GAP (HAP commands) - Chapter 26: Coxeter diagrams and graphs of groups

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26 Coxeter diagrams and graphs of groups
26.1
26.1-1 CoxeterDiagramComponents
26.1-2 CoxeterDiagramDegree
26.1-3 CoxeterDiagramDisplay
26.1-4 CoxeterDiagramFpArtinGroup
26.1-5 CoxeterDiagramFpCoxeterGroup
26.1-6 CoxeterDiagramIsSpherical
26.1-7 CoxeterDiagramMatrix
26.1-8 CoxeterSubDiagram
26.1-9 CoxeterDiagramVertices
26.1-10 EvenSubgroup
26.1-11 GraphOfGroupsDisplay
26.1-12 GraphOfResolutions
26.1-13 GraphOfGroups
26.1-14 GraphOfResolutionsDisplay
26.1-15 GraphOfGroupsTest
26.1-16 TreeOfGroupsToContractibleGcomplex
26.1-17 TreeOfResolutionsToContractibleGcomplex

### 26 Coxeter diagrams and graphs of groups

#### 26.1

##### 26.1-1 CoxeterDiagramComponents

`‣ CoxeterDiagramComponents`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns a list \([D_1, ..., D_d]\) of the maximal connected subgraphs \(D_i\).

Examples:

##### 26.1-2 CoxeterDiagramDegree

`‣ CoxeterDiagramDegree`( D, v ) | ( function )

Inputs a Coxeter diagram \(D\) and vertex \(v\). It returns the degree of \(v\) (i.e. the number of edges incident with \(v\)).

Examples:

##### 26.1-3 CoxeterDiagramDisplay

`‣ CoxeterDiagramDisplay`( D ) | ( function )

`‣ CoxeterDiagramDisplay`( D, str ) | ( function )

Inputs a Coxeter diagram \(D\) and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using a second argument \(str\)="mozilla".

This function requires Graphviz software.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7

##### 26.1-4 CoxeterDiagramFpArtinGroup

`‣ CoxeterDiagramFpArtinGroup`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns the corresponding finitely presented Artin group.

Examples: 1

##### 26.1-5 CoxeterDiagramFpCoxeterGroup

`‣ CoxeterDiagramFpCoxeterGroup`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns the corresponding finitely presented Coxeter group.

Examples: 1

##### 26.1-6 CoxeterDiagramIsSpherical

`‣ CoxeterDiagramIsSpherical`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns "true" if the associated Coxeter groups is finite, and returns "false" otherwise.

Examples: 1

##### 26.1-7 CoxeterDiagramMatrix

`‣ CoxeterDiagramMatrix`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns a matrix representation of it. The matrix is given as a function \(DiagramMatrix(D)(i,j)\) where \(i,j\) can range over the vertices.

Examples:

##### 26.1-8 CoxeterSubDiagram

`‣ CoxeterSubDiagram`( D, V ) | ( function )

Inputs a Coxeter diagram \(D\) and a subset \(V\) of its vertices. It returns the full sub-diagram of \(D\) with vertex set \(V\).

Examples:

##### 26.1-9 CoxeterDiagramVertices

`‣ CoxeterDiagramVertices`( D ) | ( function )

Inputs a Coxeter diagram \(D\) and returns its set of vertices.

Examples:

##### 26.1-10 EvenSubgroup

`‣ EvenSubgroup`( G ) | ( function )

Inputs a group \(G\) and returns a subgroup \(G^+\). The subgroup is that generated by all products \(xy\) where \(x\) and \(y\) range over the generating set for \(G\) stored by GAP. The subgroup is probably only meaningful when \(G\) is an Artin or Coxeter group.

Examples: 1 , 2

##### 26.1-11 GraphOfGroupsDisplay

`‣ GraphOfGroupsDisplay`( D ) | ( function )

`‣ GraphOfGroupsDisplay`( D, str ) | ( function )

Inputs a graph of groups \(D\) and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram. An alternative browser can be set using the second argument \(str\)="mozilla".

This function requires Graphviz software.

Examples: 1 , 2 , 3 , 4

##### 26.1-12 GraphOfResolutions

`‣ GraphOfResolutions`( D, n ) | ( function )

Inputs a graph of groups \(D\) and a positive integer \(n\). It returns a graph of resolutions, each resolution being of length \(n\). It uses the function ResolutionGenericGroup() to produce the resolutions.

Examples:

##### 26.1-13 GraphOfGroups

`‣ GraphOfGroups`( D ) | ( function )

Inputs a graph of resolutions \(D\) and returns the corresponding graph of groups.

Examples: 1 , 2 , 3 , 4

##### 26.1-14 GraphOfResolutionsDisplay

`‣ GraphOfResolutionsDisplay`( D ) | ( function )

Inputs a graph of resolutions \(D\) and displays it as a .gif file. It uses the Mozilla web browser as a default to view the diagram.

This function requires Graphviz software.

Examples:

##### 26.1-15 GraphOfGroupsTest

`‣ GraphOfGroupsTest`( D ) | ( function )

Inputs an object \(D\) and itries to test whether it is a Graph of Groups. However, it DOES NOT test the injectivity of any homomorphisms. It returns true if \(D\) passes the test, and false otherwise.

Note that there is no function \(IsHapGraphOfGroups()\) because no special data type has been created for these graphs.

Examples:

##### 26.1-16 TreeOfGroupsToContractibleGcomplex

`‣ TreeOfGroupsToContractibleGcomplex`( D, G ) | ( function )

Inputs a graph of groups \(D\) which is a tree, and also inputs the fundamental group \(G\) of the tree in a form which contains each of the groups in the graph as subgroups. It returns a corresponding contractible G-complex.

Examples:

##### 26.1-17 TreeOfResolutionsToContractibleGcomplex

`‣ TreeOfResolutionsToContractibleGcomplex`( D, G ) | ( function )

Inputs a graph of resolutions \(D\) which is a tree, and also inputs the fundamental group \(G\) of the tree in a form which contains each of the groups in the graph as subgroups. It returns a corresponding contractible G-complex. The resolutions are stored as a component of the contractible \(G\)-complex.

Examples:

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