Dear Bill Chin, dear forum,
> I have defined a polynomial ring Q[x_1,...,x_5] over the rationals. How do > you define an action of S_5 such that gx_i=x_g(i) for all g in S_5, i.e. an > action permuting the vertices? I am interested in this and other actions on > the polynomial ring. > > I have defined > > Q:=Rationals > R:=PolynomialRing(Q,5) > G:=Group[(12),(1,2,3,4,5)] With these definitions (correcting the syntax for the group definition, of course) you can use the action `OnIndeterminates' to compute orbits or stabilizers him and, for example. gap> ind:=IndeterminatesOfPolynomialRing(R); [ x_1, x_2, x_3, x_4, x_5 ] gap> pol:=ind[2]*ind[3]+ind[4]*ind[5]; x_2*x_3+x_4*x_5 gap> s:=Stabilizer(G,pol,OnIndeterminates); Group([ (2,4)(3,5), (4,5) ]) gap> Size(s); 8 gap> Orbit(G,pol,OnIndeterminates); [ x_2*x_3+x_4*x_5, x_1*x_3+x_4*x_5, x_1*x_5+x_3*x_4, x_1*x_5+x_2*x_4, x_2*x_5+x_3*x_4, x_1*x_2+x_4*x_5, x_1*x_4+x_2*x_5, x_1*x_2+x_3*x_5, x_1*x_5+x_2*x_3, x_1*x_3+x_2*x_5, x_1*x_4+x_2*x_3, x_1*x_2+x_3*x_4, x_1*x_3+x_2*x_4, x_2*x_4+x_3*x_5, x_1*x_4+x_3*x_5 ] gap> u:=Subgroup(G,[(1,2,3),(2,3,4)]); Group([ (1,2,3), (2,3,4) ]) gap> o:=Orbit(u,ind[1]*ind[2],OnIndeterminates); [ x_1*x_2, x_2*x_3, x_1*x_3, x_3*x_4, x_1*x_4, x_2*x_4 ] gap> inv:=Sum(o); x_1*x_2+x_1*x_3+x_1*x_4+x_2*x_3+x_2*x_4+x_3*x_4 gap> s:=Stabilizer(G,inv,OnIndeterminates); Group([ (1,2), (1,2,3,4) ]) gap> Size(s); 24 gap> Index(s,u); 2 I hope this is of help. Best wishes, Alexander Hulpke -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
