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 <td style="vertical-align: top;"><a
href="aboutCocycles.html">Previous</a> 
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 <td
style="vertical-align: top; color: rgb(0, 0, 102); text-align: center;"><big>About HAP: Groups acting on polytopes 
 </big></td>
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href="bieberbach.html">next</a> 
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style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center;"><big>Polytopes, Resolutions and Group
Presentations</big> 
 </div>
 
Suppose
that
G
is
a permutation group of degree n or a finite group of
n×n orthogonal real matrices. In both cases we can try to use the
action of G on Euclidean n-space Rn to
obtain
homological
information
about
G.
We do this by choosing a
vector v in Rn
which is fixed by no non-trivial element in G. Such a v is said to be
in general position and
always exists. We then form the convex hull of the vectors in the orbit
of v
under the action of G. This hull is a convex polytope which we denote
by P(G,v). Its combinatorial
structure can be accessed using the <a
href="http://www.polymake.org/doku.php">Polymake</a>
computational geometry software package. 
 
Consider for example the symmetric group S4 acting on R4
by permuting basis vectors. In this case a vector v is in general
position if its coordinates are distinct, and the polytope P(S4,v)
lies
in
the
3-dimensional
hyperplane consisting of all points whose
coordinate sum equals the sum of the coordinates in v.
Using the fact that S4 is a finite reflection group we can
show that the polytope P(S4,v) has the same combinatorial
structure for each general position vector v in R4 . The polytope
can be vizualized using the following commands (which call Polymake
functions). 
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
OrbitPolytope(SymmetricGroup(4),[1,2,3,4],["visual"]); 
 
 <div style="text-align: center;"><img
style="width: 493px; height: 304px;" alt="" src="news4poly.png"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
linearity
of the action implies that G permutes the k-faces of P(G,v)
in each dimension k. The cellular chain complex C*(P(G,v))
of the polytope is thus a ZG-equivariant chain complex. It is in fact a
ZG-resolution of the trivial module Z since the polytope is
contractible. In general this will not be a free resolution. 
 
The following commands first compute the chain complex C*(P(G,v))
for
G=S4 and v=[1,2,3,4], then show that there are 3 orbits
of edges and three orbits of facets. 
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=SymmetricGroup(4);;v:=[1,2,3,4];; 
gap> P:=PolytopalComplex(G,v); 
Non-free resolution in characteristic 0 for <matrix group of size 24
with 
2 generators> . 
No contracting homotopy available. 
 
gap> P!.dimension(0); 
1 
gap> P!.dimension(1); 
3 
gap> P!.dimension(2); 
3 
gap> P!.dimension(3); 
1 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following additional commands use the non-free resolution P to
construct three terms of a free ZG-resolution of Z for G=S4,
and then use this resolution to comute the second homology of G. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
R:=FreeGResolution(P,3); 
Resolution of length 3 in characteristic 0 for <matrix group of size
24 with 
2 generators> . 
No contracting homotopy available. 
 
gap> Homology(TensorWithIntegers(R),2); 
[ 2 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">For
groups G that are not reflection groups the combinatorial structure of
the polytope P(G,v) will depend on the choice of general position
vector v. The following commands illustrate this for the alternating
group G=A4 and three choices of general position vectors
u=[1,2,2,4], v=[1,2,3,4], w=[1,1,3,4]. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
OrbitPolytope(AlternatingGroup(4),[1,2,3,4],["visual"]); 
 
gap> OrbitPolytope(AlternatingGroup(4),[1,1,3,4],["visual"]); 
 
gap> OrbitPolytope(AlternatingGroup(4),[1,2,2,4],["visual"]); 
 
 <div style="text-align: center;"><img
style="width: 408px; height: 252px;" alt="" src="newa4first.png"><img
style="width: 409px; height: 253px;" alt="" src="newa4second.png"><img
style="width: 411px; height: 252px;" alt="" src="newa4third.png"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
resolutions obtained from P(G,v) will also vary with the choice of
general position v. For instance, continuoing with the last example we
obtain three different free resolutions for G=A4 and then
use these resolutions to obtain three different presentations of A4.

