This chapter describes functions which compute with matrix representations for pcp-groups. So far the routines in this package are only able to compute matrix representations for torsion-free nilpotent groups.
UnitriangularMatrixRepresentation( G )
computes a faithful representation of a torsion-free nilpotent group G as unipotent lowertriangular matrices over the integers. The pc-presentation for G must not contain any power relations. The algorithm is described in dGN02.
checks if the map defined by maping the i-th generator of the pcp-group G to the i-th matrix of matrices defines a homomorphism.
We call a matrix upper unitriangular if it is an upper triangular matrix with ones on the main diagonal. The weight of an upper unitriangular matrix is the number of diagonals above the main diagonal that contain zeroes only.
The subgroup of all upper unitriangular matrices of GL(n,Z) is torsion-free nilpotent. The k-th term of its lower central series is the set of all matrices of weight k−1. The Z-rank of the k-th term of the lower central series modulo the (k+1)-th term is n−k.
takes a group G generated by upper unitriangular matrices over the integers and computes a polycylic presentation for the group. The function returns an isomorphism from the matrix group to the pcp group. Note that a group generated by upper unitriangular matrices is necessarily torsion-free nilpotent.
takes a group G generated by upper unitriangular matrices over the integers and returns a recursive data structure L with the following properties: L contains a polycyclic generating sequence for G, using L one can decide if a given upper unitriangular matrix is contained in G, a given element of G can be written as a word in the polycyclic generating sequence. L is a representation of a chain of subgroups of G refining the lower centrals series of G.. It contains for each subgroup in the chain a minimal generating set.
takes the data structure returned by
SiftUpperUnitriMat and prints
the Z-rank of each the subgroup in L.
creates one level of the data structure returned by
SiftUpperUnitriMat and initialises it with weight m.
SiftUpperUnitriMat( gens, level, M )
takes the generators gens of an upper unitriangular group, the data
structure returned level by
SiftUpperUnitriMat and another upper
unitriangular matrix M. It sift M through level and adds M at
the appropriate place if M is not contained in the subgroup
represented by level.
SiftUpperUnitriMatGroup illustrates the use of
InstallGlobalFunction( "SiftUpperUnitriMatGroup", function( G ) local firstlevel, g; firstlevel := MakeNewLevel( 0 ); for g in GeneratorsOfGroup(G) do SiftUpperUnitriMat( GeneratorsOfGroup(G), firstlevel, g ); od; return firstlevel; end );
takes the data structure level returned by
and a upper unitriangular matrix M and decomposes M into a word in
the polycyclic generating sequence of level.
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