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InstallMethod( \=,
"for two nilpotent orbits",
true, [ IsNilpotentOrbit, IsNilpotentOrbit ], 0,
function( a, b )
return AmbientLieAlgebra(a) = AmbientLieAlgebra(b) and
WeightedDynkinDiagram(a) = WeightedDynkinDiagram(b);
end );
InstallMethod( NilpotentOrbit,
"for a Lie algebra and list",
true, [ IsLieAlgebra, IsList ], 0,
function( L, list )
local o, fam;
if not IsBound( L!.nilorbType ) then
fam:= NewFamily( "nilorbsfam", IsNilpotentOrbit );
fam!.invCM:= CartanMatrix( RootSystem(L) )^-1;
L!.nilorbType:= NewType( fam,
IsNilpotentOrbit and IsAttributeStoringRep );
fi;
o:= Objectify( L!.nilorbType, rec() );
SetAmbientLieAlgebra( o, L );
SetWeightedDynkinDiagram( o, list );
if HasSemiSimpleType(L) then
SetSemiSimpleType( o, SemiSimpleType(L) );
fi;
return o;
end );
InstallMethod( SL2Triple,
"for a Lie algebra and a nilpotent element",
true, [ IsLieAlgebra, IsObject ], 0,
function( L, x )
# Same as FindSl2 in the library; only returns the elements, rather
# than the subalgebra. (Advantage: GAP doesn't change the basis.)
local n, # the dimension of `L'
F, # the field of `L'
B, # basis of `L'
T, # the table of structure constants of `L'
xc, # coefficient vector
eqs, # a system of equations
i,j,k,l, # loop variables
cij, # the element `T[i][j]'
b, # the right hand side of the equation system
v, # solution of the equations
z, # element of `L'
h, # element of `L'
R, # centralizer of `x' in `L'
BR, # basis of `R'
Rvecs, # basis vectors of `R'
H, # the matrix of `ad H' restricted to `R'
e0, # coefficient vector
e1, # coefficient vector
y; # element of `L'
if not IsNilpotentElement( L, x ) then
Error( "<x> must be a nilpotent element of the Lie algebra <L>" );
fi;
n:= Dimension( L );
F:= LeftActingDomain( L );
B:= Basis( L );
T:= StructureConstantsTable( B );
xc:= Coefficients( B, x );
eqs:= NullMat( 2*n, 2*n, F );
# First we try to find elements `z' and `h' such that `[x,z]=h'
# and `[h,x]=2x' (i.e., such that two of the three defining equations
# of sl_2 are satisfied).
# This results in a system of `2n' equations for `2n' variables.
for i in [1..n] do
for j in [1..n] do
cij:= T[i][j];
for k in [1..Length(cij[1])] do
l:= cij[1][k];
eqs[i][l] := eqs[i][l] + xc[j]*cij[2][k];
eqs[n+i][n+l]:= eqs[n+i][n+l] + xc[j]*cij[2][k];
od;
od;
eqs[n+i][i]:= One( F );
od;
b:= [];
for i in [1..n] do
b[i]:= Zero( F );
b[n+i]:= 2*One( F )*xc[i];
od;
v:= SolutionMat( eqs, b );
if v = fail then
# There is no sl_2 containing <x>.
return fail;
fi;
z:= LinearCombination( B, v{ [ 1 .. n ] } );
h:= LinearCombination( B, v{ [ n+1 .. 2*n ] } );
R:= LieCentralizer( L, SubalgebraNC( L, [ x ] ) );
BR:= Basis( R );
Rvecs:= BasisVectors( BR );
# `ad h' maps `R' into `R'. `H' will be the matrix of that map.
H:= List( Rvecs, v -> Coefficients( BR, h * v ) );
# By the proof of the lemma of Jacobson-Morozov (see Jacobson,
# Lie Algebras, p. 98) there is an element `e1' in `R' such that
# `(H+2)e1=e0' where `e0=[h,z]+2z'.
# If we set `y=z-e1' then `x,h,y' will span a subalgebra of `L'
# isomorphic to sl_2.
H:= H+2*IdentityMat( Dimension( R ), F );
e0:= Coefficients( BR, h * z + 2*z );
e1:= SolutionMat( H, e0 );
if e1 = fail then
# There is no sl_2 containing <x>.
return fail;
fi;
y:= z-LinearCombination(Rvecs,e1);
return [y,h,x];
end );
InstallMethod( PrintObj,
"for nilpotent orbit",
true,
[ IsNilpotentOrbit ], 0,
function( o )
Print("<nilpotent orbit in Lie algebra");
if HasSemiSimpleType(o) then
Print( " of type ", SemiSimpleType(o) );
fi;
Print(">");
end );
InstallMethod( RandomSL2Triple,
"for a nilpotent orbit",
true, [ IsNilpotentOrbit], 0,
function( o )
local Ci, L, c, H, v, h, adh, sp, i, j, k, e, f, found, found_good, co,
x, eqns, sol, allcft1;
Ci:= FamilyObj( o )!.invCM;
L:= AmbientLieAlgebra( o );
c:= ChevalleyBasis(L);
H:= c[3];
v:= Ci*WeightedDynkinDiagram( o );
h:= Sum( List( [1..Length(H)], x -> v[x]*H[x] ) );
adh:= AdjointMatrix( Basis(L), h );
sp:= NullspaceMat( TransposedMat( adh-2*adh^0 ) );
e:= List( sp, x -> Sum( List( [1..Dimension(L)], ii -> x[ii]*Basis(L)[ii] ) ) );
sp:= NullspaceMat( TransposedMat( adh+2*adh^0 ) );
f:= List( sp, x -> Sum( List( [1..Dimension(L)], ii -> x[ii]*Basis(L)[ii] ) ) );
allcft1:= false;
while not allcft1 do
found:= false;
j:= 1;
while not found do
co:= List( [1..Length(e)], x-> Random( Rationals ) );
x:= Sum( List( [1..Length(e)], ii -> co[ii]*e[ii] ) );
sp:= Subspace( L, List( f, y -> x*y ) );
if Dimension(sp) = Length(e) and h in sp then
found:= true;
else
j:= j+1;
if j > 100 then
Error("Tried hard but found no representative of the nilpotent orbit. Are you sure the weighted Dynkin diagram really corresponds to a nilpotent orbit?");
fi;
fi;
od;
for k in [Length(co),Length(co)-1..1] do
i:= -1; found_good:= false;
while not found_good do
i:= i+1;
co[k]:= i;
x:= Sum( List( [1..Length(e)], ii -> co[ii]*e[ii] ) );
sp:= Subspace( L, List( f, y -> x*y ) );
if Dimension(sp) = Length(e) and h in sp then
found_good:= true;
fi;
od;
od;
allcft1:= ForAll( co, x -> (x=0) or (x=1) );
od;
eqns:= List( f, u -> Coefficients( Basis(sp), x*u ) );
sol:= SolutionMat( eqns, Coefficients( Basis(sp), h ) );
return [Sum([1..Length(f)], ii -> sol[ii]*f[ii] ),h,x];
end );
InstallMethod( SL2Triple,
"for a nilpotent orbit",
true, [ IsNilpotentOrbit], 0,
function( o )
return RandomSL2Triple( o );
end );
SLAfcts.weighted_dyn_diags_Dn:= function( n )
local part, good, p, q, k, new, cur, m, diags, is_very_even, d,
d0, h, a, b, diag;
part:= Partitions( 2*n );
new:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
good:= true;
for k in [1..Length(q)] do
if IsEvenInt(q[k][1]) and not IsEvenInt(q[k][2]) then
good:= false;
break;
fi;
od;
if good then
Add( new, p );
fi;
od;
part:= new;
diags:= [ ];
new:= [ ];
for p in part do
is_very_even:= ForAll( p, x -> IsEvenInt(x) );
d:= [ ];
for k in p do
d0:= List( [1..k], i -> k -2*(i-1)-1 );
Append( d, d0 );
od;
Sort(d);
d:= Reversed( d );
h:= d{[1..n]};
if not is_very_even then
diag:= List( [1..n-1], i -> h[i]-h[i+1] );
Add( diag, h[n-1]+h[n] );
Add( diags, diag );
Add( new, p );
else
if IsInt(n/4) then a:= 0; else a:= 2; fi;
b:= 2-a;
diag:= List( [1..n-2], i -> h[i]-h[i+1] );
Add( diag, a ); Add( diag, b );
Add( diags, diag );
Add( new, p );
diag:= List( [1..n-2], i -> h[i]-h[i+1] );
Add( diag, b ); Add( diag, a );
Add( diags, diag );
Add( new, p );
fi;
od;
return [diags,new];
end;
SLAfcts.weighted_dyn_diags_An:= function( n )
local part, diags, p, d, k, d0, h;
part:= Partitions( n+1 );
diags:= [ ];
for p in part do
d:= [ ];
for k in p do
d0:= List( [1..k], i -> k -2*(i-1)-1 );
Append( d, d0 );
od;
Sort(d);
h:= Reversed( d );
Add( diags, List( [1..n], i -> h[i]-h[i+1] ) );
od;
return [diags,part];
end;
SLAfcts.weighted_dyn_diags_Bn:= function( n )
local part, good, p, q, k, new, cur, m, diags, is_very_even, d,
d0, h, a, b, diag;
part:= Partitions( 2*n+1 );
new:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
good:= true;
for k in [1..Length(q)] do
if IsEvenInt(q[k][1]) and not IsEvenInt(q[k][2]) then
good:= false;
break;
fi;
od;
if good then
Add( new, p );
fi;
od;
part:= new;
diags:= [ ];
for p in part do
d:= [ ];
for k in p do
d0:= List( [1..k], i -> k -2*(i-1)-1 );
Append( d, d0 );
od;
# Get rid of a zero;
for k in [1..Length(d)] do
if d[k] = 0 then
RemoveElmList( d, k );
break;
fi;
od;
Sort(d);
d:= Reversed( d );
h:= d{[1..n]};
diag:= List( [1..n-1], i -> h[i]-h[i+1] );
Add( diag, h[n] );
Add( diags, diag );
od;
return [diags,part];
end;
SLAfcts.weighted_dyn_diags_Cn:= function( n )
local part, good, p, q, k, new, cur, m, diags, is_very_even, d,
d0, h, a, b, diag;
part:= Partitions( 2*n );
new:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
good:= true;
for k in [1..Length(q)] do
if IsOddInt(q[k][1]) and not IsEvenInt(q[k][2]) then
good:= false;
break;
fi;
od;
if good then
Add( new, p );
fi;
od;
part:= new;
diags:= [ ];
for p in part do
d:= [ ];
for k in p do
d0:= List( [1..k], i -> k -2*(i-1)-1 );
Append( d, d0 );
od;
Sort(d);
d:= Reversed( d );
h:= d{[1..n]};
diag:= List( [1..n-1], i -> h[i]-h[i+1] );
Add( diag, 2*h[n] );
Add( diags, diag );
od;
return [diags,part];
end;
SLAfcts.nilporbs:= function( L )
