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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.
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.
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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