/* Sessionlog of second demo session of Magma, Wed June 27, 2007 */
iload "demo.txt";
2+2;
load "~/tue/research/paloalto/magma_demo/demo_defs.txt";
sheet_head();
sheet_contents();
prm("Magma in general");
/* Syntax */
2+2;
Random(GF(25));
Random(GF(25))*Random(GF(25));
P<x> := PolynomialRing(Rationals());
d := Xgcd(x^5 + x^3 + 3*x^2 + 3, x^10 + 3*x^7 + x^4 + 
2*x^3 + 3*x + 6); d;
d,a,b := Xgcd(x^5 + x^3 + 3*x^2 + 3, x^10 + 3*x^7 + x^4 + 
2*x^3 + 3*x + 6); 
a*(x^5+x^3+3*x^2+3) + b*(x^10+3*x^7+x^4+2*x^3+3*x+6);
//
l := [ i : i in [1..100] | IsPrime(i) ];
#l;
#[ i : i in l | i mod 4 eq 1 ];
//
sub<G | G!(1,2,3)> where G is SymmetricGroup(4);
sub<G | (1,2,3)> where G is SymmetricGroup(4);
H, phi := sub<G | (1,2,3)> where G is SymmetricGroup(4);
H;
phi;
phi(Identity(H));
#H;
#$1;
/* Error handling */
try
        for i in [3..0 by -1] do
                print 1/i;
        end for;
catch e
        print "Whoops: ", e`Object;
end try;
try for i in [3..0 by -1] do print 1/i; end for; 
catch e print "whoops"; print i; end try;
/* The profiler */
fib := func<n | (n in {1,2} ) select 1 else 
$$(n-1)+$$(n-2) >;
fib(8);
SetProfile(true);
time _ := fib(20);  
G := ProfileGraph();
SetProfile(false);
ProfilePrintByTotalTime(G);
fib := func<n | (n eq 1 or n eq 2) select 1 else 
$$(n-1)+$$(n-2) >;
fib(8); SetProfile(true); time _ := fib(20);   G := 
ProfileGraph(); SetProfile(false);
ProfilePrintByTotalTime(G);
prm("Support for Lie groups");
f := function(a,b)
 return a+b;
end function;
/* Root data */
R := RootDatum("A4" : Isogeny := "Ad");
G := GroupOfLieType(R, GF(59));
Order(G);
FactoredOrder(G);
Dimension(G);
CenterPolynomials(G);
//
DynkinDiagram(G);
CartanMatrix(G);
DynkinDiagram("D4");
DynkinDiagram("G2");
DynkinDiagram("~G2");
DynkinDiagram("~G2");
//
W := WeylGroup(G); W;
//
g := GraphAutomorphism(G, Sym(4)!(1,4)(2,3));
g;
[ g(elt<G | <i,-1> >) : i in [1..4] ];
//
x := Random(G); x;
[ IsSemisimple(x), IsUnipotent(x), IsCentral(x) ];
//By default, the Steinberg presentation is used:
// G = UTWU
x := elt<G | [*
    [ <4, 23>, <7, 31>, <10, 29> ],
    Vector([58, 43, 9, 15]),
        1, 2, 1, 3, 2, 1, 4,
    [ <9, 44>, <11, 19>, <12, 24>, <3, 23> ]
*]>; x;
//But it is easy to go to other presentations:
sr := StandardRepresentation(G);
X := sr(x); X;
Determinant(X);
//And, due to an algorithm by Cohen, Murray, and
//  Taylor, we can go back as well:
X @@ sr;
X @@ sr eq x;
//Galois Cohomology:
// (Due to Haller)
// Compute the cohomology of A_3(5^2)
q := 5; k := GF(q); K := GF(q^2);
G := GroupOfLieType( "A3", K : Isogeny:="SC" );
A := AutomorphismGroup(G);
AGRP := GammaGroup( k, A );
Gamma,m := ActingGroup(AGRP); Gamma;
m;
time GaloisCohomology(AGRP);
// And now the cocycle defining the group 2A_3(5) 
// and check for two elements if they are contained
// in 2A_3(5):
c := OneCocycle( AGRP, [GraphAutomorphism(G, 
Sym(3)!(1,3))] );
x := Random(G);
IsInTwistedForm( x, c );
x := elt< G | <1,y>, <3,y @ m(Gamma.1)> > where y is 
Random(K);
IsInTwistedForm( x, c );
//The twisted group 2 A_3(5) as a subgroup of A_3(5^2).