# This for simple types only (at least for the moment).
# REMARK: nonzero nilpotent orbits!
# This is only for the standard simple Lie alg!
local type, list, elms, orbs, i, j, o, u, a, sl2;
type:= SemiSimpleType( L );
if type = "G2" then
list:= [ [1,0], [0,1], [0,2], [2,2] ];
elms:=
[ [ [ 10, 1 ], [ 13, 2, 14, 3 ], [ 4, 1 ] ],
[ [ 12, 1 ], [ 13, 1, 14, 2 ], [ 6, 1 ] ],
[ [ 8, 1, 10, 1 ], [ 13, 2, 14, 4 ], [ 2, 1, 4, 1 ] ],
[ [ 7, 6, 8, 10 ], [ 13, 6, 14, 10 ], [ 1, 1, 2, 1 ] ] ];
elif type = "F4" then
list:= [ [1,0,0,0], [0,0,0,1], [0,1,0,0], [2,0,0,0], [0,0,0,2],
[0,0,1,0], [2,0,0,1], [0,1,0,1], [1,0,1,0], [0,2,0,0], [2,2,0,0],
[1,0,1,2], [0,2,0,2], [2,2,0,2], [2,2,2,2] ];
# need to permute, to account for slightly strange numbering of the Dynkin
# diagram in GAP:
list:= List( list, x -> Permuted( x, (1,2,4) ) );
elms:=
[ [ [ 48, 1 ], [ 49, 1, 50, 2, 51, 2, 52, 3 ], [ 24, 1 ] ],
[ [ 45, 1 ], [ 49, 2, 50, 2, 51, 3, 52, 4 ], [ 21, 1 ] ],
[ [ 41, 1, 46, 1 ], [ 49, 2, 50, 3, 51, 4, 52, 6 ], [ 17, 1, 22, 1 ] ],
[ [ 40, 2, 42, 2 ], [ 49, 2, 50, 4, 51, 4, 52, 6 ], [ 16, 1, 18, 1 ] ],
[ [ 35, 2, 36, 2 ], [ 49, 4, 50, 4, 51, 6, 52, 8 ], [ 11, 1, 12, 1 ] ],
[ [ 38, 1, 39, 2, 40, 2 ], [ 49, 3, 50, 4, 51, 6, 52, 8 ],
[ 14, 1, 15, 1, 16, 1 ] ],
[ [ 33, 3, 39, 4 ], [ 49, 4, 50, 6, 51, 7, 52, 10 ], [ 9, 1, 15, 1 ] ],
[ [ 32, 2, 38, 2, 40, 1 ], [ 49, 4, 50, 5, 51, 7, 52, 10 ],
[ 8, 1, 14, 1, 16, 1 ] ],
[ [ 34, 4, 35, 3, 39, 1 ], [ 49, 4, 50, 6, 51, 8, 52, 11 ],
[ 10, 1, 11, 1, 15, 1 ] ],
[ [ 32, 2, 33, 2, 34, 2, 42, 2 ], [ 49, 4, 50, 6, 51, 8, 52, 12 ],
[ 8, 1, 9, 1, 10, 1, 18, 1 ] ],
[ [ 26, 10, 32, 6, 34, 6 ], [ 49, 6, 50, 10, 51, 12, 52, 18 ],
[ 2, 1, 8, 1, 10, 1 ] ],
[ [ 25, 8, 33, 5, 34, 9 ], [ 49, 8, 50, 10, 51, 14, 52, 19 ],
[ 1, 1, 9, 1, 10, 1 ] ],
[ [ 29, 8, 30, 9, 31, 5, 37, 1 ], [ 49, 8, 50, 10, 51, 14, 52, 20 ],
[ 5, 1, 6, 1, 7, 1, 13, 1 ] ],
[ [ 26, 14, 28, 18, 29, 10, 34, 8 ], [ 49, 10, 50, 14, 51, 18, 52, 26 ],
[ 2, 1, 4, 1, 5, 1, 10, 1 ] ],
[ [ 25, 16, 26, 22, 27, 30, 28, 42 ], [ 49, 16, 50, 22, 51, 30, 52, 42 ],
[ 1, 1, 2, 1, 3, 1, 4, 1 ] ] ];
elif type = "E6" then
list:= [ [0,1,0,0,0,0], [1,0,0,0,0,1], [0,0,0,1,0,0],
[0,2,0,0,0,0], [1,1,0,0,0,1], [2,0,0,0,0,2], [0,0,1,0,1,0],
[1,2,0,0,0,1], [1,0,0,1,0,1], [0,1,1,0,1,0], [0,0,0,2,0,0],
[2,2,0,0,0,2], [0,2,0,2,0,0], [1,1,1,0,1,1], [2,1,1,0,1,2],
[1,2,1,0,1,1], [2,0,0,2,0,2], [2,2,0,2,0,2], [2,2,2,0,2,2],
[2,2,2,2,2,2] ];
elms:=
[ [ [ 72, 1 ], [ 73, 1, 74, 2, 75, 2, 76, 3, 77, 2, 78, 1 ], [ 36, 1 ] ],
[ [ 66, 1, 70, 1 ], [ 73, 2, 74, 2, 75, 3, 76, 4, 77, 3, 78, 2 ],
[ 30, 1, 34, 1 ] ],
[ [ 60, 1, 68, 1, 69, 1 ], [ 73, 2, 74, 3, 75, 4, 76, 6, 77, 4, 78, 2 ],
[ 24, 1, 32, 1, 33, 1 ] ],
[ [ 61, 2, 62, 2 ], [ 73, 2, 74, 4, 75, 4, 76, 6, 77, 4, 78, 2 ],
[ 25, 1, 26, 1 ] ],
[ [ 59, 1, 61, 2, 62, 2 ], [ 73, 3, 74, 4, 75, 5, 76, 7, 77, 5, 78, 3 ],
[ 23, 1, 25, 1, 26, 1 ] ],
[ [ 53, 2, 54, 2, 56, 2, 57, 2 ], [ 73, 4, 74, 4, 75, 6, 76, 8, 77, 6, 78,
4 ], [ 17, 1, 18, 1, 20, 1, 21, 1 ] ],
[ [ 58, 1, 59, 2, 60, 2, 61, 1 ],
[ 73, 3, 74, 4, 75, 6, 76, 8, 77, 6, 78, 3 ],
[ 22, 1, 23, 1, 24, 1, 25, 1 ] ],
[ [ 49, 3, 50, 3, 59, 4 ], [ 73, 4, 74, 6, 75, 7, 76, 10, 77, 7, 78, 4 ],
[ 13, 1, 14, 1, 23, 1 ] ],
[ [ 48, 2, 56, 2, 57, 2, 58, 2, 60, 1 ], [ 73, 4, 74, 5, 75, 7, 76, 10, 77,
7, 78, 4 ], [ 12, 1, 20, 1, 21, 1, 22, 1, 24, 1 ] ],
[ [ 51, 4, 53, 3, 56, 3, 59, 1 ],
[ 73, 4, 74, 6, 75, 8, 76, 11, 77, 8, 78, 4 ],
[ 15, 1, 17, 1, 20, 1, 23, 1 ] ],
[ [ 45, 1, 52, 4, 53, 3, 54, 1, 55, 3 ],
[ 73, 4, 74, 6, 75, 8, 76, 12, 77, 8, 78, 4 ],
[ 9, 1, 16, 1, 17, 1, 18, 1, 19, 1 ] ],
[ [ 47, 6, 48, 6, 49, 4, 50, 4 ],
[ 73, 6, 74, 8, 75, 10, 76, 14, 77, 10, 78, 6 ],
[ 11, 1, 12, 1, 13, 1, 14, 1 ] ],
[ [ 38, 10, 48, 6, 51, 6, 52, 6 ], [ 73, 6, 74, 10, 75, 12, 76, 18, 77, 12,
78, 6 ], [ 2, 1, 12, 1, 15, 1, 16, 1 ] ],
[ [ 47, 6, 48, 6, 49, 4, 50, 4, 51, 1 ],
[ 73, 6, 74, 8, 75, 11, 76, 15, 77, 11, 78, 6 ],
[ 11, 1, 12, 1, 13, 1, 14, 1, 15, 1 ] ],
[ [ 37, 8, 42, 8, 49, 5, 50, 5, 51, 9 ],
[ 73, 8, 74, 10, 75, 14, 76, 19, 77, 14, 78, 8 ],
[ 1, 1, 6, 1, 13, 1, 14, 1, 15, 1 ] ],
[ [ 38, -5, 43, 5, 44, 10, 47, 7, 48, 2, 51, 6, 52, 2 ],
[ 73, 7, 74, 10, 75, 13, 76, 18, 77, 13, 78, 7 ],
[ 7, 1, 8, 1, 11, 1, 12, 1, 15, 1 ] ],
[ [ 37, 8, 44, 5, 45, 9, 46, 1, 47, 8, 55, 5 ],
[ 73, 8, 74, 10, 75, 14, 76, 20, 77, 14, 78, 8 ],
[ 1, 1, 8, 1, 9, 1, 10, 1, 11, 1, 19, 1 ] ],
[ [ 38, 14, 42, 10, 43, 10, 45, 8, 46, 18 ],
[ 73, 10, 74, 14, 75, 18, 76, 26, 77, 18, 78, 10 ],
[ 2, 1, 6, 1, 7, 1, 9, 1, 10, 1 ] ],
[ [ 37, 12, 38, 8, 39, -8, 41, 22, 42, 12, 44, 8, 45, 22, 46, 8 ],
[ 73, 12, 74, 16, 75, 22, 76, 30, 77, 22, 78, 12 ],
[ 1, 1, 2, 1, 5, 1, 6, 1, 8, 1, 9, 1 ] ],
[ [ 37, 16, 38, 22, 39, 30, 40, 42, 41, 30, 42, 16 ],
[ 73, 16, 74, 22, 75, 30, 76, 42, 77, 30, 78, 16 ],
[ 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1 ] ] ];
elif type = "E7" then
list:= [ [1,0,0,0,0,0,0], [0,0,0,0,0,1,0], [0,0,0,0,0,0,2], [0,0,1,0,0,0,0],
[2,0,0,0,0,0,0], [0,1,0,0,0,0,1], [1,0,0,0,0,1,0], [0,0,0,1,0,0,0],
[2,0,0,0,0,1,0], [0,0,0,0,0,2,0], [0,2,0,0,0,0,0], [2,0,0,0,0,0,2],
[0,0,1,0,0,1,0], [1,0,0,1,0,0,0], [0,0,2,0,0,0,0], [1,0,0,0,1,0,1],
[2,0,2,0,0,0,0], [0,1,1,0,0,0,1], [0,0,0,1,0,1,0], [2,0,0,0,0,2,0],
[0,0,0,0,2,0,0], [2,0,0,0,0,2,2], [2,1,1,0,0,0,1], [1,0,0,1,0,1,0],
[2,0,0,1,0,1,0], [0,0,0,2,0,0,0], [1,0,0,1,0,2,0], [1,0,0,1,0,1,2],
[2,0,0,0,2,0,0], [0,1,1,0,1,0,2], [0,0,2,0,0,2,0], [2,0,2,0,0,2,0],
[0,0,0,2,0,0,2], [0,0,0,2,0,2,0], [2,1,1,0,1,1,0], [2,1,1,0,1,0,2],
[2,0,0,2,0,0,2], [2,1,1,0,1,2,2], [2,0,0,2,0,2,0], [2,0,2,2,0,2,0],
[2,0,0,2,0,2,2], [2,2,2,0,2,0,2], [2,2,2,0,2,2,2], [2,2,2,2,2,2,2] ];
elms:=
[ [ [ 126, 1 ], [ 127, 2, 128, 2, 129, 3, 130, 4, 131, 3, 132, 2, 133, 1 ],
[ 63, 1 ] ],
[ [ 120, 1, 123, 1 ], [ 127, 2, 128, 3, 129, 4, 130, 6, 131, 5, 132, 4,
133, 2 ], [ 57, 1, 60, 1 ] ],
[ [ 110, 1, 111, 1, 112, 1 ], [ 127, 2, 128, 3, 129, 4, 130, 6, 131, 5,
132, 4, 133, 3 ], [ 47, 1, 48, 1, 49, 1 ] ],
[ [ 105, 1, 119, 1, 122, 1 ],
[ 127, 3, 128, 4, 129, 6, 130, 8, 131, 6, 132, 4, 133, 2 ],
[ 42, 1, 56, 1, 59, 1 ] ],
[ [ 107, 2, 109, 2 ], [ 127, 4, 128, 4, 129, 6, 130, 8, 131, 6, 132, 4,
133, 2 ], [ 44, 1, 46, 1 ] ],
[ [ 108, 1, 110, 1, 115, 1, 116, 1 ], [ 127, 3, 128, 5, 129, 6, 130, 9,
131, 7, 132, 5, 133, 3 ], [ 45, 1, 47, 1, 52, 1, 53, 1 ] ],
[ [ 107, 2, 109, 2, 112, 1 ],
[ 127, 4, 128, 5, 129, 7, 130, 10, 131, 8, 132, 6, 133, 3 ],
[ 44, 1, 46, 1, 49, 1 ] ],
[ [ 104, 2, 105, 1, 106, 2, 114, 1 ], [ 127, 4, 128, 6, 129, 8, 130, 12,
131, 9, 132, 6, 133, 3 ], [ 41, 1, 42, 1, 43, 1, 51, 1 ] ],
[ [ 83, 3, 84, 3, 112, 4 ],
[ 127, 6, 128, 7, 129, 10, 130, 14, 131, 11, 132, 8, 133, 4 ],
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[ 241, 20, 242, 30, 243, 40, 244, 60, 245, 49, 246, 38, 247, 26, 248,
14 ], [ 8, 1, 20, 1, 32, 1, 34, 1, 35, 1, 37, 1, 38, 1 ] ],
[ [ 121, 12, 128, 16, 140, 7, 153, 7, 154, 15, 155, 15, 164, 12 ],
[ 241, 24, 242, 34, 243, 46, 244, 68, 245, 56, 246, 44, 247, 30, 248,
16 ], [ 1, 1, 8, 1, 20, 1, 33, 1, 34, 1, 35, 1, 44, 1 ] ],
[ [ 128, 14, 146, 10, 147, 2, 150, 2, 152, 10, 159, 18, 161, 8 ],
[ 241, 20, 242, 30, 243, 40, 244, 60, 245, 50, 246, 38, 247, 26, 248,
14 ], [ 8, 1, 26, 1, 27, 1, 30, 1, 32, 1, 39, 1, 41, 1 ] ],
[ [ 121, 28, 128, 18, 152, 15, 153, 15, 154, 10, 155, 24 ],
[ 241, 28, 242, 40, 243, 54, 244, 79, 245, 64, 246, 49, 247, 34, 248,
18 ], [ 1, 1, 8, 1, 32, 1, 33, 1, 34, 1, 35, 1 ] ],
[ [ 121, 16, 127, 42, 128, 22, 146, 30, 147, 30, 164, 16 ],
[ 241, 32, 242, 46, 243, 62, 244, 92, 245, 76, 246, 60, 247, 42, 248,
22 ], [ 1, 1, 7, 1, 8, 1, 26, 1, 27, 1, 44, 1 ] ],
[ [ 143, 18, 144, 4, 146, 3, 147, 10, 148, 10, 149, 14, 152, 8, 161, 3 ],
[ 241, 22, 242, 32, 243, 43, 244, 64, 245, 52, 246, 40, 247, 27, 248,
