48.25 MatRepresentationsPGroup

MatRepresentationsPGroup( G )
MatRepresentationsPGroup( G [, int ] )

MatRepresentationsPGroup( G ) returns a list of homomorphisms from the finite polycyclic group G to irreducible complex matrix groups. These matrix groups form a system of representatives of the complex irreducible representations of G.

MatRepresentationsPGroup( G, int ) returns only the int-th representation.

Let G be a finite polycyclic group with an abelian normal subgroup N such that the factorgroup <G> / <N> is supersolvable. MatRepresentationsPGroup uses the algorithm described in Bau91. Note that for such groups all such representations are equivalent to monomial ones, and in fact MatRepresentationsPGroup only returns monomial representations.

If G has not the property stated above, a system of representatives of irreducible representations and characters only for the factor group <G> / <M> can be computed using this algorithm, where M is the derived subgroup of the supersolvable residuum of G. In this case first a warning is printed. MatRepresentationsPGroup returns the irreducible representations of G with kernel containing M then.

    gap> g:= SolvableGroup( 6, 2 );
    S3
    gap> MatRepresentationsPGroup( g );
    [ GroupHomomorphismByImages( S3, Group( [ [ 1 ] ] ), [ a, b ],
        [ [ [ 1 ] ], [ [ 1 ] ] ] ), GroupHomomorphismByImages( S3, Group(
        [ [ -1 ] ] ), [ a, b ], [ [ [ -1 ] ], [ [ 1 ] ] ] ),
      GroupHomomorphismByImages( S3, Group( [ [ 0, 1 ], [ 1, 0 ] ],
        [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ), [ a, b ],
        [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ] ) ]

CharTablePGroup can be used to compute the character table of a group with the above properties (see CharTablePGroup).

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