[aclib] 3 The catalog of almost crystallographic groups

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# 3 The catalog of almost crystallographic groups

### Sections

1. Rational matrix groups
2. Polycyclically presented groups
3. More about the type and the defining parameters
4. The electronic versus the printed library

This chapter introduces the access functions to the catalog of 3- and 4-dimensional crystallographic groups. This catalog is an electronic version of the classification obtained in KD.

## 3.1 Rational matrix groups

The following three main functions are available to access the library of almost crystallographic groups as rational matrix groups.

- `AlmostCrystallographicGroup(` dim`,` type`,` parameters`)`
- `AlmostCrystallographicDim3(` type`,` parameters`)`
- `AlmostCrystallographicDim4(` type`,` parameters`)`

  dim is the dimension of the required group. Thus dim must be either 3 or 4. The inputs type and parameters are used to define the desired group as described in KD. We outline the possible choices for type and parameters here briefly. A more extended description is given later in Section More about almost crystallographic groups or can be obtained from KD.

  type specifies the type of the required group. There are 17 types in dimension 3 and 95 types in dimension 4. The input type can either be an integer defining the position of the desired type among all types; that is, in this case type is a number in [1..17] in dimension 3 or a number in [1..95] in dimension 4. Alternatively, type can be a string defining the desired type. In dimension 3 the possible strings are `"01"`, `"02"`, …, `"17"`. In dimension 4 the possible strings are listed in the list `ACDim4Types` and thus can be accessed from GAP.

  parameters is a list of integers. Its length depends on the type of the chosen group. The lists `ACDim3Param` and `ACDim4Param` contain at position i the length of the parameter list for the type number i. Every list of integers of this length is a valid parameter input. Alternatively, one can input `false` instead of a parameter list. Then GAP will chose a random parameter list of suitable length.

      gap> G := AlmostCrystallographicGroup( 4, 50, [ 1, -4, 1, 2 ] );
      <matrix group of size infinity with 5 generators>
      gap> DimensionOfMatrixGroup( G );
      5
      gap> FieldOfMatrixGroup( G );
      Rationals
      gap> GeneratorsOfGroup( G );
      [ [ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
            [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
        [ [ 1, 1/2, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ],
            [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
        [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
            [ 0, 0, 0, 1, 1 ], [ 0, 0, 0, 0, 1 ] ],
        [ [ 1, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
            [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
        [ [ 1, -4, 1, 0, 1/2 ], [ 0, 0, -1, 0, 0 ], [ 0, 1, 0, 0, 0 ],
            [ 0, 0, 0, 1, 1/4 ], [ 0, 0, 0, 0, 1 ] ] ]
      gap> G.1;
      [ [ 1, 0, -1/2, 0, 0 ], [ 0, 1, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ],
        [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ]
      gap> ACDim4Types[50];
      "076"
      gap> ACDim4Param[50];
      4

  ## 3.2 Polycyclically presented groups

  All the almost crystallographic groups considered in this package are polycyclic. Hence they have a polycyclic presentation and this can be used to facilitate efficient computations with the groups. To obtain the polycyclic presentation of an almost crystallographic group we supply the following functions. Note that the share package Polycyclic must be installed to use these functions.

- `AlmostCrystallographicPcpGroup(` dim`,` type`,` parameters`)`
- `AlmostCrystallographicPcpDim3(` type`,` parameters`)`
- `AlmostCrystallographicPcpDim4(` type`,` parameters`)`

  The input is the same as for the corresponding matrix group functions. The output is a pcp group isomorphic to the corresponding matrix group. An explicit isomorphism from an almost crystallographic matrix group to the corresponding pcp group can be obtained by the following function.

- `IsomorphismPcpGroup(` G`)`

  We can use the polycyclic presentations of almost crystallographic groups to exhibit structure information on these groups. For example, we can determine their Fitting subgroup and ask group-theoretic questions about this nilpotent group. The factor G / Fit(G) of an almost crystallographic group G is called holonomy group. We provide access to this factor of a pcp group via the following functions. Let G be an almost crystallographic pcp group.