14 ], [ 23, 1, 24, 1, 26, 1, 27, 1, 28, 1, 29, 1, 32, 1, 41, 1 ] ],
[ [ 134, 15, 143, 12, 144, 12, 146, 7, 147, 7, 149, 15, 152, 16 ],
[ 241, 24, 242, 35, 243, 47, 244, 70, 245, 57, 246, 44, 247, 30, 248,
15 ], [ 14, 1, 23, 1, 24, 1, 26, 1, 27, 1, 29, 1, 32, 1 ] ],
[ [ 135, 16, 136, 12, 146, 15, 147, 15, 148, 7, 150, 12, 152, 1, 161, 7 ],
[ 241, 24, 242, 35, 243, 47, 244, 70, 245, 57, 246, 44, 247, 30, 248,
16 ], [ 15, 1, 16, 1, 26, 1, 27, 1, 28, 1, 30, 1, 32, 1, 41, 1 ] ],
[ [ 129, 28, 135, 18, 146, 24, 147, 10, 148, 15, 152, 15, 161, 1 ],
[ 241, 28, 242, 40, 243, 54, 244, 80, 245, 65, 246, 50, 247, 34, 248,
18 ], [ 9, 1, 15, 1, 26, 1, 27, 1, 28, 1, 32, 1, 41, 1 ] ],
[ [ 124, 8, 128, 12, 132, -4, 135, 16, 143, 6, 145, 14, 146, 12, 150, 4,
151, 18, 154, 4, 155, 10, 159, -4 ],
[ 241, 24, 242, 36, 243, 48, 244, 72, 245, 58, 246, 44, 247, 30, 248,
16 ], [ 4, 1, 15, 1, 23, 1, 25, 1, 26, 1, 31, 1, 34, 1, 35, 1 ] ],
[ [ 138, 10, 139, 24, 140, 15, 141, 15, 142, 18, 143, 28, 153, 1, 169, 1 ],
[ 241, 28, 242, 40, 243, 54, 244, 80, 245, 66, 246, 50, 247, 34, 248,
18 ], [ 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 33, 1, 49, 1 ] ],
[ [ 127, 42, 128, 22, 143, 16, 144, 16, 146, 30, 147, 30, 152, 1 ],
[ 241, 32, 242, 47, 243, 63, 244, 94, 245, 77, 246, 60, 247, 42, 248,
22 ], [ 7, 1, 8, 1, 23, 1, 24, 1, 26, 1, 27, 1, 32, 1 ] ],
[ [ 127, 28, 128, 22, 134, 42, 137, 17, 138, 28, 143, 3, 144, 32, 146, 3,
147, 15, 151, -3 ],
[ 241, 32, 242, 48, 243, 64, 244, 95, 245, 78, 246, 60, 247, 42, 248,
22 ], [ 8, 1, 14, 1, 17, 1, 18, 1, 24, 1, 26, 1, 27, 1 ] ],
[ [ 124, 8, 134, 18, 135, 18, 143, 8, 144, 20, 145, 14, 146, 20, 147, 14 ],
[ 241, 28, 242, 42, 243, 56, 244, 84, 245, 68, 246, 52, 247, 36, 248,
18 ], [ 4, 1, 14, 1, 15, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1 ] ],
[ [ 121, 36, 133, 21, 137, 40, 138, 12, 139, 30, 142, 22, 148, 21 ],
[ 241, 36, 242, 52, 243, 70, 244, 103, 245, 84, 246, 64, 247, 43, 248,
22 ], [ 1, 1, 13, 1, 17, 1, 18, 1, 19, 1, 22, 1, 28, 1 ] ],
[ [ 124, 16, 128, 22, 134, 42, 143, 2, 144, 30, 145, 30, 146, 16, 147, 2 ],
[ 241, 32, 242, 48, 243, 64, 244, 96, 245, 78, 246, 60, 247, 42, 248,
22 ], [ 4, 1, 8, 1, 14, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1 ] ],
[ [ 121, 40, 127, 15, 128, 26, 134, 50, 137, 21, 138, 15, 139, 57, 146, 22,
147, -22 ],
[ 241, 40, 242, 58, 243, 78, 244, 115, 245, 94, 246, 72, 247, 50, 248,
26 ], [ 1, 1, 8, 1, 14, 1, 17, 1, 18, 1, 19, 1, 26, 1 ] ],
[ [ 121, 36, 124, 12, 135, 2, 137, 30, 138, 2, 139, 40, 141, 22, 142, 22,
146, 20, 147, -20 ],
[ 241, 36, 242, 52, 243, 70, 244, 104, 245, 84, 246, 64, 247, 44, 248,
22 ], [ 1, 1, 4, 1, 17, 1, 18, 1, 19, 1, 21, 1, 22, 1, 26, 1 ] ],
[ [ 124, 21, 127, 15, 128, 26, 129, 40, 134, 50, 137, 1, 138, 57, 139, 15,
146, -22, 147, 22 ],
[ 241, 40, 242, 58, 243, 78, 244, 116, 245, 94, 246, 72, 247, 50, 248,
26 ], [ 4, 1, 8, 1, 9, 1, 14, 1, 17, 1, 18, 1, 19, 1, 27, 1 ] ],
[ [ 121, 52, 126, 96, 127, 66, 128, 34, 137, 27, 138, 49, 139, 75 ],
[ 241, 52, 242, 76, 243, 102, 244, 151, 245, 124, 246, 96, 247, 66,
248, 34 ], [ 1, 1, 6, 1, 7, 1, 8, 1, 17, 1, 18, 1, 19, 1 ] ],
[ [ 121, 44, 124, 14, 128, 28, 133, 26, 134, 54, 137, 36, 138, 28, 139, 50 ]
, [ 241, 44, 242, 64, 243, 86, 244, 128, 245, 104, 246, 80, 247, 54,
248, 28 ], [ 1, 1, 4, 1, 8, 1, 13, 1, 14, 1, 17, 1, 18, 1, 19, 1 ] ]
,
[ [ 126, 96, 127, 66, 128, 34, 129, 52, 130, 27, 131, 1, 132, 75, 145, 49 ],
[ 241, 52, 242, 76, 243, 102, 244, 152, 245, 124, 246, 96, 247, 66,
248, 34 ], [ 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 25, 1 ] ],
[ [ 121, 60, 122, 66, 123, -66, 125, 108, 127, 22, 128, 38, 130, 22, 131,
118, 132, 22, 134, 74, 140, 34 ],
[ 241, 60, 242, 88, 243, 118, 244, 174, 245, 142, 246, 108, 247, 74,
248, 38 ], [ 1, 1, 2, 1, 5, 1, 8, 1, 10, 1, 11, 1, 14, 1, 20, 1 ] ],
[ [ 121, 72, 122, 38, 123, -38, 125, 172, 126, 132, 127, 90, 128, 46, 130,
68, 131, 142, 132, 68 ],
[ 241, 72, 242, 106, 243, 142, 244, 210, 245, 172, 246, 132, 247, 90,
248, 46 ], [ 1, 1, 2, 1, 5, 1, 6, 1, 7, 1, 8, 1, 10, 1, 11, 1 ] ],
[ [ 121, 92, 122, 136, 123, 182, 124, 270, 125, 220, 126, 168, 127, 114,
128, 58 ],
[ 241, 92, 242, 136, 243, 182, 244, 270, 245, 220, 246, 168, 247, 114,
248, 58 ], [ 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1 ] ] ];
elif type[1] = 'A' then
list:= SLAfcts.weighted_dyn_diags_An(Length(CartanMatrix(RootSystem(L))));
orbs:= [ ];
for i in [1..Length(list[1])] do
if not IsZero(list[1][i]) then
o:= NilpotentOrbit( L, list[1][i] );
SetOrbitPartition( o, list[2][i] );
Add( orbs, o );
fi;
od;
return orbs;
elif type[1] = 'B' then
list:= SLAfcts.weighted_dyn_diags_Bn(Length(CartanMatrix(RootSystem(L))));
orbs:= [ ];
for i in [1..Length(list[1])] do
if not IsZero( list[1][i] ) then
o:= NilpotentOrbit( L, list[1][i] );
SetOrbitPartition( o, list[2][i] );
Add( orbs, o );
fi;
od;
return orbs;
elif type[1] = 'C' then
list:= SLAfcts.weighted_dyn_diags_Cn(Length(CartanMatrix(RootSystem(L))));
orbs:= [ ];
for i in [1..Length(list[1])] do
if not IsZero( list[1][i] ) then
o:= NilpotentOrbit( L, list[1][i] );
SetOrbitPartition( o, list[2][i] );
Add( orbs, o );
fi;
od;
return orbs;
elif type[1] = 'D' then
list:= SLAfcts.weighted_dyn_diags_Dn(Length(CartanMatrix(RootSystem(L))));
orbs:= [ ];
for i in [1..Length(list[1])] do
if not IsZero( list[1][i] ) then
o:= NilpotentOrbit( L, list[1][i] );
SetOrbitPartition( o, list[2][i] );
Add( orbs, o );
fi;
od;
return orbs;
fi;
orbs:= [ ];
for i in [1..Length( list )] do
o:= NilpotentOrbit( L, list[i] );
sl2:= [ ];
for u in elms[i] do
a:= Zero(L);
for j in [1,3..Length(u)-1] do
a := a + u[j+1]*Basis(L)[u[j]];
od;
Add( sl2, a );
od;
SetSL2Triple( o, sl2 );
Add( orbs, o );
od;
return orbs;
end;
SLAfcts.nilporbs1:= function( L )
# L simple.
local tL, F, K, f, o, tK, eK, p, orbs, orb, wd, new, exceptional, eL, list, i;
tL:= CartanType( CartanMatrix( RootSystem(L) ) );
if Length(tL.types) > 1 then
Error("The Lie algebra <L> is not simple.");
fi;
F:= LeftActingDomain(L);
K:= SimpleLieAlgebra( tL.types[1][1], tL.types[1][2], F );
exceptional:= tL.types[1][1] in ["E","F","G"];
if exceptional then
f:= IsomorphismOfSemisimpleLieAlgebras( K, L );
fi;
o:= SLAfcts.nilporbs( K );
tK:= CartanType( CartanMatrix( RootSystem(K) ) );
eK:= tK.enumeration[1];
eL:= tL.enumeration[1];
list:= [ ];
for i in [1..Length(eK)] do list[eK[i]]:= eL[i]; od;
p:= PermList( list );
orbs:= [ ];
for orb in o do
wd:= Permuted( WeightedDynkinDiagram(orb), p );
new:= NilpotentOrbit( L, wd );
if exceptional then
SetSL2Triple( new, List( SL2Triple( orb ), x -> Image( f, x ) ) );
else
SetOrbitPartition( new, OrbitPartition( orb ) );
fi;
Add( orbs, new );
od;
return orbs;
end;
InstallMethod( IsSLA,
"for a Lie algebra",
true, [ IsLieAlgebra ], 0,
function( L )
local R, t, c, Ks, i, b, bounds, sm, e, pp;
R:= RootSystem(L);
if R = fail then return fail; fi;
t:= CartanType( CartanMatrix( RootSystem(L) ) );
if Length( t.types ) = 1 then
return true;
else
c:= CanonicalGenerators(R);
Ks:= [ ];
for i in [1..Length(t.enumeration)] do
Add( Ks, UnderlyingLieAlgebra( SLAfcts.rtsys( L,
List( c, u -> u{t.enumeration[i]} ) ) ) );
od;
b:= Concatenation( List( Ks, U -> BasisVectors( Basis(U) ) ) );
bounds:= [ [1,Dimension(Ks[1])] ];
for i in [2..Length(Ks)] do
sm:= Sum( [1..i-1], j -> Dimension( Ks[j] ) );
Add( bounds, [sm+1,sm+Dimension(Ks[i])] );
od;
e:= Concatenation( t.enumeration );
pp:= [ ];
for i in [1..Length(e)] do
Add( pp, Position(e,i) );
od;
SetSLAComponents( L, rec( Ks:= Ks, pp:= pp, bounds:= bounds,
bas:= Basis( L, b ) ) );
return false;
fi;
end );
InstallOtherMethod( NilpotentOrbits,
"for a Lie algebra",
true, [ IsLieAlgebra ], 10,
function( L )
