text:
#############################################################################
##
#W In this file we prepare functions to loop over the 2,3 or 4 dimensional
#W crystallographic groups as matrix groups or pcp groups.
##

cryst4params:=
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  [ 7, 2, 2, 1 ], [ 7, 2, 2, 2 ], [ 7, 2, 3, 1 ], [ 7, 2, 3, 2 ],
  [ 7, 3, 1, 1 ], [ 7, 3, 1, 2 ], [ 7, 3, 1, 3 ], [ 7, 3, 2, 1 ],
  [ 7, 3, 2, 2 ], [ 7, 3, 3, 1 ], [ 7, 3, 3, 2 ], [ 7, 4, 1, 1 ],
  [ 7, 4, 1, 2 ], [ 7, 4, 2, 1 ], [ 7, 4, 2, 2 ], [ 7, 4, 3, 1 ],
  [ 7, 4, 3, 2 ], [ 7, 5, 1, 1 ], [ 7, 5, 1, 2 ], [ 7, 5, 1, 3 ],
  [ 7, 5, 1, 4 ], [ 7, 5, 2, 1 ], [ 7, 5, 2, 2 ], [ 7, 5, 2, 3 ],
  [ 7, 5, 2, 4 ], [ 7, 5, 3, 1 ], [ 7, 5, 3, 2 ] ];

cryst2params :=
[ [ 1, 1, 1, 1 ], [ 1, 2, 1, 1 ], [ 2, 1, 1, 1 ], [ 2, 1, 1, 2 ],
  [ 2, 1, 2, 1 ], [ 2, 2, 1, 1 ], [ 2, 2, 1, 2 ], [ 2, 2, 1, 3 ],
  [ 2, 2, 2, 1 ], [ 3, 1, 1, 1 ], [ 3, 2, 1, 1 ], [ 3, 2, 1, 2 ],
  [ 4, 1, 1, 1 ], [ 4, 2, 1, 1 ], [ 4, 2, 2, 1 ], [ 4, 3, 1, 1 ],
  [ 4, 4, 1, 1 ] ];

NumberSpaceGroups := function( dim )
    if dim = 4 then return 4783; fi;
    if dim = 3 then return 219; fi;
    if dim = 2 then return 17; fi;
end;

SpaceGroup := function( dim, nr )
    local type;
    if dim = 4 then
        type := cryst4params[nr];
        return SpaceGroupBBNWZ( 4, type[1], type[2], type[3], type[4] );
    elif dim = 3 then
        type := cryst3params[nr];
        return SpaceGroupBBNWZ( 3, type[1], type[2], type[3], type[4] );
    elif dim = 2 then
        type := cryst2params[nr];
        return SpaceGroupBBNWZ( 2, type[1], type[2], type[3], type[4] );
    else
        return fail;
    fi;
end;

PointGroupByNumber := function( dim, nr )
    local type;
    if dim = 4 then
        type := cryst4params[nr];
        return MatGroupZClass( 4, type[1], type[2], type[3] );
    elif dim = 3 then
        type := cryst3params[nr];
        return MatGroupZClass( 3, type[1], type[2], type[3] );
    elif dim = 2 then
        type := cryst2params[nr];
        return MatGroupZClass( 2, type[1], type[2], type[3] );
    fi;
end;

SpaceGroupPcpGroup := function( dim, nr )
    local G, iso, H, F, n;
    G := SpaceGroup( dim, nr );
    iso := IsomorphismPcpGroup( G );
    if IsBool( iso ) then return fail; fi;
    H := Image( iso );
    n := Length( Igs(H) );
    F := Subgroup( H, Igs(H){[n-dim+1..n]} );
    SetFittingSubgroup( H, F );
    return H;
end;

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text:
local reps,tori,labels,I,II,ords,blocks;

reps:=[[[1,1],[2,1]],
       [[2,1],[5,1]],
       [[1,1]],
       [[2,1]]];

I:=[[],[1],[],[]];

#
# The parabolic where the representative is distinguished
#
II:=[[1,2],
     [1,2],
     [1],
     [2]];

labels:=["G_2","G_2(a_1)","\\tilde A_1","A_1"];

tori:=[
    [2,2],
    [0,2],
    [2,-3],
    [-1,2]
    ];

blocks:=[[[8, 1], [6, 1]], [[4, 2], [3, 2]], [[2, 6], [1, 2]], [[2, 6], [1, 2]]];

