48.41 MatAutomorphisms

MatAutomorphisms( mat, maps, subgroup )

returns the permutation group record representing the matrix automorphisms of the matrix mat that respect all lists in the list maps, i.e. representing the group of those permutations of columns of mat which acts on the set of rows of mat and additionally fixes all lists in maps.

subgroup is a list of permutation generators of a subgroup of this group. E.g. generators of the Galois automorphisms of a matrix of ordinary characters may be entered here.

    gap> t:= CharTable( "Dihedral", 8 );;
    gap> PrintCharTable( t );
    rec( identifier := "D8", name := "D8", size := 8, order :=
    8, centralizers := [ 8, 4, 8, 4, 4 ], orders := [ 1, 4, 2, 2, 2
     ], powermap := [ , [ 1, 3, 1, 1, 1 ] ], irreducibles :=
    [ [ 1, 1, 1, 1, 1 ], [ 1, 1, 1, -1, -1 ], [ 1, -1, 1, 1, -1 ],
      [ 1, -1, 1, -1, 1 ], [ 2, 0, -2, 0, 0 ] ], classparam :=
    [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 2, 0 ], [ 2, 1 ] ], irredinfo :=
    [ rec(
          charparam := [ 1, 0 ] ), rec(
          charparam := [ 1, 1 ] ), rec(
          charparam := [ 1, 2 ] ), rec(
          charparam := [ 1, 3 ] ), rec(
          charparam := [ 2, 1 ] )
     ], text := "computed using generic character table for dihedral group\
    s", classes := [ 1, 2, 1, 2, 2
     ], operations := CharTableOps, fusions := [  ], fusionsource :=
    [  ], projections := [  ], projectionsource := [  ] )
    gap> MatAutomorphisms( t.irreducibles, [], Group( () ) );
    Group( (4,5), (2,4) )
    gap> MatAutomorphisms( t.irreducibles, [ t.orders ], Group( () ) );
    Group( (4,5) )

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