The first version of the package, written for GAP 3, formed part of Anne Heyworth's thesis [Hey99] in 1999, but was not made generally available.
Version, kan 0.91, was prepared to run under GAP 4.4.6, in July 2005.
Version, kan 0.94, differed in two significant ways:
this manual is prepared using the GAPDoc package;
the test file kan/tst/kan\_manual.tst sets the AssertionLevel to 0 to avoid recursion in the Automata package.
The latest version, kan 0.95, of 9th October 2007, just fixed file protections and added a CHANGES file.
There are too many items to list here, but some of the most important are as follows.
Implement iterators and enumerators for double cosets.
At present the methods for DoubleCosetsNC and RightCosetsNC in this package return automata, rather than lists of cosets or coset enumerators. This needs to be fixed.
Provide methods for operations such as DoubleCosetRepsAndSizes.
Convert the rest of the GAP 3 kan package to GAP 4.
> DoubleCosetsAutomaton( G, U, V ) | ( operation ) |
> RightCosetsAutomaton( G, V ) | ( operation ) |
Alternative methods for DoubleCosetsNC(G,U,V) and RightCosetsNC(G,V) should be provided in the cases where the group G has a rewriting system or is known to be infinite. At present the functions RightCosetsAutomaton and DoubleCosetsAutomaton return minimized automata, and Iterators for these are not yet available.
gap> F := FreeGroup(2);;
gap> rels := [ F.2^2, (F.1*F.2)^2 ];;
gap> G4 := F/rels;;
gap> genG4 := GeneratorsOfGroup( G4 );;
gap> a := genG4[1]; b := genG4[2];;
gap> U := Subgroup( G4, [a^2] );;
gap> V := Subgroup( G4, [b] );;
gap> dc4 := DoubleCosetsAutomaton( G4, U, V );;
gap> Print( dc4 );
Automaton("det",5,"HKaAbB",[ [ 2, 2, 2, 5, 2 ], [ 2, 2, 1, 2, 1 ], [ 2, 2, 2, \
2, 3 ], [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2 ] ],[ 4 ],[ 1 ])\
;;
gap> rc4 := RightCosetsAutomaton( G4, V );;
gap> Print( rc4 );
Automaton("det",6,"HKaAbB",[ [ 2, 2, 2, 6, 2, 2 ], [ 2, 2, 1, 2, 1, 1 ], [ 2, \
2, 3, 2, 2, 3 ], [ 2, 2, 2, 2, 5, 5 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, \
2 ] ],[ 4 ],[ 1 ]);;
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