R := RootDatum("A3" : Twist := 2);
G := TwistedGroupOfLieType(R,5,25);
G;
BaseRing(G);
DefRing(G);
iload "demo.txt";
load "~/tue/research/paloalto/magma_demo/demo_defs.txt";
sheet_head();
sheet_contents();
prm("Magma in general");
/* Syntax */
2+2;
P<x> := PolynomialRing(Rationals());
P<x> := PolynomialRing(Rationals());
d := Xgcd(x^5 + x^3 + 3*x^2 + 3, x^10 + 3*x^7 + x^4 + 
2*x^3 + 3*x + 6); d;
d,a,b := Xgcd(x^5 + x^3 + 3*x^2 + 3, x^10 + 3*x^7 + x^4 + 
2*x^3 + 3*x + 6); 
a*(x^5+x^3+3*x^2+3) + b*(x^10+3*x^7+x^4+2*x^3+3*x+6);
//
l := [ i : i in [1..100] | IsPrime(i) ];
l;
#l;
#[ i : i in l | i mod 4 eq 1 ];
//
sub<G | G!(1,2,3)> where G is SymmetricGroup(4);
G := SymmetricGroup(4);
H,phi := sub<G | (1,2,3)>;
phi;
#H;
Random(H);
Random(H);
Random(H);
phi(Random(H));
phi(H!(1,3,2));
GmodH, psi := G/H;
#$1;
/* Error handling */
try
        for i in [3..0 by -1] do
                print 1/i;
        end for;
catch e
        print "Whoops: ", e`Object;
end try;
try for i in [3..0 by -1] do print 1/i; end for; catch e; 
end try;
for i in [3..0 by -1] do print 1/i; end for;
for i in [300..0 by -1] do print 1/i; end for;
/* The profiler */
fib := func<n | (n in {1,2} ) select 1 else 
$$(n-1)+$$(n-2) >;
fib := function(n)
 if n in {1,2} then return 1;
else return $$(n-1) + $$(n-2);
end if;
end function;
fib(8);
SetProfile(true);
time _ := fib(20);  
G := ProfileGraph();
SetProfile(false);
ProfilePrintByTotalTime(G);
fib := func<n | (n eq 1 or n eq 2) select 1 else 
$$(n-1)+$$(n-2) >;
fib(8); SetProfile(true); time _ := fib(20);   G := 
ProfileGraph(); SetProfile(false);
ProfilePrintByTotalTime(G);
prm("Support for Lie groups");
/* Root data */
R := RootDatum("A4" : Isogeny := "Ad");
G := GroupOfLieType("A4", GF(59) : Isogeny := "Ad");
G := GroupOfLieType(R, GF(59));
Order(G);
FactoredOrder(G);
Dimension(G);
CenterPolynomials(G);
//
DynkinDiagram(G);
DynkinDiagram("G2");
DynkinDiagram("D4");
CartanMatrix(G);
GeneralLinearGroup(GF(59),4);
GeneralLinearGroup(4,GF(59));
GL := GeneralLinearGroup(4,GF(59)); CenterPolynomials(GL);
//
W := WeylGroup(G); W;
//
g := GraphAutomorphism(G, Sym(4)!(1,4)(2,3));
g;
[ g(elt<G | <i,-1> >) : i in [1..4] ];
//
x := Random(G); x;
[ IsSemisimple(x), IsUnipotent(x), IsCentral(x) ];
//By default, the Steinberg presentation is used:
// G = UTWU
x := elt<G | [*
    [ <4, 23>, <7, 31>, <10, 29> ],
    Vector([58, 43, 9, 15]),
        1, 2, 1, 3, 2, 1, 4,
    [ <9, 44>, <11, 19>, <12, 24>, <3, 23> ]
*]>; x;
//But it is easy to go to other presentations:
sr := StandardRepresentation(G);
X := sr(x); X;
Determinant(X);
//And, due to an algorithm by Cohen, Murray, and
//  Taylor, we can go back as well:
X @@ sr;
X @@ sr eq x;
x @ StandardRepresentation(G);
(x @ StandardRepresentation(G)) @@ 
StandardRepresentation(G);
StandardRepresentation(G)(x);
rrh := RowReductionHomomorphism(StandardRepresentation(G)\
);
rrh( X );
//Galois Cohomology:
// (Due to Haller)
// Compute the cohomology of A_3(5^2)
q := 5; k := GF(q); K := GF(q^2);
G := GroupOfLieType( "A3", K : Isogeny:="SC" );
A := AutomorphismGroup(G);
AGRP := GammaGroup( k, A );
Gamma,m := ActingGroup(AGRP); Gamma;
m;
time GaloisCohomology(AGRP);
// And now the cocycle defining the group 2A_3(5) 
// and check for two elements if they are contained
// in 2A_3(5):
c := OneCocycle( AGRP, [GraphAutomorphism(G, 
Sym(3)!(1,3))] );
x := Random(G);
IsInTwistedForm( x, c );
x := elt< G | <1,y>, <3,y @ m(Gamma.1)> > where y is 
Random(K);
IsInTwistedForm( x, c );
//The twisted group 2 A_3(5) as a subgroup of A_3(5^2).
