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| A
Coxeter group is a finitely presented group obtained from an Artin
presentation by imposing the extra relations x2=1 on all
Artin generators x. The resolution for certain (and conjecturally all)
Artin groups described on the preceding page leads to a resolution for
all Coxeter groups. At present this resolution has only been
implemented in HAP for finite Coxeter groups and only in dimensions
less than or equal to n where n denotes the number of generators in the
Coxeter presentation. |
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| The
following commands compute 7 terms of a free resolution for the
symmetric group on 8 letters, and then use this resolution to construct
a resolution for its alternating subgroup of index 2. |
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| gap>
D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,3]]];; gap> R:=ResolutionCoxeterGroup(D,7);; gap> S:=ResolutionFiniteSubgroup(R,AlternatingGroup(8));; gap> Homology(TensorWithIntegers(S),4); [ 2 ] |
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