# L semi-simple.
local Ks, wds, wds0, w1, w2, o, wd, K, p, e, pp, i, orbs;
if IsSLA(L) then
return SLAfcts.nilporbs1(L);
else
Ks:= SLAComponents( L ).Ks;
wds:= [ [] ];
for K in Ks do
o:= SLAfcts.nilporbs1(K);
wd:= [ List( [1..Length(CartanMatrix(RootSystem(K)))], i -> 0 ) ];
Append( wd, List( o, WeightedDynkinDiagram ) );
wds0:= [ ];
for w1 in wds do
for w2 in wd do
Add( wds0, Concatenation(w1,w2) );
od;
od;
wds:= wds0;
od;
p:= PositionProperty( wds, x -> ForAll( x, IsZero ) );
RemoveElmList( wds, p );
wds0:= [ ];
pp:= SLAComponents(L).pp;
for w1 in wds do
Add( wds0, w1{pp} );
od;
wds:= wds0;
orbs:= List( wds, w -> NilpotentOrbit( L, w ) );
fi;
return orbs;
end );
InstallMethod( SignatureTable,
"for Lie algebra", true, [IsLieAlgebra], 0,
function( L )
local o, R, p, tab, x, w, max, dims, r, u, wt, dc, char, it, i,
Ci, h, ev, pos, tp, res, en;
o:= NilpotentOrbits(L);
R:= RootSystem(L);
tp:= CartanType( CartanMatrix(R) ).types[1];
if tp[1] in [ "A", "B", "C", "E", "F", "G" ] then
p:= PositiveRootsNF(R);
tab:= [ ];
for x in o do
w:= WeightedDynkinDiagram(x);
max:= p[Length(p)]*w;
if not IsInt( max ) then # hack to make it work with SqrtField...
max:= max![1][1][1];
fi;
dims:= List([1..max+1], u -> 0 );
for r in p do
u:= r*w+1;
if not IsInt( u ) then
u:= u![1][1][1];
fi;
dims[u]:= dims[u]+1;
od;
dims[1]:= 2*dims[1]+Length(CartanMatrix(R));
Add( tab, [ dims, w ] );
od;
return rec( tipo:= "notD", tab:= tab );
else
en:= CartanType( CartanMatrix(R) ).enumeration[1];
wt:= List( CartanMatrix( R ), x -> 0 );
wt[en[1]]:= 1;
dc:= DominantCharacter( L, wt );
char:= [[],[]];
for i in [1..Length(dc[1])] do
it:= WeylOrbitIterator( WeylGroup(R), dc[1][i] );
while not IsDoneIterator( it ) do
Add( char[1], NextIterator( it ) );
Add( char[2], dc[2][i] );
od;
od;
Ci:= FamilyObj(o[1])!.invCM;
tab:= [ ];
for x in o do
h:= Ci*WeightedDynkinDiagram(x);
dims:= [ ];
for i in [1..Length(char[1])] do
ev:= h*char[1][i];
pos:= PositionProperty( dims, y -> y[1]=ev);
if pos = fail then
Add( dims, [ev, char[2][i]] );
else
dims[pos][2]:= dims[pos][2]+char[2][i];
fi;
od;
Sort( dims, function(a,b) return a[1] < b[1]; end );
Add( tab, [dims, WeightedDynkinDiagram(x)] );
od;
res:= rec( tipo:= "D", char1:= char, tab1:= tab, V1:=
HighestWeightModule( L, wt ) );
wt:= List( CartanMatrix( R ), x -> 0 );
wt[en[Length(wt)]]:= 1;
dc:= DominantCharacter( L, wt );
char:= [[],[]];
for i in [1..Length(dc[1])] do
it:= WeylOrbitIterator( WeylGroup(R), dc[1][i] );
while not IsDoneIterator( it ) do
Add( char[1], NextIterator( it ) );
Add( char[2], dc[2][i] );
od;
od;
Ci:= FamilyObj(o[1])!.invCM;
tab:= [ ];
for x in o do
h:= Ci*WeightedDynkinDiagram(x);
dims:= [ ];
for i in [1..Length(char[1])] do
ev:= h*char[1][i];
pos:= PositionProperty( dims, y -> y[1]=ev);
if pos = fail then
Add( dims, [ev, char[2][i]] );
else
dims[pos][2]:= dims[pos][2]+char[2][i];
fi;
od;
Sort( dims, function(a,b) return a[1] < b[1]; end );
Add( tab, [dims, WeightedDynkinDiagram(x)] );
od;
res.char2:= char; res.tab2:= tab;
res.V2:= HighestWeightModule( L, wt );
return res;
fi;
end );
SLAfcts.wdd:= function( L, x )
local sl2, K, H, ch, t, cc, tableM, T, adh, possibles,
k, dim, rank, poss0, l, c1, c2, fac, f, inds1, inds2, ev, pos, id;
if not x in L then Error("<x> does not lie in <L>"); fi;
if IsZero(x) then
return List( CartanMatrix( RootSystem(L) ), x -> 0 );
fi;
T:= SignatureTable(L);
sl2:= SL2Triple( L, x );
if T.tipo = "notD" then
T:= T.tab;
adh:= TransposedMat( AdjointMatrix( Basis(L), sl2[2] ) );
possibles:= [ ];
dim:= Dimension(L);
rank:= RankMat(adh);
for k in [1..Length(T)] do
if T[k][1][1] = dim-rank then
Add( possibles, T[k] );
fi;
od;
k:= 1;
while Length(possibles) > 1 do
rank:= RankMat( adh-k*adh^0 );
poss0:= [ ];
for l in [1..Length(possibles)] do
if possibles[l][1][k+1] = dim-rank then
Add( poss0, possibles[l] );
fi;
od;
possibles:= poss0;
k:= k+1;
od;
return possibles[1][2];
else
adh:= MatrixOfAction( Basis( T.V1 ), sl2[2] );
id:= adh^0;
c1:= [ ];
k:= 0; ev:=0;
while ev < Dimension(T.V1) do
f:= Dimension(T.V1)-RankMat( adh - k*id );
if f > 0 then
Add( c1, [ k, f ] );
if k=0 then
ev:= ev + f;
else
ev:= ev+2*f;
Add( c1, [ -k, f ] );
fi;
fi;
k:= k+1;
od;
Sort( c1, function(a,b) return a[1]<b[1]; end );
inds1:= Filtered( [1..Length(T.tab1)], x -> T.tab1[x][1]=c1 );
if Length(inds1) = 1 then
return T.tab1[ inds1[1] ][2];
fi;
adh:= MatrixOfAction( Basis( T.V2 ), sl2[2] );
id:= adh^0;
c2:= [ ];
k:= 0; ev:=0;
while ev < Dimension(T.V2) do
f:= Dimension(T.V2)-RankMat( adh - k*id );
if f > 0 then
Add( c2, [ k, f ] );
if k=0 then
ev:= ev + f;
else
ev:= ev+2*f;
Add( c2, [ -k, f ] );
fi;
fi;
k:= k+1;
od;
Sort( c2, function(a,b) return a[1]<b[1]; end );
inds2:= Filtered( [1..Length(T.tab2)], x -> T.tab2[x][1]=c2 );
return T.tab1[ Intersection( inds1, inds2 )[1] ][2];
fi;
end;
InstallOtherMethod( WeightedDynkinDiagram,
"for a semisimple Lie algebra and a nilpotent element",
true, [ IsLieAlgebra, IsObject ], 0,
function(L,x)
local r, wd, cf, i, c0, u;
if IsSLA(L) then
return SLAfcts.wdd(L,x);
else
r:= SLAComponents( L );
wd:= [ ];
cf:= Coefficients( r.bas, x );
for i in [1..Length(r.Ks)] do
c0:= cf{[r.bounds[i][1]..r.bounds[i][2]]};
u:= Sum( [1..Length(c0)], k -> c0[k]*Basis(r.Ks[i])[k] );
Append( wd, SLAfcts.wdd( r.Ks[i], u ) );
od;
return wd{r.pp};
fi;
end );
InstallMethod( SL2Grading,
"for Lie algebra and element", true, [IsLieAlgebra, IsObject], 0,
function( L, h )
# Here o is a nilpotent orbit. We return the grading of the Lie
# algebra corresponding to "h". Three lists are returned:
# first the components with grades 1,2,3,... then a list with
# the components with grades -1,-2,-3... Then the zero component.
local adh, pos, neg, k, done, sp;
adh:= AdjointMatrix( Basis(L), h );
pos:= [ ];
neg:= [ ];
k:= 1;
done:= false;
while not done do
sp:= NullspaceMat( TransposedMat( adh-k*adh^0 ) );
Add( pos, List( sp, c -> LinearCombination( Basis(L), c ) ) );
sp:= NullspaceMat( TransposedMat( adh+k*adh^0 ) );
Add( neg, List( sp, c -> LinearCombination( Basis(L), c ) ) );
if Length(pos)>=2 and (pos[ Length(pos) ] = [ ] and pos[ Length(pos)-1 ] = [ ]) then
done:= true;
else
k:= k+1;
fi;
od;
while pos[Length(pos)] = [ ] do Unbind( pos[Length(pos)] ); od;
while neg[Length(neg)] = [ ] do Unbind( neg[Length(neg)] ); od;
return [ pos, neg, ShallowCopy( BasisVectors(
Basis( LieCentralizer( L, Subalgebra( L,[h] ) ) ) ) ) ];
end );
SLAfcts.rigid_diags_Dn:= function( n )
local part, good, p, q, cur, k, m, is_even, no_gap, odd_good, u, diags,
u0, i, d, L, h, Ci, dims, h0, mat, dim;
part:= Partitions( 2*n );
good:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
is_even:= true;
for k in [1..Length(q)] do
if IsEvenInt( q[k][1] ) and not IsEvenInt( q[k][2] ) then
is_even:= false;
break;
fi;
od;
no_gap:= true;
for k in [1..Length(q)-1] do
if q[k][1] > q[k+1][1]+1 then
no_gap:= false;
break;
fi;
od;
if no_gap then
no_gap:= q[Length(q)][1] = 1;
fi;
odd_good:= true;
for k in [1..Length(q)] do
if IsOddInt(q[k][1]) and q[k][2] = 2 then
odd_good:= false;
break;
fi;
od;
if is_even and no_gap and odd_good then Add( good, q ); fi;
od;
diags:= [ ];
for q in good do
u:= [ ];
for k in [1..Length(q)] do
u0:= [ ];
d:= q[k][1]-1;
while d >= 0 do
Add( u0, d );
d:= d-2;
od;
for i in [1..q[k][2]] do Append( u, u0 ); od;
od;
Sort(u);
u:= Reversed( u );
d:= [ ];
for k in [1..n-1] do
Add( d, u[k]-u[k+1] );
od;
Add( d, u[n-1]+u[n] );
Add( diags, d );
od;
L:= SimpleLieAlgebra( "D", n, Rationals );
h:= ChevalleyBasis(L)[3];
Ci:= CartanMatrix( RootSystem(L) )^-1;
dims:= [ ];
for d in diags do
h0:= (Ci*d)*h;
mat:= AdjointMatrix( Basis(L), h0 );
dim:= Dimension(L);
dim:= dim - Length( NullspaceMat(mat) );
dim:= dim - Length( NullspaceMat( mat-IdentityMat( Dimension(L) ) ) );
Add( dims, dim );
od;
return [diags,dims,good];
end;
SLAfcts.rigid_diags_Bn:= function( n )
local part, good, p, q, cur, k, m, is_even, no_gap, odd_good, u, diags,
u0, i, d, L, h, Ci, dims, h0, mat, dim;
part:= Partitions( 2*n+1 );
good:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
is_even:= true;
for k in [1..Length(q)] do
if IsEvenInt( q[k][1] ) and not IsEvenInt( q[k][2] ) then
is_even:= false;
break;
fi;
od;
no_gap:= true;
for k in [1..Length(q)-1] do
if q[k][1] > q[k+1][1]+1 then
no_gap:= false;
break;
fi;
od;
if no_gap then
no_gap:= q[Length(q)][1] = 1;
fi;
odd_good:= true;
for k in [1..Length(q)] do
if IsOddInt(q[k][1]) and q[k][2] = 2 then
odd_good:= false;
break;
fi;
od;
if is_even and no_gap and odd_good then Add( good, q ); fi;
od;
diags:= [ ];
for q in good do
u:= [ ];
for k in [1..Length(q)] do
u0:= [ ];
d:= q[k][1]-1;
while d >= 0 do
Add( u0, d );
d:= d-2;
od;
for i in [1..q[k][2]] do Append( u, u0 ); od;
od;