#ords:=[];
ords:=[8,4,2,2];

return [reps,tori,labels,I,II,ords,blocks];

ext: gi


text:
#
# usage:
# Read("~/Workspace/Chevalley.gap/init.gi"); Read(Filename(home_dir,"load.gi")); Read(Filename(f4_dir_char3,"_init_char3.gi")); Read(Filename(f4_dir_char3,"F4(a3).gi"));
#
# Read(Filename(f4_dir_char3,"F4(a3).gi"));
#

label:="F_4(a_3)";
orb_nr:=Position(labels,label);
#orb:=AllClasses(orbs)[6];
orb:=AllClasses(orbs)[orb_nr];
info:=infos[orb_nr];

#
# ------------------------------------------------------------------------
#

Print("Consider the class ",Label(orb)," in characteristic ",Characteristic(ring(sys)),":\n");
Print("\tBorel representative \n\t",coefficients(BorelRep(orb)),"\n");
Print("\tconnected C_U(u) \n\t",coefficients(info[1]),"\n");
Print("\tconnected double C_U(u) in Levi \n\t",coefficients(info[2]),"\n");
Print("\tconnected double C_U(u) \n\t",coefficients(info[3]),"\n");

Z0:=info[2];
# move Z0 to match the representative
Z0:=FromPositiveBorel(orb,Z0);
# adjust ordering of roots
Z0:=Unipotent(chevalleyAdj(Z0),coefficients(Z0),[1..Length(positiveRoots(sys))]);
Print("Z0 in position of Representative:\n\tZ0=",coefficients(Z0),"\n");

Print("Component group is S_4.\n");
Print("On Z0:\n");
n1:=[3];
a1Z0:=ApplyRootsReflections(Z0,n1);
u1:=Unipotent(chevalleyAdj(Z0),[[2,-1],[6,1]]);
a1Z0:=Conj(a1Z0,u1);
#a1Z0:=ConjugateByTori(a1Z0,[2,4],[-1,-1]);
# need to convert to gap numbering, see dataF4 file
a1Z0:=ConjugateByTori(a1Z0,[1,4],[-1,-1]);
Print("\ta^Z0=",coefficients(a1Z0),"\n");
Print("\th_3 has weights ",TorusWeights([3],a1Z0),"\n");

Print("\ta^Z0=",coefficients(a1Z0),"\n");

#
# ------------------------------------------------------------------------
#

tmp:=handle6char3();
Print("\nComponent group check:\n");
Print("\ta^u= \n",coefficients(tmp[1]),"\n");
Print("\tconnected center= \n",coefficients(tmp[2]),"\n");


text:
# Copyright (c) 2018-2021, Carnegie Mellon University
# See LICENSE for details

#F PolyDTT( <dtt-nonterminal> )
#F Parameters:       <nonterminal for a (optionally skew) DCT or DST of type 1--8>
#F Definition:       Let <t> be an (otionally skew) DCT or DST of type 1--8. Then
#F                   Transform( "PolyDTT", <t> ) represents the (n x n)-matrix
#F                   obtained from dividing every row in <t> by its first
#F                   entry.
#F                   This transform is called the polynomial version of
#F                   the DCT or DST.
#F Note:             The transpose of PolyDTT(<t>) is
#F                   TPolyDTT(<t>^T).
#F Example:          PolyDTT(DCT2(8)).
#F
Class(PolyDTT, NonTerminal, rec(
    _short_print := true,
    abbrevs := [ nt -> Checked(IsNonTerminal(nt),
	    ObjId(nt) in
	    [ DCT1, DCT3, DCT5, DCT7, # PolyDTT(T)=T for these
	      DCT2, DCT4, DCT6, DCT8,
	      DST1, DST2, DST3, DST4,
	      DST5, DST6, DST7, DST8,
	      SkewDTT],
	    [ Copy(nt) ]
	)],
    dims := self >> self.params[1].dimensions,
    terminate := self >> Mat(PolynomialDTT(MatSPL(TerminateSPL(self.params[1])))),
    isReal := True
));

