GAP (HAPcryst) - Chapter 1: Introduction

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1 Introduction
1.1 Abstract and Notation
1.1-1 The natural action of crystallographic groups
1.2 Requirements
1.2-1 Recommendation concerning polymake
1.3 Global Variables
1.3-1 InfoHAPcryst

### 1 Introduction

#### 1.1 Abstract and Notation

HAPcryst is an extension for "Homological Algebra Programming" (HAP, [Ell]) by Graham Ellis. It uses geometric methods to calculate resolutions for crystallographic groups. In this manual, we will use the terms "space group" and "crystallographic group" synonymous. As usual in GAP, group elements are supposed to act from the right. To emphasize this fact, some functions have names ending in "OnRight" (namely those, which rely on the action from the right). This is also meant to make work with HAPcryst and cryst [EGN] easier.

The functions called "somethingStandardSpaceGroup" are supposed to work for standard crystallographic groups on left and right some time in the future. Currently only the versions acting on right are implemented. As in cryst [EGN], space groups are represented as affine linear groups. For the computations in HAPcryst, crystallographic groups have to be in "standard form". That is, the translation basis has to be the standard basis of the space. This implies that the linear part of a group element need not be orthogonal with respect to the usual scalar product.

##### 1.1-1 The natural action of crystallographic groups

There is some confusion about the way crystallographic groups are written. This concerns the question if we act on left or on right and if vectors are of the form `[1,...]` or `[...,1]`.

As mentioned, HAPcryst handles affine crystallographic groups on right (and maybe later also on left) acting on vectors of the form [...,1].

BUT: The functions in HAPcryst do not take augmented vectors as input (no leading or ending ones). The handling of vectors is done internally. So in HAPcryst, a crystallographic group is a group of n× n matrices which acts on a vector space of dimension n-1 whose elements are vectors of length n-1 (not n). Example:

    gap> G:=SpaceGroup(3,4); #This group acts on 3-Space
    SpaceGroupOnRightBBNWZ( 3, 2, 1, 1, 2 )
    gap> Display(Representative(G));
    [ [  1,  0,  0,  0 ],
      [  0,  1,  0,  0 ],
      [  0,  0,  1,  0 ],
      [  0,  0,  0,  1 ] ]
    gap> OrbitStabilizerInUnitCubeOnRight(G,[1/2,0,0]);
    rec( orbit := [ [ 1/2, 0, 0 ], [ 1/2, 1/2, 0 ] ],
      stabilizer := Group([ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ],
              [ 0, 0, 0, 1 ] ] ]) )

#### 1.2 Requirements

The following GAP packages are required

- polymaking which in turn depends on the computational geometry software polymake.

- HAP

- Cryst

The following GAP packages are not required but highly recommended:

- carat

- CrystCat

- GAPDoc is needed to display the online manual

##### 1.2-1 Recommendation concerning polymake

Calculating resolutions of Bieberbach groups involves convex hull computations. polymake by default uses cdd to compute convex hulls. Experiments suggest that lrs is the more suitable algorithm for the computations done in HAPcryst than the default cdd. You can change the behaviour of by editing the file "yourhomedirectory/.polymake/prefer.pl". It should contain a section like this (just make sure lrs is before cdd, the position of beneath_beyond does not matter):

    #########################################
    application polytope;

    prefer "*.convex_hull  lrs, beneath_beyond, cdd";

#### 1.3 Global Variables

HAPcryst itself does only have one global variable, namely `InfoHAPcryst` (1.3-1). The location of files generated for interaction with polymake are determined by the value of `POLYMAKE_DATA_DIR` (polymaking: POLYMAKE_DATA_DIR) which is a global variable of polymaking.

##### 1.3-1 InfoHAPcryst

`‣ InfoHAPcryst` | ( info class )

At a level of 1, only the most important messages are printed. At level 2, additional information is displayed, and level 3 is even more verbose. At level 0, HAPcryst remains silent.

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