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The
cohomology of a group G with coefficients in a ZG-module A is defined
as:
When the abelian group underlying A is free of rank n we can encode A as a group homomorphism A:G → GLn(Z). When G is a permutation group of degree n the free abelian group Zn admits a canonical G-action defined by g·(x1, x2,
... , xn) = (xg'(1) , xg'(2) , ... , xg'(n))
where g'=g-1, for g in G and xi in Z. This canonical permutation module A can be constructed for any permutation group G using the HAP command PermToMatrixGroup(). For example: |
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gap>
G:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(G,5); [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 0, 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, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ] ] |
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The
following commands show that:
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gap>
Alt5:=AlternatingGroup(5);; gap> A:=PermToMatrixGroup(SymmetricGroup(5),5);; gap> R:=ResolutionFiniteGroup(Alt5,7);; gap> TR:=HomToIntegralModule(R,A);; gap> Cohomology(TR,6); [ 2, 6 ] gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]]];; gap> R:=ResolutionArtinGroup(D,10);; gap> Brd5:=R.group;; Brd5Gens:=GeneratorsOfGroup(Brd5);; gap> ImGens:=[Image(A,(1,2)),Image(A,(2,3)),Image(A,(3,4)),Image(A,(4,5))];; gap> B:=GroupHomomorphismByImages(Brd5,Image(A),Brd5Gens,ImGens);; gap> EvBrd5:=EvenSubgroup(Brd5);; gap> S:=ResolutionSubgroup(R,EvBrd5);; gap> TS:=HomToIntegralModule(S,B);; gap> Cohomology(TS,3); [ 2, 6, 0, 0, 0 ] |
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Such
commands can be used, for instance, to produce strong evidence in
support of the following.
The following commands support the conjecture. (It might be possible to use the techniques of [M. Salvetti, "Cohomology of Coxeter groups", Topology and its Applications 118 (2002), 199-208] to prove the conjecture.) |
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gap>
R:=[];;C:=[];;D:=[];; gap> for n in [2..5] do > R[n]:=ResolutionFiniteGroup(SymmetricGroup(n),6); > C[n]:=HomToIntegralModule(R[n],PermToMatrixGroup(SymmetricGroup(n),n)); > D[n]:=HomToIntegers(R[n]); > od; gap> for k in [0..5] do > for n in [2..4] do > Print("Twisted Cohomology: H",k,"(S",n+1,") = ",Cohomology(C[n+1],k),"\n"); > Print("Integral Cohomology: H",k,"(S",n,") = ",Cohomology(D[n],k),"\n\n"); > od;od; Twisted Cohomology: H0(S3) = [ 0 ] Integral Cohomology: H0(S2) = [ 0 ] Twisted Cohomology: H0(S4) = [ 0 ] Integral Cohomology: H0(S3) = [ 0 ] Twisted Cohomology: H0(S5) = [ 0 ] Integral Cohomology: H0(S4) = [ 0 ] Twisted Cohomology: H1(S3) = [ ] Integral Cohomology: H1(S2) = [ ] Twisted Cohomology: H1(S4) = [ ] Integral Cohomology: H1(S3) = [ ] Twisted Cohomology: H1(S5) = [ ] Integral Cohomology: H1(S4) = [ ] Twisted Cohomology: H2(S3) = [ 2 ] Integral Cohomology: H2(S2) = [ 2 ] Twisted Cohomology: H2(S4) = [ 2 ] Integral Cohomology: H2(S3) = [ 2 ] Twisted Cohomology: H2(S5) = [ 2 ] Integral Cohomology: H2(S4) = [ 2 ] Twisted Cohomology: H3(S3) = [ ] Integral Cohomology: H3(S2) = [ ] Twisted Cohomology: H3(S4) = [ ] Integral Cohomology: H3(S3) = [ ] Twisted Cohomology: H3(S5) = [ 2 ] Integral Cohomology: H3(S4) = [ 2 ] Twisted Cohomology: H4(S3) = [ 2 ] Integral Cohomology: H4(S2) = [ 2 ] Twisted Cohomology: H4(S4) = [ 6 ] Integral Cohomology: H4(S3) = [ 6 ] Twisted Cohomology: H4(S5) = [ 2, 12 ] Integral Cohomology: H4(S4) = [ 2, 12 ] Twisted Cohomology: H5(S3) = [ ] Integral Cohomology: H5(S2) = [ ] Twisted Cohomology: H5(S4) = [ ] Integral Cohomology: H5(S3) = [ ] Twisted Cohomology: H5(S5) = [ 2 ] Integral Cohomology: H5(S4) = [ 2 ] |
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