- `HolonomyGroup(` G`)`
- `NaturalHomomorphismOnHolonomyGroup(` G`)`

  The following example shows applications of these functions.

      gap> G := AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );
      Pcp-group with orders [ 4, 0, 0, 0, 0 ]
      gap> Cgs(G);
      [ g1, g2, g3, g4, g5 ]

      gap> F := FittingSubgroup( G );
      Pcp-group with orders [ 0, 0, 0, 0 ]
      gap> Centre(F);
      Pcp-group with orders [ 0, 0 ]
      gap> LowerCentralSeries(F);
      [ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0 ],
        Pcp-group with orders [  ] ]
      gap> UpperCentralSeries(F);
      [ Pcp-group with orders [ 0, 0, 0, 0 ], Pcp-group with orders [ 0, 0 ],
        Pcp-group with orders [  ] ]
      gap> MinimalGeneratingSet(F);
      [ g2, g3, g4 ]

      gap> H := HolonomyGroup( G );
      Pcp-group with orders [ 4 ]
      gap> hom := NaturalHomomorphismOnHolonomyGroup( G );
      [ g1, g2, g3, g4, g5 ] -> [ g1, identity, identity, identity, identity ]
      gap> U := Subgroup( H, [Pcp(H)[1]^2] );
      Pcp-group with orders [ 2 ]
      gap> PreImage( hom, U );
      Pcp-group with orders [ 2, 0, 0, 0, 0 ]

  ## 3.3 More about the type and the defining parameters

  Each group from this library knows that it is almost crystallographic and, additionally, it knows its type and defining parameters.

- `AlmostCrystallographicInfo(` G`)`

  This attribute is set for groups from the library only. It is not possible at current to determine the type and the defining parameters for an arbitrary almost crystallographic groups which is not defined by the library access functions.

      gap> G := AlmostCrystallographicGroup( 4, 70, false );
      <matrix group of size infinity with 5 generators>
      gap> IsAlmostCrystallographic(G);
      true
      gap> AlmostCrystallographicInfo(G);
      rec( dim := 4, type := 70, param := [ 1, -4, 1, 2, -3 ] )

      gap> G := AlmostCrystallographicPcpGroup( 4, 70, false );
      Pcp-group with orders [ 6, 0, 0, 0, 0 ]
      gap> IsAlmostCrystallographic(G);
      true
      gap> AlmostCrystallographicInfo(G);
      rec( dim := 4, type := 70, param := [ -3, 2, 5, 1, 0 ] )

  We consider the types of almost crystallographic groups in more detail. The almost crystallographic groups in dimensions 3 and 4 fall into three families

  (1)
    3-dimensional almost crystallographic groups.
  (2)
    4-dimensional almost crystallographic groups with a Fitting subgroup of class 2.
  (3)
    4-dimensional almost crystallographic groups with a Fitting subgroup of class 3.

  These families are split up further into subfamilies in KD and to each subfamily is assigned a type; that is, a string which is used to identify the subfamily. As mentioned above, for the 3-dimensional almost crystallographic groups the type is a string representing the numbers from 1 to 17, i.e. the available types are `"01"`, `"02"`, …, `"17"`.

  For the 4-dimensional almost crystallographic groups with a Fitting subgroup of class 2 the type is a string of 3 or 4 characters. In general, a string of 3 characters representing the number of the table entry in KD is used. So possible types are `"001"`, `"002"`, …. The reader is warned however that not all possible numbers are used, e.g. there are no groups of type `"016"`. Also, the types do not appear in their natural order in KD. Moreover, for certain numbers there is more than one family of groups listed in KD. For example, the 3 families of groups corresponding to number 19 on pages 179-180 of KD have types `"019"`, `"019b"` and `"019c"` (the order is the one given in KD).

  For the last category of groups, the 4-dimensional almost crystallographic groups with a Fitting subgroup of class 3, the type is a string of 2 or 3 characters, where the first character is always the letter `"B"`. This `"B"` is followed by the number of the table entry as found in KD, eventually followed by a `"b"` or `"c"` as in the previous case.

