GAP (XMod) - Chapter 9: Crossed cubes and Cat\(^3\)-groups

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9 Crossed cubes and Cat\(^3\)-groups
9.1 Functions for (pre-)cat\(^3\)-groups
9.1-1 Cat3Group
9.1-2 Front3DimensionalGroup
9.2 Enumerating cat\(^3\)-groups with a given source
9.2-1 AllCat3GroupTriples
9.3 Definition and constructions for cat\(^n\)-groups and their morphisms
9.3-1 PreCatnGroup

### 9 Crossed cubes and Cat\(^3\)-groups

The term 4d-group refers to a set of equivalent categories of which the most common are the categories of crossed cubes and cat\(^3\)-groups. A 4d-mapping is a function between two 4d-groups which preserves all the structure.

The material in this chapter should be considered very experimental. As yet there are no functions for crossed cubes.

#### 9.1 Functions for (pre-)cat\(^3\)-groups

We shall use the following standard orientation of a cat\(^3\)-group \(\calE\) on a group \(G\). \(\calE\) contains \(8\) groups; \(12\) cat\(^1\)-groups and \(6\) cat\(^2\)-groups forming the vertices; edges and faces of a cube, as shown in the following diagram.

\[ \vcenter{\xymatrix{ & H \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[dr] <+0.6ex> & & & & N \ar[llll] <+0.6ex> \ar[dddd] <+0.6ex> \ar[dddd] <+0.0ex> \ar[dr] <+0.6ex> & \\ & & G \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> & & & & R \ar[llll] <+0.6ex> \ar[dddd] <+0.5ex> \ar[dddd] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\ & & & & & & \\ & & & & & & \\ & M \ar[uuuu] <+0.6ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[dr] <+0.6ex> & & & & L \ar[uuuu] <+0.6ex> \ar[llll] <+0.6ex> \ar[dr] <+0.6ex> & \\ & & Q \ar[uuuu] <+0.6ex> \ar[rrrr] <+0.5ex> \ar[rrrr] <+0.0ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> & & & & P \ar[uuuu] <+0.6ex> \ar[llll] <+0.6ex> \ar[ul] <+0.5ex> \ar[ul] <+0.0ex> \\ }} \]

By definition, \(\calE\) is generated by three commuting cat\(^1\)-groups \((G \Rightarrow R), (G \Rightarrow Q)\) and \((G \Rightarrow H)\), but it is more convenient to think of \(\calE\) as generated by two cat\(^2\)-groups

- front\((\calE)\), generated by \((G \Rightarrow R)\) and \((G \Rightarrow Q)\);

- left\((\calE)\), generated by \((G \Rightarrow Q)\) and \((G \Rightarrow H)\).

Because the tail, head and embedding maps all commute, it follows that up\((\calE)\), generated by \((G \Rightarrow H)\) and \((G \Rightarrow R)\), is a third cat\(^2\)-group. The three remaining faces (cat\(^2\)-groups) right\((\calE)\), down\((\calE)\) and back\((\calE)\) are then easily constructed. We shall always use the order [front,up,left,right,down,back] for the six faces.

##### 9.1-1 Cat3Group

`‣ Cat3Group`( args ) | ( function )

`‣ PreCat3Group`( args ) | ( function )

`‣ IsCat3Group`( C ) | ( property )

`‣ PreCat3GroupByPreCat2Groups`( L ) | ( operation )

The global functions `Cat3Group` and `PreCat3Group` are normally take as arguments a pair of cat\(^2\)-groups or a trio of cat\(^1\)-groups. In subsection `AllCat2GroupsIterator` (8.5-4) the list of pairs `CatnGroupLists(d12).pairs` contains the three entries `[6,8],[8,11]` and `[6,11]`. It follows that the sixth, eighth and eleventh cat\(^1\)-groups for `d12` generate a cat\(^3\)-group.

    gap> all1 := AllCat1Groups( d12 );;
    gap> C68 := Cat2Group( all1[6], all1[8] );;
    gap> C811 := Cat2Group( all1[8], all1[11] );;
    gap> C3Ga := Cat3Group( C68, C811 );
    cat3-group with generating (pre-)cat1-groups:
    1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
    2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
    3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
    gap> C3Gb := Cat3Group( all1[6], all1[8], all1[11] );
    cat3-group with generating (pre-)cat1-groups:
    1 : [d12 => Group( [ (), (1,6)(2,5)(3,4) ] )]
    2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )]
    3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
    gap> C3Ga = C3Gb;
    true

##### 9.1-2 Front3DimensionalGroup

`‣ Front3DimensionalGroup`( C3 ) | ( attribute )

`‣ Up3DimensionalGroup`( C3 ) | ( attribute )

`‣ Left3DimensionalGroup`( C3 ) | ( attribute )

`‣ Right3DimensionalGroup`( C3 ) | ( attribute )

`‣ Down3DimensionalGroup`( C3 ) | ( attribute )

`‣ Back3DimensionalGroup`( C3 ) | ( attribute )

The six faces of a cat\(^3\)-group are stored as these attributes.

    gap> C116 := Cat2Group( all1[11], all1[6] );;
    gap> Up3DimensionalGroup( C3Ga ) = C116;
    true

#### 9.2 Enumerating cat\(^3\)-groups with a given source

Once the list `CatnGroupLists(G).pairs` has been obtained we may seek all triples \([i,j],[j,k]\) and \([k,i]\) or \([i,k]\) of pairs in this list and then, for each such triple, construct a cat\(^3\)-group generated by the \(i\)-th, \(j\)-th and \(k\)-th cat\(^1\)-group on \(G\).

