About the symmorphic space group determination algorithm.

[hongyi-zhao]

The algorithm of IsSymmorphicSpaceGroup implemented in cryst package is as follows:

cryst/gap/cryst.gi

Lines 537 to 543 in fbf54fb

	#############################################################################
	##
	#M IsSymmorphicSpaceGroup( S ) . . . . . . . . . . . . . . .is S symmorphic?
	##
	InstallMethod( IsSymmorphicSpaceGroup,
	"generic method", true, [ IsAffineCrystGroupOnLeftOrRight ], 0,
	S -> CocVecs( S ) = [] );

cryst/gap/max.gi

Lines 467 to 504 in fbf54fb

	#############################################################################
	##
	#F CocycleInfo( <S> ) . . . . . . . . . . .information about extension type
	##
	CocycleInfo := function( S )
	local P, iso, F, d, gens, mats, coc, rel, new;
	P := PointGroup( S );
	iso := IsomorphismFpGroup( P );
	F := Image( iso );
	d := DimensionOfMatrixGroup( S ) - 1;
	gens := GeneratorsOfGroup( FreeGroupOfFpGroup( F ) );
	mats := iso!.preimagesInAffineCrystGroup;
	coc := [];
	for rel in RelatorsOfFpGroup( F ) do
	new := MappedWord( rel, gens, mats );
	Add( coc, new[d+1]{[1..d]} );
	od;
	return coc;
	end;
	#############################################################################
	##
	#M CocVecs( <S> ) . . Cocycles of extension of point group with translations
	##
	InstallMethod( CocVecs, true, [ IsAffineCrystGroupOnRight ], 0,
	function( S )
	return ReducedLatticeBasis( CocycleInfo( S ) );
	end );
	InstallMethod( CocVecs, true, [ IsAffineCrystGroupOnLeft ], 0,
	function( S )
	return CocVecs( TransposedMatrixGroup( S ) );
	end );

But I also noticed the following definition in the paper by Bettina Eick and Bernd Souvignier:

It seems that this definition does not completely correspond to the algorithm above. Any tips/comments will be appreciated.

Regards,
Zhao

[hongyi-zhao]

See #39 (comment) for the related discussion. Considering that a system of nonprimitive translations (SNoT) is completely determined by its values on generators of the point group, as shown in the screenshot below, so a space group is symmorphic if and only if the SNoT of the preimage of the generators of the point group in the corresponding space group is 0 mod lattice basis.

While the current implementation of IsSymmorphicSpaceGroup in Cryst package is based on the condition that the SNoT of the preimage of the relators of the point group in the corresponding space group is 0 mod lattice basis.