# get rid of a zero...
for k in [1..Length(u)] do
if u[k] = 0 then RemoveElmList( u, k ); break; fi;
od;
Sort(u);
u:= Reversed( u );
d:= [ ];
for k in [1..n-1] do
Add( d, u[k]-u[k+1] );
od;
Add( d, u[n] );
Add( diags, d );
od;
L:= SimpleLieAlgebra( "B", n, Rationals );
h:= ChevalleyBasis(L)[3];
Ci:= CartanMatrix( RootSystem(L) )^-1;
dims:= [ ];
for d in diags do
h0:= (Ci*d)*h;
mat:= AdjointMatrix( Basis(L), h0 );
dim:= Dimension(L);
dim:= dim - Length( NullspaceMat(mat) );
dim:= dim - Length( NullspaceMat( mat-IdentityMat( Dimension(L) ) ) );
Add( dims, dim );
od;
return [diags,dims,good];
end;
SLAfcts.rigid_diags_Cn:= function( n )
local part, good, p, q, cur, k, m, is_even, no_gap, odd_good, u, diags,
u0, i, d, L, h, Ci, dims, h0, mat, dim;
part:= Partitions( 2*n );
good:= [ ];
for p in part do
q:= [ ];
cur:= p[1];
k:= 1;
m:= 0;
while k <= Length(p) do
m:= m+1;
if k < Length(p) and p[k+1] <> cur then
Add( q, [ cur, m ] );
cur:= p[k+1];
m:= 0;
fi;
k:= k+1;
od;
Add( q, [ cur,m ] );
is_even:= true;
for k in [1..Length(q)] do
if IsOddInt( q[k][1] ) and not IsEvenInt( q[k][2] ) then
is_even:= false;
break;
fi;
od;
no_gap:= true;
for k in [1..Length(q)-1] do
if q[k][1] > q[k+1][1]+1 then
no_gap:= false;
break;
fi;
od;
if no_gap then
no_gap:= q[Length(q)][1] = 1;
fi;
odd_good:= true;
for k in [1..Length(q)] do
if IsEvenInt(q[k][1]) and q[k][2] = 2 then
odd_good:= false;
break;
fi;
od;
if is_even and no_gap and odd_good then Add( good, q ); fi;
od;
diags:= [ ];
for q in good do
u:= [ ];
for k in [1..Length(q)] do
u0:= [ ];
d:= q[k][1]-1;
while d >= 0 do
Add( u0, d );
d:= d-2;
od;
for i in [1..q[k][2]] do Append( u, u0 ); od;
od;
Sort(u);
u:= Reversed( u );
d:= [ ];
for k in [1..n-1] do
Add( d, u[k]-u[k+1] );
od;
Add( d, 2*u[n] );
Add( diags, d );
od;
L:= SimpleLieAlgebra( "C", n, Rationals );
h:= ChevalleyBasis(L)[3];
Ci:= CartanMatrix( RootSystem(L) )^-1;
dims:= [ ];
for d in diags do
h0:= (Ci*d)*h;
mat:= AdjointMatrix( Basis(L), h0 );
dim:= Dimension(L);
dim:= dim - Length( NullspaceMat(mat) );
dim:= dim - Length( NullspaceMat( mat-IdentityMat( Dimension(L) ) ) );
Add( dims, dim );
od;
return [diags,dims,good];
end;
SLAfcts.indtableAn:= function( L ) # L of type An....
local o, rank, p, T, d, sd, i, k, dt, pos;
o:= NilpotentOrbits(L);
rank:= Length( CartanMatrix( RootSystem(L) ) );
p:= [ List( [1..rank+1], x -> 1 ) ];
Append( p, List( o, OrbitPartition ) );
T:= [ ];
for d in p do
if Length(d) > 1 then
sd:= List( [1..rank], x -> 0 );
for i in [1..Length(d)-1] do
k:= Sum( d{[1..i]} );
sd[k]:= 2;
od;
dt:=[ ];
for i in [1..rank+1] do
dt[i]:= Length( Filtered( [1..Length(d)], j -> d[j] >= i ) );
od;
for i in [1..Length(dt)] do if dt[i] = 0 then Unbind( dt[i] ); fi; od;
dt:= Filtered( dt, x -> IsBound(x) );
pos:= Position( p, dt );
Add( T, [ sd, WeightedDynkinDiagram(o[pos-1]) ] );
fi;
od;
return T;
end;
InstallMethod( InducedNilpotentOrbits,
"for Lie algebra", true, [IsLieAlgebra], 0,
function( L )
local tp, T, rank, i, S, pos, o, e, f, wt;
if not IsSLA(L) then
Error("currently this only works for simple Lie algebras.");
fi;
tp:= CartanType( CartanMatrix( RootSystem(L) ) );
rank:= tp.types[1][2];
if tp.types[1][1] = "A" then
T:= SLAfcts.indtableAn( SimpleLieAlgebra( "A", rank, Rationals ) );
elif tp.types[1][1] = "B" then
T:= SLAfcts.SB[rank];
elif tp.types[1][1] = "C" then
T:= SLAfcts.SC[rank];
elif tp.types[1][1] = "D" then
T:= SLAfcts.SD[rank];
elif tp.types[1][1] = "E" then
T:= SLAfcts.SE[rank];
elif tp.types[1][1] = "F" then
T:= SLAfcts.SF[rank];
elif tp.types[1][1] = "G" then
T:= SLAfcts.SG[rank];
fi;
wt:= List( [1..rank], x -> 2 );
T:= Concatenation( [[ ShallowCopy(wt), ShallowCopy(wt) ]], T );
e:= tp.enumeration[1];
f:= [ ];
for i in [1..rank] do f[ e[i] ]:= i; od;
for i in [1..Length(T)] do
T[i][1]:= T[i][1]{f};
T[i][2]:= T[i][2]{f};
od;
S:= [ ];
o:= NilpotentOrbits(L);
for i in [1..Length(T)] do
pos:= PositionProperty( o, x -> WeightedDynkinDiagram(x) = T[i][2] );
Add( S, rec( sheetdiag:= T[i][1], norbit:= o[pos] ) );
od;
return S;
end );
InstallMethod( RigidNilpotentOrbits,
"for Lie algebra", true, [IsLieAlgebra], 0,
function( L )
local tp, T, rank, i, S, pos, o, e, f;
if not IsSLA(L) then
Error("currently this only works for simple Lie algebras.");
fi;
tp:= CartanType( CartanMatrix( RootSystem(L) ) );
if tp.types[1][1] = "A" then
T:= [ ];
else
rank:= tp.types[1][2];
if tp.types[1][1] = "B" then
T:= SLAfcts.rigid_diags_Bn( rank )[1];
elif tp.types[1][1] = "C" then
T:= SLAfcts.rigid_diags_Cn( rank )[1];
elif tp.types[1][1] = "D" then
T:= SLAfcts.rigid_diags_Dn( rank )[1];
elif tp.types[1][1] = "E" then
if rank = 6 then
T:= [ [1,0,0,1,0,1], [0,0,0,1,0,0], [0,1,0,0,0,0] ];
elif rank = 7 then
T:= [ [1,0,0,1,0,0,0], [0,0,1,0,0,1,0],
[0,0,0,1,0,0,0], [0,1,0,0,0,0,1],
[0,0,1,0,0,0,0], [0,0,0,0,0,1,0],
[0,1,0,0,0,0,0] ];
elif rank = 8 then
T:= [ [ 0, 0, 0, 0, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 1, 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 0, 0, 0, 1 ],
[ 0, 0, 1, 0, 0, 1, 0, 1 ] ];
fi;
elif tp.types[1][1] = "F" then
T:= [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 1 ] ];
elif tp.types[1][1] = "G" then
T:= [ [ 1, 0 ], [ 0, 1 ] ];
fi;
e:= tp.enumeration[1];
f:= [ ];
for i in [1..rank] do f[ e[i] ]:= i; od;
for i in [1..Length(T)] do
T[i]:= T[i]{f};
od;
fi;
o:= NilpotentOrbits(L);
return Filtered( o, x -> WeightedDynkinDiagram(x) in T );
end );
InstallOtherMethod( Dimension,
"for a nilpotent orbit", true,
[ IsNilpotentOrbit ], 0,
function( o )
local wd, L, R, posR, dim, r0;
wd:= WeightedDynkinDiagram(o);
L:= AmbientLieAlgebra(o);
R:= RootSystem(L);
posR:= PositiveRootsNF(R);
dim:= Dimension(L);
r0:= Length( Filtered( posR, r -> wd*r = 0 ) );
dim:= dim-Length(CartanMatrix(R))-2*r0;
r0:= Length( Filtered( posR, r -> wd*r = 1 ) );
return dim-r0;
end );
InstallMethod( IsRegular,
"for a nilpotent orbit", true,
[ IsNilpotentOrbit ], 0,
function( o )
return ForAll( WeightedDynkinDiagram(o), x -> x=2 );
end );
InstallMethod( RegularNilpotentOrbit,
"for a semisimple Lie algebra", true,
[ IsLieAlgebra ], 0,
function( L )
return Filtered( NilpotentOrbits(L), IsRegular )[1];
end );
InstallMethod( IsDistinguished,
"for a nilpotent orbit", true,
[ IsNilpotentOrbit ], 0,
function( o )
local wd, L, R, posR, d0, r0;
wd:= WeightedDynkinDiagram(o);
L:= AmbientLieAlgebra(o);
R:= RootSystem(L);
posR:= PositiveRootsNF(R);
r0:= Length( Filtered( posR, r -> wd*r = 0 ) );
d0:= Length(CartanMatrix(R))+2*r0;
r0:= Length( Filtered( posR, r -> wd*r = 2 ) );
return d0=r0;
end );
InstallMethod( DistinguishedNilpotentOrbits,
"for a semisimple Lie algebra", true,
[ IsLieAlgebra ], 0,
function( L )
return Filtered( NilpotentOrbits(L), IsDistinguished );
end );
InstallMethod( DisplayWeightedDynkinDiagram,
"for a nilpotent orbit",
true, [ IsNilpotentOrbit ], 0,
function(o )
local C;
C:= CartanMatrix(RootSystem(AmbientLieAlgebra(o)));
SLAfcts.printdyndiag(C,WeightedDynkinDiagram(o));
end);
InstallOtherMethod( DisplayWeightedDynkinDiagram,
"for a Lie algebra and a nipotent element",
true, [ IsLieAlgebra, IsObject ], 0,
function( L, x )
local C;
C:= CartanMatrix(RootSystem(L));
SLAfcts.printdyndiag(C,WeightedDynkinDiagram(L,x));
end);
InstallMethod( RichardsonOrbits,
"for a semisimple Lie algebra", true,
[ IsLieAlgebra ], 0,
function( L )
local ind, rich, wdd, i;
ind:= InducedNilpotentOrbits(L);
rich:= [ ];
wdd:= [ ];
for i in [1..Length(ind)] do
if not (1 in ind[i].sheetdiag) then
if not WeightedDynkinDiagram(ind[i].norbit) in wdd then
Add( rich, ind[i].norbit );
Add( wdd, WeightedDynkinDiagram(ind[i].norbit) );
fi;
fi;
od;
return rich;
end );
SLAfcts.diagmat:= function( F, H ) # F : field
# H torus, find a matrix C such that
# C^-1*u*C is diagonal for all u in H
local spaces, i, h, sp0, s, sp, A, fct, f, V, bas, C, one,
fct0, num, fam, cf, b, c, r, n, U;
one := One(F);
n:= Length( H[1] );
if ForAll( H, IsDiagonalMat ) then # nothing to do..
return IdentityMat(n)*one;
fi;
V:= F^n;
spaces:= [ ShallowCopy( BasisVectors( Basis(V) ) ) ];
for i in [1..Length(H)] do
h:= H[i];
sp0:= [ ];
for s in spaces do
sp:= Basis( Subspace( V, s ), s );
A:= List( s, u -> Coefficients( sp, h*u ) );
fct:= Factors( MinimalPolynomial(F,A) );
num := IndeterminateNumberOfUnivariateLaurentPolynomial(fct[1]);
fam := FamilyObj( fct[1] );
fct0:= [ ];
for f in fct do
if Degree(f) = 1 then
Add( fct0, f );
elif Degree(f) = 2 then # we just take square roots...