Class(fr, Exp, rec(
    ev := self >> let(
	m := self.args[1].ev(),
	i := self.args[2].ev(),
	r := self.args[3].ev(),
	Cond(i mod 2 = 0, (r + 2 * Int(i/2)) / m,
	     i mod 2 = 1, (2 - r + 2 * Int(i/2)) / m))));

#F Rules
#F -----
RulesFor(PolyDTT, rec(
    PolyDTT_ToNormal := rec(
	isApplicable := P -> ObjId(P[1]) in [DCT1, DCT3, DCT5, DCT7],
	allChildren := P -> [P[1]],
	rule := P -> P[1]
    ),

    #F PolyDTT_Base2: (base case for non-skew DTTs of size 2)
    #F
    #F   pDCT2_2 = F_2
    #F   pDCT4_2 = F_2 * [[1, -1], [0, sqrt(2)]]
    #F   pDCT6_2 = [ [ 1, 1 ], [ 1, -2 ] ]
    #F   pDCT8_2 = [ [ 1, cos(2*pi/5) ], [ 1, cos(4*pi/5) ] ]
    #F   pDST4_2 = F_2 * [[1, 1], [0, sqrt(2)]]
    #F
    #F   Pueschel/Moura: Discrete Cosine and Sine Transforms, in preparation
    #F
    PolyDTT_Base2 := rec (
	info           := "pDTT_2 -> F_2",
	isApplicable   := P -> Rows(P[1]) = 2 and ObjId(P[1]) <> SkewDTT,
	allChildren    := P -> [[ ]],
	rule := (P, C) -> let(nt := ObjId(P[1]), Cond(
		nt = DCT2, F(2),
		nt = DCT4, F(2) * Mat([[1, -1], [0,sqrt(2)]]),
		nt = DCT6, Mat([[1,1], [1,-2]]),
		nt = DCT8, Mat([[1,2*CosPi(2/5)], [1,2*CosPi(4/5)]]),
		nt = DST2, F(2),
		nt = DST3, F(2) * Diag([1, sqrt(2)]),
		nt = DST4, F(2) * Mat([[1, 1], [0, sqrt(2)]]),
		nt = DST5, Mat([[1,2*CosPi(2/5)], [1,2*CosPi(4/5)]]),
		nt = DST6, Mat([[1,2*CosPi(1/5)], [1,2*CosPi(3/5)]]),
		nt = DST7, Mat([[1,2*CosPi(1/5)], [1,2*CosPi(3/5)]]),
		nt = DST8, Mat([[1,2], [1,-1]]),
		Error("unrecognized <L.symbol>"))
	)
    ),

   PolyDTT_SkewBase2 := rec (
	info           := "pDTT_2 -> F_2",
	isApplicable   := P -> Rows(P[1]) = 2 and ObjId(P[1]) = SkewDTT and
	                       ObjId(P[1].params[1]) = DST3,
	allChildren    := P -> [[ ]],
	rule := (P, C) -> let(skewnt := ObjId(P[1].params[1]), r := P[1].params[2],
	    Cond(
		skewnt = DST3, F(2) * Diag(1, 2*CosPi(r/2)),
		Error("unrecognized <L.symbol>"))
	)
   ),

   PolyDTT_SkewDST3_CT := rec(
       isApplicable := P -> ObjId(P[1]) = SkewDTT and
                            ObjId(P[1].params[1]) = DST3 and
                            Rows(P[1]) > 2 and not IsPrime(Rows(P[1])),

       allChildren := P -> let(
	   N := Rows(P[1].params[1]), r := P[1].params[2],
	   List(DivisorPairs(N), d->
	       let(i := Ind(d[2]),
		   ri := fr(d[2], i, r),
		   [ PolyDTT(SkewDTT(DST3(d[1]), ri)),
		     PolyDTT(SkewDTT(DST3(d[2]), r)) ]))),

       rule := (P,C) -> let(
	   MN := Rows(P[1]),  N := Rows(C[1]),  M := Rows(C[2]),
	   r := P[1].params[2],
	   i := C[1].root.params.params[2].args[2],

	    K(MN, N) *
	    IterDirectSum(i, i.range, C[1]) *
	    Tensor(C[2], I(N)) *
	    B_DST3_U(MN, M)
       )
   )
));