R := RootDatum("A3" : Twist := 2);
G := TwistedGroupOfLieType(R,5,25);
G;
BaseRing(G);
DefRing(G);
UntwistedOvergroup(G);
prm("LiE");
/* "Combinatorics with representations" */
A2 := RootDatum("A2" : Isogeny := "SC");
L := LieAlgebra(A2, Rationals());
_,_,T := StandardBasis(L);
rho := AdjointRepresentation(L);
V := VectorSpace(Rationals(), 8);
//Now V is an L-module by v --> x.v = rho(x).v
MT := [ Transpose(Matrix(rho(t))) : t in T ]; MT;
//Weights are: (-1,1), (1,-2), ... ,(1,1)
//Dominant weights: (0,0),(0,0),(1,1)
rho1 := HighestWeightRepresentation(L, [1,1]);
rho2 := HighestWeightRepresentation(L, [2,0]);
MT := [ Transpose(TensorProduct(Matrix(rho1(t)),
        Matrix(rho2(t)))) : t in T ] ;
{* Vector([(B[i]*m)[i] : m in MT]):i in [1..#B] *} 
        where B is Basis(VectorSpace(Rationals(),48));
//But these dimensions quickly grow!
//We can find out a lot about the representations
//  by just considering the highest weights
//LiE, by Van Leeuwen, Cohen, Lisser
//  Ported to Magma in summer 2006
drho1 := LieRepresentationDecomposition(A2, [1,1]);
drho2 := LieRepresentationDecomposition(A2, [2,0]);
dtp := Tensor(drho1, drho2);
dtp:Maximal;
RepresentationDimension(dtp);
[ RepresentationDimension(LieRepresentationDecomposition(\
A2, w)) 
        : w in dtp`WtsMps[1] ];
&+$1;
prm("THE END");
//
//
//
//
//
//
//
//
//Branch
//Branch
A1T1 := RootDatum("A1T1" : Isogeny := "SC");
M := RestrictionMatrix(A2, A1T1); M;
M := RestrictionMatrix(A2, A1T1);
D := Branch(A1T1, dtp, M);
RepresentationDimension(D);
D:Maximal;
E8 := RootDatum("E8" : Isogeny := "SC");
iload "";
E8 := RootDatum("E8" : Isogeny := "SC");
D1 := LieRepresentationDecomposition(E8, 
[1,0,0,0,0,0,0,0]);
D2 := LieRepresentationDecomposition(E8, 
[0,1,0,0,0,0,0,0]);
time D3 := Tensor(D1, D2);
D3;
D3:Maximal;
http://www-math.univ-poitiers.fr/~maavl/LiE/
;;
W := CoxeterGroup("A2");
KLPolynomial(W!(1,2),W!(1,2));
x := Random(W); y := Random(W);
W;
W!(1,2);
W!(1,4)(2,3)(5,6);
KLPolynomial(x,y);
W := CoxeterGroup("A4");
x := Random(W); y := Random(W);
KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);
x := Random(W); y := Random(W); KLPolynomial(x,y);