38.2 Basic Actions

returns pnt ^ g. This is for example the action of a permutation group on points, or the action of a group on its elements via conjugation. The action of a matrix group on vectors from the right is described by both OnPoints and OnRight (see OnRight).

returns pnt * g. This is for example the action of a group on its elements via right multiplication, or the action of a group on the cosets of a subgroup. The action of a matrix group on vectors from the right is described by both OnPoints (see OnPoints) and OnRight.

returns g ^-1 * pnt. Forming the inverse is necessary to make this a proper action, as in GAP groups always act from the right.

(OnLeftInverse is used for example in the representation of a right coset as an external set (see External Sets), that is a right coset Ug is an external set for the group U acting on it via OnLeftInverse.)

Let set be a proper set (see Sorted Lists and Sets). OnSets returns the proper set formed by the images OnPoints( pnt, g ) of all points pnt of set.

OnSets is for example used to compute the action of a permutation group on blocks.

(OnTuples is an action on lists that preserves the ordering of entries, see OnTuples.)

Let tup be a list. OnTuples returns the list formed by the images OnPoints( pnt, g ) for all points pnt of tup.

(OnSets is an action on lists that additionally sorts the entries of the result, see OnSets.)

is a special case of OnTuples (see OnTuples) for lists tup of length 2.

Action on sets of sets; for the special case that the sets are pairwise disjoint, it is possible to use OnSetsDisjointSets (see OnSetsDisjointSets).

Action on sets of pairwise disjoint sets (see also OnSetsSets).

Action on sets of tuples.

Action on tuples of sets.

Action on tuples of tuples

gap> g:=Group((1,2,3),(2,3,4));;
gap> Orbit(g,1,OnPoints);              
[ 1, 2, 3, 4 ]
gap> Orbit(g,(),OnRight);               
[ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3), (1,4,2), 
  (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ]
gap> Orbit(g,[1,2],OnPairs);
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ], [ 1, 4 ], 
  [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ]
gap> Orbit(g,[1,2],OnSets); 
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ]

gap> Orbit(g,[[1,2],[3,4]],OnSetsSets);
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 1, 3 ], [ 2, 4 ] ] ]
gap> Orbit(g,[[1,2],[3,4]],OnTuplesSets);
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 2, 4 ] ], 
  [ [ 3, 4 ], [ 1, 2 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ]
gap> Orbit(g,[[1,2],[3,4]],OnSetsTuples);
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], [ [ 1, 3 ], [ 4, 2 ] ], 
  [ [ 2, 4 ], [ 3, 1 ] ], [ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ]
gap> Orbit(g,[[1,2],[3,4]],OnTuplesTuples);
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], [ [ 1, 3 ], [ 4, 2 ] ], 
  [ [ 3, 1 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], [ [ 2, 1 ], [ 4, 3 ] ], 
  [ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ], [ [ 4, 2 ], [ 1, 3 ] ], 
  [ [ 3, 2 ], [ 4, 1 ] ], [ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ]

Let vec be a normed row vector, that is, its first nonzero entry is normed to the identity of the relevant field, OnLines returns the row vector obtained from normalizing OnRight( vec, g ) by scalar multiplication from the left. This action corresponds to the projective action of a matrix group on 1-dimensional subspaces.

gap> gl:=GL(2,5);;v:=[1,0]*Z(5)^0;  
[ Z(5)^0, 0*Z(5) ]
gap> h:=Action(gl,Orbit(gl,v,OnLines),OnLines);
Group([ (2,3,5,6), (1,2,4)(3,6,5) ])

A permutation perm acts on the multivariate polynomial poly by permuting the indeterminates as it permutes points.

The operation Permuted can be used to implement a permutation action on a domain:

gap> g:=Group((1,2,3),(1,2));;
gap> dom:=[ "a", "b", "c" ];;
gap> Orbit(g,dom,Permuted);
[ [ "a", "b", "c" ], [ "c", "a", "b" ], [ "b", "a", "c" ], [ "b", "c", "a" ], 
  [ "a", "c", "b" ], [ "c", "b", "a" ] ]

implements the operation of a matrix group on subspaces of a vector space. bas must be a list of (linearly independent) vectors which forms a basis of the subspace in Hermite normal form. mat is an element of the acting matrix group. The function returns a mutable matrix which gives the basis of the image of the subspace in Hermite normal form. (In other words: it triangulizes the product of bas with mat.)

If one needs an action for which no acting function is provided by the library it can be implemented via a GAP function that conforms to the syntax

actfun(omega,g)

For example one could define the following function that acts on pairs of polynomials via OnIntereminates:

OnIndeterminatesPairs:=function(polypair,g)
  return [OnIndeterminates(polypair[1],g),
          OnIndeterminates(polypair[2],g)];
end;

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