[GAP Forum] Cryst and CrystCat

[Franz Gaehler]

Dear Moritz

> To get a crystallographic group from the databases in Cryst and
> CrystCat we have two commands available:
> 
> S := SpaceGroupIT(dim, nr)
> 
> and
> 
> S := SpaceGroupBBNWZ(dim, nr).
> 
> In both cases, what we get is a matrix group in GL(dim+1, Q).
> 
> How can I find out the basis that is used for this representation? Is
> that given by InternalBasis(S) which should be the same as
> TranslationBasis(S)?
> 
> I have read the documentation of both packages but find it still
> impossible to answer this fundamental question. So any help from
> someone more knowledgeable will be greatly appreciated.

There is no simple answer, I'm afraid. These packages are about
algebra, not geometry. Each space group type is given as a 
representative of an affine conjugacy class of (affine) matrix 
groups. Choosing a representative has of course to do with
choosing an origin and a basis of affine space.

The space group tables included in Cryst and CrystCat come from 
the International Tables of Crystallography (IT) and from the 
book of Brown et al. The short answer is, that our choice of 
basis and origin is exactly the same as in those sources.

Crystallographers call the choice of a basis and origin a
"setting", and for each space group type they have one or
two standard settings, which are described in the IT.
SpaceGroupIT returns a space group in one of those standard
settings. If there are two, one can choose which one with 
an optional third parameter. Roughly speaking, crystallographers
like to work with a highly symmetric basis, even if this basis
does not generate the full translation lattice. For instance,
cubic crystals are always expressed with an orthogonal basis,
even if the crystal is face centered. TranslationBasis then
determines some basis (always the same) of the full translation 
lattice (with respect to the original basis), which in the face 
centered cubic case will contain also vectors with half-integer 
components. InternalBasis also contains a basis of the translation 
lattice, but should be used only for internal purposes.

SpaceGroupBBNWZ returns each space group in the representation
chosen in the BBNWZ book. These groups are always expressed with
a basis of the full lattice of translation symmetries, so that
TranslationBasis here always returns the standard basis of Z^d.
How this lattice looks like geometrically is discussed to some
extent in the book, but is not part of the package.

This is probably not the answer you wanted to hear, but if you
want to know more about the geometry of the basis chosen, you
will have to look up the original sources.

Best regards, Franz
_____________________________________________________________________________
Dr. Franz Gähler          Phone  +49 521 / 106   3876
Faculty of Mathematics    Fax    +49 521 / 106 153876
Bielefeld University      Email  gaehler at math.uni-bielefeld.de
D-33615 Bielefeld         http://www.math.uni-bielefeld.de/~gaehler/