 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
P:=PolytopalComplex(AlternatingGroup(4),[1,2,3,4]);; 
gap> R:=FreeGResolution(P,3);; 
gap> Y:=PresentationOfResolution(R);; 
gap> Y.freeGroup; 
<free group on the generators [ f1, f2, f3 ]> 
gap> Y.relators; 
[ f2^-2, f1*f3*f2^-1, f1^3, f3^3 ] 
 
 
gap> P:=PolytopalComplex(AlternatingGroup(4),[1,1,3,4]);; 
gap> R:=FreeGResolution(P,3);; 
gap> Y:=PresentationOfResolution(R);; 
gap> Y.freeGroup; 
<free group on the generators [ f1, f2 ]> 
gap> Y.relators; 
[ f1^-2, f2^3, f1^3*f2^-3 ] 
 
 
gap> P:=PolytopalComplex(AlternatingGroup(4),[1,2,2,4]);; 
gap> R:=FreeGResolution(P,3);; 
gap> Y:=PresentationOfResolution(R);; 
gap> Y.freeGroup; 
<free group on the generators [ f1, f2 ]> 
gap> Y.relators; 
[ (f1*f2^-1)^2, f1^3, f2^3 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Orbit
polytopes
and
associated finite presentations for other 3-dimensional
isometry groups are available <a href="web.pdf">here</a>. 
 </td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center; font-weight: bold;"><big>Periodic
Resolutions 
 </big></div>
 
The
cellular chain complex C*(P(G,v)) is a complex of ZG-modules
which, thanks to the contractibility of the polytope, has trivial
homology in all but its top and bottom dimensions; its homology groups
are infinite cyclic in both the top and bottom dimensions. Thus
infinitely many copies of C*(P(G,v)) can be spliced together
to form an infinite periodic ZG-resolution P* of Z. 
 
In general P* is not a free ZG-resolution. But sometimes it
is free, and the homology of G is then periodic with period equal to
the dimension of P(G,v). The resolution P* is free if all
faces of the polytope (except the single top dimensional face) have
trivial stabilizer group. 
 
For example, the usual 2-dimensional complex representation of the
group Q of quaternions can be regarded as a 4-dimensional real
representation. The group Q has order eight, and the 1-skeleton of the
4-dimensional polytope P(Q,v) can be pictured using the following
commands.</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
A:=[[0,-1,0,0,],[1,0,0,0,],[0,0,0,1],[0,0,-1,0]];; 
gap> B:=[[0,0,-1,0],[0,0,0,-1],[1,0,0,0],[0,1,0,0]];; 
gap> Q:=Group([A,B]);; 
gap> OrbitPolytope(Q,[1,0,0,0],["visual_graph"]); 
 
 <div style="text-align: center;"><img
style="width: 302px; height: 262px;" alt="" src="quat.png"> 
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following additional commands show that the polytope P(Q,v) yields a
free ZQ-resolution of period 4. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
P:=PolytopalComplex(Q,[1,0,0,0]);; 
gap> for k in [1..3] do 
> for n in [1..P!.dimension(k)] do 
> Print(Order(P!.stabilizer(k,n)),"\n"); 
> od;od; 
1 
1 
1 
1 
1 
1 
1 
1 
1 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following additional commands show that the quaternion group has third
integral homology H3(Q,Z)=Z8 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
TP:=TensorWithIntegers(P);; 
gap> Homology(TP,3); 
[ 8 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">The
following
additional
command
yields
the
satisfying
group presentation 
 
 <div style="text-align: center;">Q = < i, j, k : ij=k, jk=i,
ki=j, ikj=1> 
 
 <div style="text-align: left;">for the quaternion group Q. 
 </div>
 </div>
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
PresentationOfResolution(P); 
rec( freeGroup := <free group on the generators [ f1, f2, f3 ]>, 
relators := [ f2*f3^-1*f1^-1, f3*f2*f1^-1, f1*f2*f3, f1*f3^-1*f2 ] ) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">
 <div style="text-align: center;"><big style="font-weight: bold;">Resolutions
for
Finite
Reflection
Groups</big> 
 </div>
 
For
a finite reflection group G the structure of the polytope P(G,v) is
well-understood and can be constructed via theoretical methods and
without recourse to expensive convex hull
computations. The following commands construct a free resolution, via
theoretical methods, for the
finite reflection group of type B6 and order 46080.
The resolution is then used to compute the integral homology of G in
degree 4. 
 </td>
 </tr>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,4]]];; 
gap> CoxeterDiagramDisplay(D); 
 
 <div
style="text-align: center; background-color: rgb(255, 255, 204);"><img
style="width: 179px; height: 193px;" alt="" src="diagb6.png"> 
 
 <div style="text-align: left;">gap>
R:=ResolutionCoxeterGroup(D,5);; 
 </div>
 <div style="text-align: left;">gap>
Homology(TensorWithIntegers(R),4); 
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] 
 </div>
 </div>
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style="color: rgb(0, 0, 102);" href="aboutCocycles.html">Previous
Page</a> 
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href="aboutContents.html">Contents</a> 
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