##### 9.2-1 AllCat3GroupTriples

`‣ AllCat3GroupTriples`( G ) | ( operation )

`‣ AllCat3GroupsNumber`( G ) | ( attribute )

`‣ AllCat3Groups`( G ) | ( operation )

The list of triples returned by the operation `AllCat3GroupTriples` is saved as `CatnGroupLists(G).cat3triples`. The length of this list is the number of cat\(^3\)-groups on \(G\), and is saved as `CatnGroupNumbers(G).cat3`.

As yet there is no operation `AllCat3GroupsUpToIsomorphism(G)`.

    gap> triples := AllCat3GroupTriples( d12 );;
    gap> CatnGroupNumbers( d12 ).cat3;
    94
    gap> triples[46];
    [ 5, 7, 11 ]
    gap> all1 := AllCat1Groups( d12 );;
    gap> Cat3Group( all1[5], all1[7], all1[11] );
    cat3-group with generating (pre-)cat1-groups:
    1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
    2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
    3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]

#### 9.3 Definition and constructions for cat\(^n\)-groups and their morphisms

In this chapter and the previous one we are interested in cat\(^2\)-groups and cat\(^3\)-groups, and it is convenient in this section to give the more general definition. There are three equivalent descriptions of a cat\(^n\)-group.

A cat\(^n\)-group consists of the following.

- \(2^n\) groups \(G_A\), one for each subset \(A\) of \([n]\), the vertices of an \(n\)-cube.

- Group homomorphisms forming \(n2^{n-1}\) commuting cat\(^1\)-groups,

  \[ \calC_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\ G_A \to G_{A \setminus \{i\}}), \quad\mbox{for all} \quad A \subseteq [n],~ i \in A, \]

  the edges of the cube.

- These cat\(^1\)-groups combine (in sets of \(4\)) to form \(n(n-1)2^{n-3}\) cat\(^2\)-groups \(\calC_{A,\{i,j\}}\) for all \(\{i,j\} \subseteq A \subseteq [n],~ i \neq j\), the faces of the cube.

Note that, since the \(t_{A,i}, h_{A,i}\) and \(e_{A,i}\) commute, composite homomorphisms \(t_{A,B}, h_{A,B} : G_A \to G_{A \setminus B}\) and \(e_{A,B} : G_{A \setminus B} \to G_A\) are well defined for all \(B \subseteq A \subseteq [n]\).

Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner [ES87].

A cat\(^n\)-group \(\calC\) consists of \(2^n\) groups \(G_A\), one for each subset \(A\) of \([n]\), and \(3n\) homomorphisms

\[ t_{[n],i}, h_{[n],i} : G_{[n]} \to G_{[n] \setminus \{i\}},~ e_{[n],i} : G_{[n] \setminus \{i\}} \to G_{[n]}, \]

satisfying the following axioms for all \(1 \leqslant i \leqslant n\),}

- the \(\calC_{[n],i} ~=~ (e_{[n],i};\; t_{[n],i},\; h_{[n],i} \ :\ G_{[n]} \to G_{[n] \setminus \{i\}})~\) are commuting cat\(^1\)-groups, so that:

- \( (e_1 \circ t_1) \circ (e_2 \circ t_2) = (e_2 \circ t_2) \circ (e_1 \circ t_1), \quad (e_1 \circ h_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ h_1), \)

- \( (e_1 \circ t_1) \circ (e_2 \circ h_2) = (e_2 \circ h_2) \circ (e_1 \circ t_1), \quad (e_2 \circ t_2) \circ (e_1 \circ h_1) = (e_1 \circ h_1) \circ (e_2 \circ t_2). \)

Our third description defines a cat\(^n\)-group as a "cat\(^1\)-group of cat\(^{(n-1)}\)-groups".

A cat\(^n\)-group \(\calC\) consists of two cat\(^{(n-1)}\)-groups:

- \(\calA\) with groups \(G_A,\; A \subseteq [n-1]\), and homomorphisms \(\ddot{t}_{A,i}, \ddot{h}_{A,i}, \ddot{e}_{A,i}\),

- \(\calB\) with groups \(H_B,\; B \subseteq [n-1]\), and homomorphisms \(\dot{t}_{B,i}, \dot{h}_{B,i}, \dot{e}_{B,i}\), and

- cat\(^{(n-1)}\)-morphisms \(t,h : \calA \to \calB\) and \(e : \calB \to \calA\) subject to the following conditions:

  \[ (t \circ e) ~\mbox{and}~ (h \circ e) ~\mbox{are the identity mapping on}~ \calB, \qquad [\ker t, \ker h] = \{ 1_{\calA} \}. \]

##### 9.3-1 PreCatnGroup

`‣ PreCatnGroup`( L ) | ( operation )

`‣ CatnGroup`( L ) | ( operation )

The operation `(Pre)CatnGroup` expects as input a list of cat\(^1\)-groups.

    gap> PC4 := PreCatnGroup( [ all1[5], all1[7], all1[11], all1[12] ] );
    (pre-)cat4-group with generating (pre-)cat1-groups:
    1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )]
    2 : [d12 => Group( [ (1,4)(2,5)(3,6), (2,6)(3,5) ] )]
    3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )]
    4 : [d12 => Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] )]
    gap> IsCatnGroup( PC4 );
    true
    gap> HigherDimension( PC4 );
    5

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