GAP (HAP commands) - Chapter 18: Orbit polytopes and fundamental domains

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18 Orbit polytopes and fundamental domains
18.1
18.1-1 CoxeterComplex
18.1-2 ContractibleGcomplex
18.1-3 QuotientOfContractibleGcomplex
18.1-4 TruncatedGComplex
18.1-5 FundamentalDomainStandardSpaceGroup
18.1-6 OrbitPolytope
18.1-7 PolytopalComplex
18.1-8 PolytopalGenerators
18.1-9 VectorStabilizer

### 18 Orbit polytopes and fundamental domains

#### 18.1

##### 18.1-1 CoxeterComplex

`‣ CoxeterComplex`( D ) | ( function )

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

Inputs a Coxeter diagram \(D\) of finite type. It returns a non-free ZW-resolution for the associated Coxeter group \(W\). The non-free resolution is obtained from the permutahedron of type \(W\). A positive integer \(n\) can be entered as an optional second variable; just the first \(n\) terms of the non-free resolution are then returned.

Examples: 1

##### 18.1-2 ContractibleGcomplex

`‣ ContractibleGcomplex`( str ) | ( function )

Inputs one of the following strings \(str\)=:
"SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , "PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" , "PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)"
or one of the following strings
"SL(2,O-2)" , "SL(2,O-7)" , "SL(2,O-11)" , "SL(2,O-19)" , "SL(2,O-43)" , "SL(2,O-67)" , "SL(2,O-163)"
It returns a non-free ZG-resolution for the group \(G\) described by the string. The stabilizer groups of cells are finite. (Subscripts _b , _c , _d denote alternative non-free ZG-resolutions for a given group G.)
Data for the first list of non-free resolutions was provided provided by Mathieu Dutour. Data for the second list was provided by Alexander Rahm.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8

##### 18.1-3 QuotientOfContractibleGcomplex

`‣ QuotientOfContractibleGcomplex`( C, D ) | ( function )

Inputs a non-free \(ZG\)-resolution \(C\) and a finite subgroup \(D\) of \(G\) which is a subgroup of each cell stabilizer group for \(C\). Each element of \(D\) must preserves the orientation of any cell stabilized by it. It returns the corresponding non-free \(Z(G/D)\)-resolution. (So, for instance, from the \(SL(2,O)\) complex \(C=ContractibleGcomplex("SL(2,O-2)");\) we can construct a \(PSL(2,O)\)-complex using this function.)

Examples: 1

##### 18.1-4 TruncatedGComplex

`‣ TruncatedGComplex`( R, m, n ) | ( function )

Inputs a non-free \(ZG\)-resolution \(R\) and two positive integers \(m \) and \( n \). It returns the non-free \(ZG\)-resolution consisting of those modules in \(R\) of degree at least \(m\) and at most \(n\).

Examples:

##### 18.1-5 FundamentalDomainStandardSpaceGroup

`‣ FundamentalDomainStandardSpaceGroup`( v, G ) | ( function )

Inputs a crystallographic group G (represented using AffineCrystGroupOnRight as in the GAP package Cryst). It also inputs a choice of vector v in the euclidean space \(R^n\) on which \(G\) acts. It returns the Dirichlet-Voronoi fundamental cell for the action of \(G\) on euclidean space corresponding to the vector \(v\). The fundamental cell is a fundamental domain if \(G\) is Bieberbach. The fundamental cell/domain is returned as a Polymake object. Currently the function only applies to certain crystallographic groups. See the manuals to HAPcryst and HAPpolymake for full details.

This function is part of the HAPcryst package written by Marc Roeder and is thus only available if HAPcryst is loaded.

The function requires the use of Polymake software.

Examples: 1 , 2

##### 18.1-6 OrbitPolytope

`‣ OrbitPolytope`( G, v, L ) | ( function )

Inputs a permutation group or matrix group \(G\) of degree \(n\) and a rational vector \(v\) of length \(n\). In both cases there is a natural action of \(G\) on \(v\). Let \(P(G,v)\) be the convex polytope arising as the convex hull of the Euclidean points in the orbit of \(v\) under the action of \(G\). The function also inputs a sublist \(L\) of the following list of strings:

["dimension","vertex_degree", "visual_graph", "schlegel","visual"]

Depending on the sublist, the function:

- prints the dimension of the orbit polytope \(P(G,v)\);

- prints the degree of a vertex in the graph of \(P(G,v)\);

- visualizes the graph of \(P(G,v)\);

- visualizes the Schlegel diagram of \(P(G,v)\);

- visualizes \(P(G,v)\) if the polytope is of dimension 2 or 3.

The function uses Polymake software.

Examples: 1 , 2

##### 18.1-7 PolytopalComplex

`‣ PolytopalComplex`( G, v ) | ( function )

`‣ PolytopalComplex`( G, v, n ) | ( function )

Inputs a permutation group or matrix group \(G\) of degree \(n\) and a rational vector \(v\) of length \(n\). In both cases there is a natural action of \(G\) on \(v\). Let \(P(G,v)\) be the convex polytope arising as the convex hull of the Euclidean points in the orbit of \(v\) under the action of \(G\). The cellular chain complex \(C_*=C_*(P(G,v))\) is an exact sequence of (not necessarily free) \(ZG\)-modules. The function returns a component object \(R\) with components:

- \(R!.dimension(k)\) is a function which returns the number of \(G\)-orbits of the \(k\)-dimensional faces in \(P(G,v)\). If each \(k\)-face has trivial stabilizer subgroup in \(G\) then \(C_k\) is a free \(ZG\)-module of rank \(R.dimension(k)\).

- \(R!.stabilizer(k,n)\) is a function which returns the stabilizer subgroup for a face in the \(n\)-th orbit of \(k\)-faces.

- If all faces of dimension <\(k+1\) have trivial stabilizer group then the first \(k\) terms of \(C_*\) constitute part of a free \(ZG\)-resolution. The boundary map is described by the function \(boundary(k,n)\) . (If some faces have non-trivial stabilizer group then \(C_*\) is not free and no attempt is made to determine signs for the boundary map.)

- \(R!.elements\), \(R!.group\), \(R!.properties\) are as in a \(ZG\)-resolution.

If an optional third input variable \(n\) is used, then only the first \(n\) terms of the resolution \(C_*\) will be computed.

The function uses Polymake software.

Examples: 1 , 2

##### 18.1-8 PolytopalGenerators

`‣ PolytopalGenerators`( G, v ) | ( function )

Inputs a permutation group or matrix group \(G\) of degree \(n\) and a rational vector \(v\) of length \(n\). In both cases there is a natural action of \(G\) on \(v\), and the vector \(v\) must be chosen so that it has trivial stabilizer subgroup in \(G\). Let \(P(G,v)\) be the convex polytope arising as the convex hull of the Euclidean points in the orbit of \(v\) under the action of \(G\). The function returns a record \(P\) with components:

- \(P.generators\) is a list of all those elements \(g\) in \(G\) such that \(g\cdot v\) has an edge in common with \(v\). The list is a generating set for \(G\).

- \(P.vector\) is the vector \(v\).

- \(P.hasseDiagram\) is the Hasse diagram of the cone at \(v\).

The function uses Polymake software. The function is joint work with Seamus Kelly.

Examples:

##### 18.1-9 VectorStabilizer

`‣ VectorStabilizer`( G, v ) | ( function )

Inputs a permutation group or matrix group \(G\) of degree \(n\) and a rational vector of degree \(n\). In both cases there is a natural action of \(G\) on \(v\) and the function returns the group of elements in \(G\) that fix \(v\).

Examples:

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