cf := CoefficientsOfUnivariatePolynomial(f);
b := cf[2];
c := cf[1];
r := (-b+Sqrt(b^2-4*c))/2;
if not r in F then Print("diagmat: cannot do this over ",F,"\n"); fi;
Add( fct0, PolynomialByExtRep( fam,
[ [], -r, [num,1], one] ) );
r := (-b-Sqrt(b^2-4*c))/2;
if not r in F then Print("diagmat: cannot do this over ",F,"\n"); fi;
Add( fct0, PolynomialByExtRep( fam,
[ [], -r, [num,1], one] ) );
else
Error("not split!");
return fail;
fi;
od;
for f in fct0 do
U:= NullspaceMat( Value( f, A ) );
Add( sp0, List( U, x -> LinearCombination( s, x ) ) );
od;
od;
spaces:= sp0;
od;
bas:= Concatenation( spaces );
C:= List( bas, x -> Coefficients( Basis(V), x ) );
return TransposedMat(C);
end;
SLAfcts.toruspar:= function( F, H ) # H list of mats giving the Lie algebra of torus
# output: parametrization of torus
local C, ad, add, diag, lat, n, A, d, r, S, P, Q, names, i, j, R,
indet, mats, m;
n:= Length( H[1] );
C:= SLAfcts.diagmat(F,H);
add:= List( H, u -> C^-1*u*C );
diag:= List( add, a -> List( [1..n], i -> a[i][i] ) );
lat:= NullspaceMat( TransposedMat(diag) );
if lat = [ ] then lat:= [ List( [1..n], i -> Zero(F) ) ]; fi;
# saturation...
A:= lat;
if A <> 0*A then
d:= Lcm( List( Filtered( Flat(A), x -> not IsZero(x)), DenominatorRat));
A:= d*A;
fi;
r:=SmithNormalFormIntegerMatTransforms(A);
Q:= r.coltrans; P:= r.rowtrans;
S:= List( r.normal, ShallowCopy );
for i in [1..Length(S)] do
if S[i][i] > 0 then S[i][i]:= 1; fi;
od;
lat:= P^-1*S*Q^-1;
lat:= NullspaceMat( TransposedMat(lat) );
# another saturation...
A:= lat;
d:= Lcm( List( Filtered( Flat(A), x -> not IsZero(x) ), DenominatorRat ) );
A:= d*A;
r:=SmithNormalFormIntegerMatTransforms(A);
Q:= r.coltrans; P:= r.rowtrans;
S:= List( r.normal, ShallowCopy );
for i in [1..Length(S)] do
if S[i][i] > 0 then S[i][i]:= 1; fi;
od;
lat:= P^-1*S*Q^-1;
names:= [ ];
for i in [1..Length(lat)] do
Add( names, Concatenation( "a", String(i) ) );
od;
R:= PolynomialRing( F, names );
indet:= IndeterminatesOfPolynomialRing( R );
mats:= [ ];
for i in [1..Length(lat)] do
m:= NullMat( n, n, R );
for j in [1..n] do
m[j][j]:= indet[i]^lat[i][j];
od;
#Add( mats, C*m*C^-1 );
Add( mats, m );
od;
A:= TransposedMat(lat);
Q:= HermiteNormalFormIntegerMatTransform( A ).rowtrans;
return rec( mats:= mats, C:= C, indets:= indet, exps:=A, Q:= Q );
end;
SLAfcts.valpars:= function( r, A )
# here r is the output of torusparam_gen
# A is a matrix in the torus (so the nonsplit torus)
# with elements T(t1,...,tm)
# we return a the vector [t1,..,tm] such that
# T(t1,...,tm)=A
local D, Q, t, m;
D:= r.C^-1*A*r.C;
if not IsDiagonalMat( D ) then return fail; fi;
D:= List( [1..Length(D)], i -> D[i][i] );
Q:= r.Q;
t:= List( Q, s -> Product( List( [1..Length(s)], i -> D[i]^s[i] ) ) );
m:= Length(r.exps[1]);
if not ForAll([m+1..Length(t)], i -> IsOne(t[i]) ) then
return fail; # the element does not lie in the torus...
fi;
return t{[1..m]};
end;
SLAfcts.iseltof:= function( L, C, s )
# here s is an element of order 2 in the automorphism group of
# L, C is a reductive subalgebra of L, we determine whether s lies
# in the connected subgroup of Ad L, with Lie algebra ad C.
local sol, U, K, H, adH, r;
sol:= NullspaceMat( TransposedMat(s-s^0) );
U:= Subalgebra( L, List( sol, x -> x*Basis(L)), "basis" );
K:= Intersection( U, C );
H:= CartanSubalgebra( K );
if Dimension(H) < Dimension( CartanSubalgebra(C) ) then
# the restriction of s to C is not inner, hence cannot lie in the
# group
return false;
fi;
adH:= List( Basis(H), x -> AdjointMatrix( Basis(L), x ) );
r:= SLAfcts.toruspar( CF(840), adH );
return SLAfcts.valpars( r, s ) <> fail;
end;
SLAfcts.orthLA:= function( F, n )
# for n = 2k+1 this gives the Lie algebra of type B_k, k\geq 2
# for n = 2k this gives the Lie algebra of type D_k, k\geq 4.
# both in standard nxn matrix rep.
local bas, i, j, m;
bas:= [ ];
for i in [1..n-1] do
for j in [i+1..n] do
if not (i=(n+1)/2 and j=(n+1)/2) then
m:= NullMat(n,n);
m[i][n+1-j]:= 1; m[j][n+1-i]:= -1;
Add( bas, m );
fi;
od;
od;
return LieAlgebra( F, bas, "basis" );
end;
SLAfcts.simpLA:= function( F, n )
# sp(n,F) (of rank n/2).
local bas, i, j, l, m;
l:= n/2;
bas:= [ ];
for i in [1..l] do
for j in [1..l] do
m:= NullMat(n,n);
m[i][j]:=1; m[l+j][l+i]:=-1;
Add( bas, m );
od;
od;
for i in [1..l] do
for j in [i..l] do
m:= NullMat(n,n);
m[i][l+j]:=1; m[j][l+i]:=1;
Add( bas, m );
od;
od;
for i in [1..l] do
for j in [i..l] do
m:= NullMat(n,n);
m[l+i][j]:=1; m[l+j][i]:=1;
Add( bas, m );
od;
od;
return LieAlgebra( F, bas, "basis" );
end;
SLAfcts.compgrp:= function( L, sl2, eps ) # L is the output of the previous fct,
# sl2 an sl2-triple...
# eps is -1 if L is simp, 1 of l=o_n...
local n, K, V, M, i, j, phi, dV, dims, d0, vv0, v0, u0, w0, bass0,
bas, BV0, gens, s, ms, psi, found, wn, eqn, sol, ww, Bww, g,
imgs, mat, l;
n:= Length( GeneratorsOfAlgebra(L)[1]![1] );
K:= SubalgebraNC( L, sl2 );
V:= LeftAlgebraModule( K, function(x,v) return x![1]*v; end,
LeftActingDomain(L)^n );
M:= NullMat( n, n );
if eps=1 then
for i in [1..n] do M[i][n+1-i]:=1; od;
else
l:= n/2;
for i in [1..l] do
M[i][l+i]:=1;
M[l+i][i]:= -1;
od;
fi;
phi:= function(v,w) #v,w in V
return v![1]*M*w![1];
end;
dV:= DirectSumDecomposition(V);
dims:= List( dV, Dimension );
if eps=1 then
d0:= Filtered( Set( dims ), IsOddInt );
else
d0:= Filtered( Set( dims ), IsEvenInt );
fi;
vv0:= [ ];
for i in [1..Length(dV)] do
v0:= Basis( dV[i] )[dims[i]];
u0:= sl2[1]^v0;
while not IsZero(u0) do
v0:= u0;
u0:= sl2[1]^v0;
od;
Add( vv0, v0 );
od;
bass0:= [ ];
for i in [1..Length(dV)] do
bas:=[];
u0:= vv0[i];
while not IsZero(u0) do
Add(bas,u0);
u0:= sl2[3]^u0;
od;
Add( bass0, bas );
od;
BV0:= Basis( V, Concatenation(bass0) );
gens:= [ ];
for s in d0 do
ms:= [ ];
for i in [1..Length(dV)] do
if dims[i] = s then
Add( ms, vv0[i] );
fi;
od;
psi:= function( v, w )
local i0, w0;
w0:= w;
for i0 in [1..s-1] do
w0:= sl2[3]^w0;
od;
return phi( v, w0 );
end;
found:= false;
for i in [1..Length(ms)] do
if not IsZero( psi( ms[i], ms[i] ) ) then
wn:= ms[i]; found:= true; break;
fi;
od;
for i in [1..Length(ms)] do
if found then break; fi;
for j in [i+1..Length(ms)] do
if not IsZero( psi( ms[i], ms[j] ) ) then
wn:= ms[i]+ms[j]; found:= true; break;
fi;
od;
od;
eqn:= [ List( ms, v -> psi( v, wn ) ) ];
sol:= NullspaceMat( TransposedMat(eqn) );
ww:= List( sol, q -> q*ms );
Add( ww, wn );
Bww:= Basis( Subspace( V, ww ), ww );
g:= function( u ) # u in the span of ms..
local cff;
cff:= ShallowCopy( Coefficients( Bww, u ) );
cff[ Length(cff) ]:= -cff[Length(cff)];
return cff*ww;
end;
imgs:= [ ];
for i in [1..Length(dV)] do
if dims[i] = s then
u0:= g(bass0[i][1]);
while not IsZero(u0) do
Add( imgs, u0 );
u0:= sl2[3]^u0;
od;
else
Append( imgs, bass0[i] );
fi;
od;
mat:= [ ];
for i in [1..Length(Basis(V))] do
u0:= Coefficients( BV0, Basis(V)[i] )*imgs;
Add( mat, Coefficients( Basis(V), u0 ) );
od;
Add( gens, TransposedMat(mat) );
od;
return gens;
end;
SLAfcts.compgrpB_D:= function( o )
local ct, n, M, f, f1, sl2, gg, elms, mats, g, mat, mats0, CC, ad, Ad,
u, L, p, odd, parts, mults, j, a, b, size, id, CG, even, eps,
