def get_transversals(base, gens): """ Return transversals for the group with BSGS base, gens """ if not base: return [] stabs = _distribute_gens_by_base(base, gens) orbits, transversals = _orbits_transversals_from_bsgs(base, stabs) transversals = [dict((x, h._array_form) for x, h in y.items()) for y in transversals] return transversals
def test_distribute_gens_by_base(): base = [0, 1, 2] gens = [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]), Permutation([0, 2, 3, 1]), Permutation([3, 2, 1, 0])] assert _distribute_gens_by_base(base, gens) == [gens, [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]), Permutation([0, 2, 3, 1])], [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2])]]
def get_transversals(base, gens): """ Return transversals for the group with BSGS base, gens """ if not base: return [] stabs = _distribute_gens_by_base(base, gens) orbits, transversals = _orbits_transversals_from_bsgs(base, stabs) transversals = [ dict((x, h._array_form) for x, h in y.items()) for y in transversals ] return transversals
def test_distribute_gens_by_base(): base = [0, 1, 2] gens = [ Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]), Permutation([0, 2, 3, 1]), Permutation([3, 2, 1, 0]) ] assert _distribute_gens_by_base(base, gens) == [ gens, [ Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2]), Permutation([0, 2, 3, 1]) ], [Permutation([0, 1, 2, 3]), Permutation([0, 1, 3, 2])] ]
def test_orbits_transversals_from_bsgs(): S = SymmetricGroup(4) S.schreier_sims() base = S.base strong_gens = S.strong_gens strong_gens_distr = _distribute_gens_by_base(base, strong_gens) result = _orbits_transversals_from_bsgs(base, strong_gens_distr) orbits = result[0] transversals = result[1] base_len = len(base) for i in range(base_len): for el in orbits[i]: assert transversals[i][el](base[i]) == el for j in range(i): assert transversals[i][el](base[j]) == base[j] order = 1 for i in range(base_len): order *= len(orbits[i]) assert S.order() == order
def test_handle_precomputed_bsgs(): A = AlternatingGroup(5) A.schreier_sims() base = A.base strong_gens = A.strong_gens result = _handle_precomputed_bsgs(base, strong_gens) strong_gens_distr = _distribute_gens_by_base(base, strong_gens) assert strong_gens_distr == result[2] transversals = result[0] orbits = result[1] base_len = len(base) for i in range(base_len): for el in orbits[i]: assert transversals[i][el](base[i]) == el for j in range(i): assert transversals[i][el](base[j]) == base[j] order = 1 for i in range(base_len): order *= len(orbits[i]) assert A.order() == order
def _verify_bsgs(group, base, gens): """ Verify the correctness of a base and strong generating set. Explanation =========== This is a naive implementation using the definition of a base and a strong generating set relative to it. There are other procedures for verifying a base and strong generating set, but this one will serve for more robust testing. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> A = AlternatingGroup(4) >>> A.schreier_sims() >>> _verify_bsgs(A, A.base, A.strong_gens) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims """ from sympy.combinatorics.perm_groups import PermutationGroup strong_gens_distr = _distribute_gens_by_base(base, gens) current_stabilizer = group for i in range(len(base)): candidate = PermutationGroup(strong_gens_distr[i]) if current_stabilizer.order() != candidate.order(): return False current_stabilizer = current_stabilizer.stabilizer(base[i]) if current_stabilizer.order() != 1: return False return True
def _verify_bsgs(group, base, gens): """ Verify the correctness of a base and strong generating set. This is a naive implementation using the definition of a base and a strong generating set relative to it. There are other procedures for verifying a base and strong generating set, but this one will serve for more robust testing. Examples ======== >>> from sympy.combinatorics.named_groups import AlternatingGroup >>> from sympy.combinatorics.testutil import _verify_bsgs >>> A = AlternatingGroup(4) >>> A.schreier_sims() >>> _verify_bsgs(A, A.base, A.strong_gens) True See Also ======== sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims """ from sympy.combinatorics.perm_groups import PermutationGroup strong_gens_distr = _distribute_gens_by_base(base, gens) current_stabilizer = group for i in range(len(base)): candidate = PermutationGroup(strong_gens_distr[i]) if current_stabilizer.order() != candidate.order(): return False current_stabilizer = current_stabilizer.stabilizer(base[i]) if current_stabilizer.order() != 1: return False return True