[GAP Forum] Re:Orbit Lengths

[Thomas Breuer]

Dear GAP Forum,

Thekiso Seretlo wrote

> Let V := FullRowSpace(GF(2),2)
> m1:=One(GF(2))*[[0,1,0],[1,0,0],[0,0,1]];
> m2:=One(GF(2))*[[0,1,0],[0,0,1],[1,0,0]];
> m:=Group(m1,m2);
> this is a matrix representation of $S3$ in GF(2)
> If we say
> OrbitLengths(m,V)
> we get [2,4,4,4] that is we show that  this has four orbits and of the
> lenghts given that is the conjugacy classes here. Moving to irreducible
> characters by Brauer we know that the number of orbits is four,
> my problem is how do we get the orbitlengths of the orbits of irreducible
> characters ?

The situation is that a matrix group G, say,
acts on a finite row vector space V, say,
by multiplication from the right.

In the semidirect product S of G and V defined by this action,
the orbits of G on V correspond to the conjugacy classes of S
that lie in the normal subgroup corresponding to V.

The induced action of an element g in G on the complex irreducible
characters of V, that is, on Hom(V,\C),
is just the transposed map of action of g on V.
It can be described by the left multiplication with g on column vectors.
So all one has to do for computing the orbit lengths in GAP
is to consider the group of transposed matrices.

Since the above example is obviously not correct
(3x3 matrices do not act on vectors of length two,
and the zero vector yields always an orbit of length one),
I show another example.

Let us consider a dihedral subgroup of order six in GL(2,3),
acting on GF(3)^2.
The two actions yield four orbits each,
but with different orbit lengths.

    gap> gens:= Z(3)^0 * [ [ [ 0, 1 ], [ -1, -1 ] ],
    >                      [ [ 0, 1 ], [ 1, 0 ] ] ];;
    gap> OrbitLengths( Group( gens ), GF(3)^2 );
    [ 1, 3, 3, 2 ]
    gap> OrbitLengths( Group( List( gens, TransposedMat ) ), GF(3)^2 );
    [ 1, 6, 1, 1 ]

All the best,
Thomas