r, bound;
# L simple of type B-D
# o is a nilpotent orbit in L
L:= AmbientLieAlgebra(o);
id:= IdentityMat( Dimension(L), LeftActingDomain(L) );
if IsBound( L!.compgrpData ) then
r:= L!.compgrpData;
M:= r.liealg;
f:= r.isomLM;
f1:= r.isomML;
ct:= r.types;
bound:= true;
else
ct:= CartanType( CartanMatrix( RootSystem(L) ) ).types;
r:= rec(types:= ct);
bound:= false;
fi;
if ct[1][1] = "B" then
n:= 2*ct[1][2]+1;
p:= OrbitPartition(o);
odd:= Filtered( p, IsOddInt );
a:= Length(Set(odd));
size:= 2^(a-1);
if not bound then
M:= SLAfcts.orthLA( LeftActingDomain(L), n );
r.liealg:= M;
fi;
eps:=1;
elif ct[1][1] = "C" then
n:= 2*ct[1][2];
p:= OrbitPartition(o);
even:= Filtered( p, IsEvenInt );
if Length(even)=0 then
size:=1;
else
parts:= [ even[1] ];
mults:= [ 1 ];
for j in [2..Length(even)] do
if even[j] = parts[ Length(parts) ] then
mults[Length(mults)]:= mults[Length(mults)]+1;
else
Add( parts, even[j] );
Add( mults, 1 );
fi;
od;
a:= Length(parts);
if ForAll( mults, IsEvenInt ) then
b:= a;
else
b:= a-1;
fi;
size:= 2^b;
fi;
if not bound then
M:= SLAfcts.simpLA( LeftActingDomain(L), n );
r.liealg:= M;
fi;
eps:=-1;
elif ct[1][1] = "D" then
n:= 2*ct[1][2];
p:= OrbitPartition(o);
odd:= Filtered( p, IsOddInt );
if Length(odd)=0 then
size:=1;
else
parts:= [ odd[1] ];
mults:= [ 1 ];
for j in [2..Length(odd)] do
if odd[j] = parts[ Length(parts) ] then
mults[Length(mults)]:= mults[Length(mults)]+1;
else
Add( parts, odd[j] );
Add( mults, 1 );
fi;
od;
a:= Length(parts);
if ForAll( mults, IsEvenInt ) then
b:= Maximum(0,a-1);
else
b:= Maximum(0,a-2);
fi;
size:= 2^b;
fi;
if not bound then
M:= SLAfcts.orthLA( LeftActingDomain(L), n );
r.liealg:= M;
fi;
eps:=1;
fi;
if size = 1 then return Group([id]); fi;
if not bound then
f:= IsomorphismOfSemisimpleLieAlgebras( L, M );
f1:= IsomorphismOfSemisimpleLieAlgebras( M, L );
r.isomLM:= f;
r.isomML:= f1;
L!.compgrpData:= r;
fi;
sl2:= List( SL2Triple(o), x -> Image( f, x ) );
gg:= SLAfcts.compgrp( M, sl2, eps );
elms:= Filtered( Elements( Group(gg) ), x -> DeterminantMat(x)=1 );
mats:= [ ];
for g in elms do
mat:= List( Basis(L), x -> Coefficients( Basis(L),
Image( f1, g*Image(f,x)*g^-1 ) ) );
Add( mats, TransposedMat(mat) );
od;
mats:= Set(mats);
CG:= Group(mats);
if Length(Elements(CG)) = size then
return CG;
else
mats0:= [ ];
CC:= LieCentralizer( L, Subalgebra(L,SL2Triple(o)) );
if Dimension(CC) > 0 then
for u in mats do
if SLAfcts.iseltof( L, CC, u ) then
Add( mats0, u );
fi;
od;
# so mats0 is the normal subgroup of elements in the idcomp
# we need to take coset reps
mats:= RightTransversal( Group(mats), Group(mats0) );
fi;
if Length(mats)=0 then
return Group( [id]);
else
return Group(mats);
fi;
fi;
end;
SLAfcts.compgrps:= [
[ "G", 2, [ [0,2],
[ 14, [ 1, 1, -1 ], [ 2, 2, 1 ], [ 3, 3, -1 ], [ 4, 4, 1 ], [ 5, 5, -1 ],
[ 6, 6, -1 ], [ 7, 7, -1 ], [ 8, 8, 1 ], [ 9, 9, -1 ], [ 10, 10, 1 ],
[ 11, 11, -1 ], [ 12, 12, -1 ], [ 13, 13, 1 ], [ 14, 14, 1 ] ],
[ 14, [ 1, 1, 1/4 ], [ 1, 7, 3/4 ], [ 1, 13, -1/2*E(3)+1/2*E(3)^2 ],
[ 1, 14, 1/4*E(3)-1/4*E(3)^2 ], [ 2, 2, -1/8 ],
[ 2, 3, 3/8*E(3)-3/8*E(3)^2 ], [ 2, 4, 9/8 ],
[ 2, 5, -3/8*E(3)+3/8*E(3)^2 ], [ 3, 2, 1/8*E(3)-1/8*E(3)^2 ],
[ 3, 3, 5/8 ], [ 3, 4, -1/8*E(3)+1/8*E(3)^2 ], [ 3, 5, 3/8 ],
[ 4, 2, 3/8 ], [ 4, 3, -1/8*E(3)+1/8*E(3)^2 ], [ 4, 4, 5/8 ],
[ 4, 5, 1/8*E(3)-1/8*E(3)^2 ], [ 5, 2, -3/8*E(3)+3/8*E(3)^2 ],
[ 5, 3, 9/8 ], [ 5, 4, 3/8*E(3)-3/8*E(3)^2 ], [ 5, 5, -1/8 ], [ 6, 6, 1 ],
[ 7, 1, 3/4 ], [ 7, 7, 1/4 ], [ 7, 13, 1/2*E(3)-1/2*E(3)^2 ],
[ 7, 14, -1/4*E(3)+1/4*E(3)^2 ], [ 8, 8, -1/8 ],
[ 8, 9, -3/8*E(3)+3/8*E(3)^2 ], [ 8, 10, 9/8 ],
[ 8, 11, 3/8*E(3)-3/8*E(3)^2 ], [ 9, 8, -1/8*E(3)+1/8*E(3)^2 ],
[ 9, 9, 5/8 ], [ 9, 10, 1/8*E(3)-1/8*E(3)^2 ], [ 9, 11, 3/8 ],
[ 10, 8, 3/8 ], [ 10, 9, 1/8*E(3)-1/8*E(3)^2 ], [ 10, 10, 5/8 ],
[ 10, 11, -1/8*E(3)+1/8*E(3)^2 ], [ 11, 8, 3/8*E(3)-3/8*E(3)^2 ],
[ 11, 9, 9/8 ], [ 11, 10, -3/8*E(3)+3/8*E(3)^2 ], [ 11, 11, -1/8 ],
[ 12, 12, 1 ], [ 13, 1, -1/4*E(3)+1/4*E(3)^2 ],
[ 13, 7, 1/4*E(3)-1/4*E(3)^2 ], [ 13, 13, -1/2 ], [ 13, 14, 3/4 ],
[ 14, 14, 1 ] ] ] ],
[ "F", 4, [ [ 1, 0, 0, 0 ],
[ 52, [ 1, 5, 1 ], [ 2, 2, 1 ], [ 3, 27, 1 ], [ 4, 10, 1 ], [ 5, 1, 1 ],
[ 6, 13, 1 ], [ 7, 7, 1 ], [ 8, 12, 1 ], [ 9, 9, 1 ], [ 10, 4, 1 ],
[ 11, 14, 1 ], [ 12, 8, 1 ], [ 13, 6, 1 ], [ 14, 11, 1 ], [ 15, 15, 1 ],
[ 16, 16, 1 ], [ 17, 19, -1 ], [ 18, 18, 1 ], [ 19, 17, -1 ],
[ 20, 22, 1 ], [ 21, 21, 1 ], [ 22, 20, 1 ], [ 23, 23, 1 ], [ 24, 24, 1 ],
[ 25, 29, 1 ], [ 26, 26, 1 ], [ 27, 3, 1 ], [ 28, 34, 1 ], [ 29, 25, 1 ],
[ 30, 37, 1 ], [ 31, 31, 1 ], [ 32, 36, 1 ], [ 33, 33, 1 ], [ 34, 28, 1 ],
[ 35, 38, 1 ], [ 36, 32, 1 ], [ 37, 30, 1 ], [ 38, 35, 1 ], [ 39, 39, 1 ],
[ 40, 40, 1 ], [ 41, 43, -1 ], [ 42, 42, 1 ], [ 43, 41, -1 ],
[ 44, 46, 1 ], [ 45, 45, 1 ], [ 46, 44, 1 ], [ 47, 47, 1 ], [ 48, 48, 1 ],
[ 49, 49, 1 ], [ 50, 50, 1 ], [ 51, 49, 1 ], [ 51, 51, -1 ], [ 51, 52, 1 ],
[ 52, 52, 1 ] ] ],
[ [ 0, 2, 0, 0 ],
[ 52, [ 1, 29, -E(3)^2 ], [ 2, 23, E(3) ], [ 3, 3, -1 ], [ 4, 34, E(3) ],
[ 5, 25, -E(3)^2 ], [ 6, 20, E(3)^2 ], [ 7, 31, E(3) ], [ 8, 36, 1 ],
[ 9, 21, -E(3)^2 ], [ 10, 28, E(3) ], [ 11, 17, -E(3) ], [ 12, 32, 1 ],
[ 13, 22, E(3)^2 ], [ 14, 19, E(3) ], [ 15, 39, E(3)^2 ], [ 16, 18, 1 ],
[ 17, 11, -E(3)^2 ], [ 18, 16, 1 ], [ 19, 14, E(3)^2 ], [ 20, 6, E(3) ],
[ 21, 9, -E(3) ], [ 22, 13, E(3) ], [ 23, 2, E(3)^2 ], [ 24, 24, -1 ],
[ 25, 5, -E(3) ], [ 26, 47, E(3)^2 ], [ 27, 27, -1 ], [ 28, 10, E(3)^2 ],
[ 29, 1, -E(3) ], [ 30, 44, E(3) ], [ 31, 7, E(3)^2 ], [ 32, 12, 1 ],
[ 33, 45, -E(3) ], [ 34, 4, E(3)^2 ], [ 35, 41, -E(3)^2 ], [ 36, 8, 1 ],
[ 37, 46, E(3) ], [ 38, 43, E(3)^2 ], [ 39, 15, E(3) ], [ 40, 42, 1 ],
[ 41, 35, -E(3) ], [ 42, 40, 1 ], [ 43, 38, E(3) ], [ 44, 30, E(3)^2 ],
[ 45, 33, -E(3)^2 ], [ 46, 37, E(3)^2 ], [ 47, 26, E(3) ], [ 48, 48, -1 ],
[ 49, 49, -1 ], [ 49, 50, 1 ], [ 50, 50, 1 ], [ 51, 49, -1 ],
[ 51, 50, 2 ], [ 51, 51, 1 ], [ 51, 52, -1 ], [ 52, 50, 3 ],
[ 52, 52, -1 ] ] ],
[ [ 1, 2, 0, 0 ],
[ 52, [ 1, 8, -E(4) ], [ 2, 16, -1 ], [ 3, 3, 1 ], [ 4, 34, 1 ],
[ 5, 12, E(4) ], [ 6, 6, 1 ], [ 7, 31, -1 ], [ 8, 1, E(4) ], [ 9, 9, 1 ],
[ 10, 28, 1 ], [ 11, 17, E(4) ], [ 12, 5, -E(4) ], [ 13, 13, 1 ],
[ 14, 19, E(4) ], [ 15, 15, 1 ], [ 16, 2, -1 ], [ 17, 11, -E(4) ],
[ 18, 23, -1 ], [ 19, 14, -E(4) ], [ 20, 20, 1 ], [ 21, 21, 1 ],
[ 22, 22, 1 ], [ 23, 18, -1 ], [ 24, 24, 1 ], [ 25, 32, E(4) ],
[ 26, 40, -1 ], [ 27, 27, 1 ], [ 28, 10, 1 ], [ 29, 36, -E(4) ],
[ 30, 30, 1 ], [ 31, 7, -1 ], [ 32, 25, -E(4) ], [ 33, 33, 1 ],
[ 34, 4, 1 ], [ 35, 41, -E(4) ], [ 36, 29, E(4) ], [ 37, 37, 1 ],
[ 38, 43, -E(4) ], [ 39, 39, 1 ], [ 40, 26, -1 ], [ 41, 35, E(4) ],
[ 42, 47, -1 ], [ 43, 38, E(4) ], [ 44, 44, 1 ], [ 45, 45, 1 ],
[ 46, 46, 1 ], [ 47, 42, -1 ], [ 48, 48, 1 ], [ 49, 49, 1 ],
[ 50, 50, 1 ], [ 51, 49, 1 ], [ 51, 50, 1 ], [ 51, 51, 1 ],
[ 51, 52, -1 ], [ 52, 49, 2 ], [ 52, 50, 2 ], [ 52, 52, -1 ] ] ],
[ [ 0, 1, 1, 0 ],
[ 52, [ 1, 1, -1 ], [ 2, 2, 1 ], [ 3, 3, -1 ], [ 4, 4, 1 ], [ 5, 5, 1 ],
[ 6, 6, 1 ], [ 7, 7, -1 ], [ 8, 8, 1 ], [ 9, 9, -1 ], [ 10, 10, 1 ],
