieu _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