  For each type of almost crystallographic group contained in the library there exists a function taking a parameter list as input and returning the desired matrix or pcp group. These functions can be accessed from GAP using the lists `ACDim3Funcs`, `ACDim4Funcs`, `ACPcpDim3Funcs` and `ACPcpDim4Funcs` which consist of the corresponding functions.

  Although we include these direct access functions here for completeness, we note that the user should in general use the higher-level functions introduced above to obtain almost crystallographic groups from the library. In particular, these low-level access functions return matrix or pcp groups, but the almost crystallographic info flags will not be attached to them.

      gap> ACDim3Funcs[15];
      function( k1, k2, k3, k4 ) ... end
      gap> ACDim3Funcs[15](1,1,1,1);
      <matrix group with 5 generators>
      gap> ACPcpDim3Funcs[1](1);
      Pcp-group with orders [ 0, 0, 0 ]

  ## 3.4 The electronic versus the printed library

  The package `aclib` can be considered as the electronic version of Chapter 7 of KD. In this section we outline the relationship between the library presented in this manual and the printed version in KD. First we consider an example. At page 175 of KD, we find the following groups in the table starting with entry ``13''.

  13. Q=P2/c

  E: 〈a,b,c,d,α,β ||
  [b,a]=1·61cm [d,a]=1
  〉
  [c,a]=d2 k1
  [d,b]=1
  [c,b]=1
  [d,c]=1
  αa=a−1αdk2
  α2=dk3
  αb=bα
  αd = d α
  αc=c−1αd−2 k6
  βa=a−1βdk1+k2
  β2=dk5
  βb=b−1βdk4
  βd = d β
  βc=c−1βd−2 k6
  αβ = cβαdk6

  λ(α)= | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |
  1
  | k1 2 | +k2
  0
  −2 k6
  | k3 2 | + | k6 2 |
  0
  −1
  0
  0
  0
  0
  0
  1
  0
  0
  0
  0
  0
  −1
  | 1 2 |
  0
  0
  0
  0
  1
  |  | ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ | λ(β)= | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |
  1
  k1+k2
  k4
  −2 k6
  | k5 2 |
  0
  −1
  0
  0
  0
  0
  0
  −1
  0
  0
  0
  0
  0
  −1
  0
  0
  0
  0
  0
  1
  |  | ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ |

  H2(Q,Z)=Z⊕(Z2)4=Z6/A,

  A={(k1,…,k6)|| k1=0, k2,…, k5 ∈ 2Z, k6 ∈ Z}

  AB-groups:

  ∀k > 0, k ≡ 0 mod 2, (k,0,1,0,1,0)

  The number ``13'' at the beginning of this entry is the type of the almost crystallographic group in this library. This family of groups with type 13 depends on 6 parameters k1, k2, …, k6 and these are the parameters list in this library. The rational matrix representation in GAP corresponds exactly to the printed version in KD where it is named λ. In the example below, we consider the group with parameters (k1,k2,k3,k4,k5,k6)=(8,0,1,0,1,0).

      gap> G:=AlmostCrystallographicDim4("013",[8,0,1,0,1,0]);
      <matrix group with 6 generators>
      gap> G.5;
      [ [ 1, 4, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
        [ 0, 0, 0, -1, 1/2 ], [ 0, 0, 0, 0, 1 ] ]
      gap> G.6;
      [ [ 1, 8, 0, 0, 1/2 ], [ 0, -1, 0, 0, 0 ], [ 0, 0, -1, 0, 0 ],
        [ 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, 1 ] ]

  For a 4-dimensional almost crystallographic group the matrix group is built up such that { a, b, c, d, α, β, γ} as described in KD forms the defining generating set of G. For certain types the elements α, β or γ may not be present. Similarly, for a 3-dimensional group we have the generating set { a, b, c, α, β} and α and β may be absent.

  To obtain a polycyclic generating sequence from the defining generators of the matrix group we have to order the elements in the generating set suitably. For this purpose we take the subsequence of (γ, β, α, a, b, c, d) of those generators which are present in the defining generating set of the matrix group. This new ordering of the generators is then used to define a polycyclic presentation of the given almost crystallographic group.

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