[ 11, 11, 1 ], [ 12, 12, -1 ], [ 13, 13, 1 ], [ 14, 14, -1 ],
[ 15, 15, 1 ], [ 16, 16, 1 ], [ 17, 17, -1 ], [ 18, 18, 1 ],
[ 19, 19, 1 ], [ 20, 20, 1 ], [ 21, 21, -1 ], [ 22, 22, 1 ],
[ 23, 23, 1 ], [ 24, 24, 1 ], [ 25, 25, -1 ], [ 26, 26, 1 ],
[ 27, 27, -1 ], [ 28, 28, 1 ], [ 29, 29, 1 ], [ 30, 30, 1 ],
[ 31, 31, -1 ], [ 32, 32, 1 ], [ 33, 33, -1 ], [ 34, 34, 1 ],
[ 35, 35, 1 ], [ 36, 36, -1 ], [ 37, 37, 1 ], [ 38, 38, -1 ],
[ 39, 39, 1 ], [ 40, 40, 1 ], [ 41, 41, -1 ], [ 42, 42, 1 ],
[ 43, 43, 1 ], [ 44, 44, 1 ], [ 45, 45, -1 ], [ 46, 46, 1 ],
[ 47, 47, 1 ], [ 48, 48, 1 ], [ 49, 49, 1 ], [ 50, 50, 1 ], [ 51, 51, 1 ],
[ 52, 52, 1 ] ] ],
[ [ 2, 0, 0, 2 ],
[ 52, [ 1, 1, -1 ], [ 2, 2, -1 ], [ 3, 3, -1 ], [ 4, 4, -1 ], [ 5, 5, 1 ],
[ 6, 6, 1 ], [ 7, 7, 1 ], [ 8, 8, -1 ], [ 9, 9, -1 ], [ 10, 10, -1 ],
[ 11, 11, 1 ], [ 12, 12, 1 ], [ 13, 13, 1 ], [ 14, 14, -1 ],
[ 15, 15, -1 ], [ 16, 16, -1 ], [ 17, 17, 1 ], [ 18, 18, 1 ],
[ 19, 19, -1 ], [ 20, 20, -1 ], [ 21, 21, 1 ], [ 22, 22, -1 ],
[ 23, 23, 1 ], [ 24, 24, -1 ], [ 25, 25, -1 ], [ 26, 26, -1 ],
[ 27, 27, -1 ], [ 28, 28, -1 ], [ 29, 29, 1 ], [ 30, 30, 1 ],
[ 31, 31, 1 ], [ 32, 32, -1 ], [ 33, 33, -1 ], [ 34, 34, -1 ],
[ 35, 35, 1 ], [ 36, 36, 1 ], [ 37, 37, 1 ], [ 38, 38, -1 ],
[ 39, 39, -1 ], [ 40, 40, -1 ], [ 41, 41, 1 ], [ 42, 42, 1 ],
[ 43, 43, -1 ], [ 44, 44, -1 ], [ 45, 45, 1 ], [ 46, 46, -1 ],
[ 47, 47, 1 ], [ 48, 48, -1 ], [ 49, 49, 1 ], [ 50, 50, 1 ],
[ 51, 51, 1 ], [ 52, 52, 1 ] ] ],
[ [ 2, 2, 0, 2 ],
[ 52, [ 1, 1, -1 ], [ 2, 2, 1 ], [ 3, 3, -1 ], [ 4, 4, 1 ], [ 5, 5, 1 ],
[ 6, 6, 1 ], [ 7, 7, -1 ], [ 8, 8, 1 ], [ 9, 9, -1 ], [ 10, 10, 1 ],
[ 11, 11, 1 ], [ 12, 12, -1 ], [ 13, 13, 1 ], [ 14, 14, -1 ],
[ 15, 15, 1 ], [ 16, 16, 1 ], [ 17, 17, -1 ], [ 18, 18, 1 ],
[ 19, 19, 1 ], [ 20, 20, 1 ], [ 21, 21, -1 ], [ 22, 22, 1 ],
[ 23, 23, 1 ], [ 24, 24, 1 ], [ 25, 25, -1 ], [ 26, 26, 1 ],
[ 27, 27, -1 ], [ 28, 28, 1 ], [ 29, 29, 1 ], [ 30, 30, 1 ],
[ 31, 31, -1 ], [ 32, 32, 1 ], [ 33, 33, -1 ], [ 34, 34, 1 ],
[ 35, 35, 1 ], [ 36, 36, -1 ], [ 37, 37, 1 ], [ 38, 38, -1 ],
[ 39, 39, 1 ], [ 40, 40, 1 ], [ 41, 41, -1 ], [ 42, 42, 1 ],
[ 43, 43, 1 ], [ 44, 44, 1 ], [ 45, 45, -1 ], [ 46, 46, 1 ],
[ 47, 47, 1 ], [ 48, 48, 1 ], [ 49, 49, 1 ], [ 50, 50, 1 ], [ 51, 51, 1 ],
[ 52, 52, 1 ] ] ],
[ [ 0, 0, 0, 2 ],
[ 52, [ 1, 1, E(3) ], [ 2, 2, E(3) ], [ 3, 3, E(3) ], [ 4, 4, E(3) ],
[ 5, 5, E(3)^2 ], [ 6, 6, E(3)^2 ], [ 7, 7, E(3)^2 ], [ 8, 8, 1 ],
[ 9, 9, 1 ], [ 10, 10, 1 ], [ 11, 11, E(3) ], [ 12, 12, E(3) ],
[ 13, 13, E(3) ], [ 14, 14, E(3)^2 ], [ 15, 15, E(3)^2 ],
[ 16, 16, E(3)^2 ], [ 17, 17, 1 ], [ 18, 18, 1 ], [ 19, 19, E(3) ],
[ 20, 20, E(3) ], [ 21, 21, E(3)^2 ], [ 22, 22, 1 ], [ 23, 23, E(3) ],
[ 24, 24, E(3)^2 ], [ 25, 25, E(3)^2 ], [ 26, 26, E(3)^2 ],
[ 27, 27, E(3)^2 ], [ 28, 28, E(3)^2 ], [ 29, 29, E(3) ], [ 30, 30, E(3) ],
[ 31, 31, E(3) ], [ 32, 32, 1 ], [ 33, 33, 1 ], [ 34, 34, 1 ],
[ 35, 35, E(3)^2 ], [ 36, 36, E(3)^2 ], [ 37, 37, E(3)^2 ],
[ 38, 38, E(3) ], [ 39, 39, E(3) ], [ 40, 40, E(3) ], [ 41, 41, 1 ],
[ 42, 42, 1 ], [ 43, 43, E(3)^2 ], [ 44, 44, E(3)^2 ], [ 45, 45, E(3) ],
[ 46, 46, 1 ], [ 47, 47, E(3)^2 ], [ 48, 48, E(3) ], [ 49, 49, 1 ],
[ 50, 50, 1 ], [ 51, 51, 1 ], [ 52, 52, 1 ] ],
[ 52, [ 1, 1, 1/9 ], [ 1, 3, 4/9 ], [ 1, 5, -2/9 ], [ 1, 25, 4/9 ],
[ 1, 27, -2/9 ], [ 1, 29, 4/9 ], [ 1, 51, 2/3 ], [ 1, 52, -4/9 ],
[ 2, 2, 1 ], [ 3, 1, 4/9 ], [ 3, 3, 1/9 ], [ 3, 5, -2/9 ], [ 3, 25, -2/9 ],
[ 3, 27, 4/9 ], [ 3, 29, 4/9 ], [ 3, 49, -2/3 ], [ 3, 52, 2/9 ],
[ 4, 4, 1/9 ], [ 4, 7, -4/9 ], [ 4, 8, -4/9 ], [ 4, 10, 4/9 ],
[ 4, 12, 8/9 ], [ 4, 15, 4/9 ], [ 5, 1, -2/9 ], [ 5, 3, -2/9 ],
[ 5, 5, 1/9 ], [ 5, 25, 4/9 ], [ 5, 27, 4/9 ], [ 5, 29, 4/9 ],
[ 5, 49, 2/3 ], [ 5, 51, -2/3 ], [ 5, 52, 2/9 ], [ 6, 6, 1/9 ],
[ 6, 9, -4/9 ], [ 6, 11, -4/9 ], [ 6, 13, 4/9 ], [ 6, 14, 8/9 ],
[ 6, 18, 4/9 ], [ 7, 4, -2/9 ], [ 7, 7, 5/9 ], [ 7, 8, 2/9 ],
[ 7, 10, -2/9 ], [ 7, 12, 2/9 ], [ 7, 15, 4/9 ], [ 8, 4, -2/9 ],
[ 8, 7, 2/9 ], [ 8, 8, 5/9 ], [ 8, 10, 4/9 ], [ 8, 12, 2/9 ],
[ 8, 15, -2/9 ], [ 9, 6, -2/9 ], [ 9, 9, 5/9 ], [ 9, 11, 2/9 ],
[ 9, 13, -2/9 ], [ 9, 14, 2/9 ], [ 9, 18, 4/9 ], [ 10, 4, 4/9 ],
[ 10, 7, -4/9 ], [ 10, 8, 8/9 ], [ 10, 10, 1/9 ], [ 10, 12, -4/9 ],
[ 10, 15, 4/9 ], [ 11, 6, -2/9 ], [ 11, 9, 2/9 ], [ 11, 11, 5/9 ],
[ 11, 13, 4/9 ], [ 11, 14, 2/9 ], [ 11, 18, -2/9 ], [ 12, 4, 4/9 ],
[ 12, 7, 2/9 ], [ 12, 8, 2/9 ], [ 12, 10, -2/9 ], [ 12, 12, 5/9 ],
[ 12, 15, -2/9 ], [ 13, 6, 4/9 ], [ 13, 9, -4/9 ], [ 13, 11, 8/9 ],
[ 13, 13, 1/9 ], [ 13, 14, -4/9 ], [ 13, 18, 4/9 ], [ 14, 6, 4/9 ],
[ 14, 9, 2/9 ], [ 14, 11, 2/9 ], [ 14, 13, -2/9 ], [ 14, 14, 5/9 ],
[ 14, 18, -2/9 ], [ 15, 4, 4/9 ], [ 15, 7, 8/9 ], [ 15, 8, -4/9 ],
[ 15, 10, 4/9 ], [ 15, 12, -4/9 ], [ 15, 15, 1/9 ], [ 16, 16, 1/9 ],
[ 16, 17, 4/9 ], [ 16, 19, -4/9 ], [ 16, 20, 4/9 ], [ 16, 21, -8/9 ],
[ 16, 22, 4/9 ], [ 17, 16, 2/9 ], [ 17, 17, 5/9 ], [ 17, 19, -2/9 ],
[ 17, 20, 2/9 ], [ 17, 21, 2/9 ], [ 17, 22, -4/9 ], [ 18, 6, 4/9 ],
[ 18, 9, 8/9 ], [ 18, 11, -4/9 ], [ 18, 13, 4/9 ], [ 18, 14, -4/9 ],
[ 18, 18, 1/9 ], [ 19, 16, -2/9 ], [ 19, 17, -2/9 ], [ 19, 19, 5/9 ],
[ 19, 20, 4/9 ], [ 19, 21, -2/9 ], [ 19, 22, -2/9 ], [ 20, 16, 4/9 ],
[ 20, 17, 4/9 ], [ 20, 19, 8/9 ], [ 20, 20, 1/9 ], [ 20, 21, 4/9 ],
[ 20, 22, 4/9 ], [ 21, 16, -4/9 ], [ 21, 17, 2/9 ], [ 21, 19, -2/9 ],
[ 21, 20, 2/9 ], [ 21, 21, 5/9 ], [ 21, 22, 2/9 ], [ 22, 16, 4/9 ],
[ 22, 17, -8/9 ], [ 22, 19, -4/9 ], [ 22, 20, 4/9 ], [ 22, 21, 4/9 ],
[ 22, 22, 1/9 ], [ 23, 23, 1 ], [ 24, 24, 1 ], [ 25, 1, 4/9 ],
[ 25, 3, -2/9 ], [ 25, 5, 4/9 ], [ 25, 25, 1/9 ], [ 25, 27, 4/9 ],
[ 25, 29, -2/9 ], [ 25, 51, 2/3 ], [ 25, 52, -4/9 ], [ 26, 26, 1 ],
[ 27, 1, -2/9 ], [ 27, 3, 4/9 ], [ 27, 5, 4/9 ], [ 27, 25, 4/9 ],
[ 27, 27, 1/9 ], [ 27, 29, -2/9 ], [ 27, 49, -2/3 ], [ 27, 52, 2/9 ],
[ 28, 28, 1/9 ], [ 28, 31, -4/9 ], [ 28, 32, -4/9 ], [ 28, 34, 4/9 ],
[ 28, 36, 8/9 ], [ 28, 39, 4/9 ], [ 29, 1, 4/9 ], [ 29, 3, 4/9 ],
[ 29, 5, 4/9 ], [ 29, 25, -2/9 ], [ 29, 27, -2/9 ], [ 29, 29, 1/9 ],
[ 29, 49, 2/3 ], [ 29, 51, -2/3 ], [ 29, 52, 2/9 ], [ 30, 30, 1/9 ],
[ 30, 33, -4/9 ], [ 30, 35, -4/9 ], [ 30, 37, 4/9 ], [ 30, 38, 8/9 ],
[ 30, 42, 4/9 ], [ 31, 28, -2/9 ], [ 31, 31, 5/9 ], [ 31, 32, 2/9 ],
[ 31, 34, -2/9 ], [ 31, 36, 2/9 ], [ 31, 39, 4/9 ], [ 32, 28, -2/9 ],
[ 32, 31, 2/9 ], [ 32, 32, 5/9 ], [ 32, 34, 4/9 ], [ 32, 36, 2/9 ],
[ 32, 39, -2/9 ], [ 33, 30, -2/9 ], [ 33, 33, 5/9 ], [ 33, 35, 2/9 ],
[ 33, 37, -2/9 ], [ 33, 38, 2/9 ], [ 33, 42, 4/9 ], [ 34, 28, 4/9 ],
[ 34, 31, -4/9 ], [ 34, 32, 8/9 ], [ 34, 34, 1/9 ], [ 34, 36, -4/9 ],
[ 34, 39, 4/9 ], [ 35, 30, -2/9 ], [ 35, 33, 2/9 ], [ 35, 35, 5/9 ],
[ 35, 37, 4/9 ], [ 35, 38, 2/9 ], [ 35, 42, -2/9 ], [ 36, 28, 4/9 ],
[ 36, 31, 2/9 ], [ 36, 32, 2/9 ], [ 36, 34, -2/9 ], [ 36, 36, 5/9 ],
[ 36, 39, -2/9 ], [ 37, 30, 4/9 ], [ 37, 33, -4/9 ], [ 37, 35, 8/9 ],
[ 37, 37, 1/9 ], [ 37, 38, -4/9 ], [ 37, 42, 4/9 ], [ 38, 30, 4/9 ],
[ 38, 33, 2/9 ], [ 38, 35, 2/9 ], [ 38, 37, -2/9 ], [ 38, 38, 5/9 ],
[ 38, 42, -2/9 ], [ 39, 28, 4/9 ], [ 39, 31, 8/9 ], [ 39, 32, -4/9 ],
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