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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 ],
[ 20, 1, 21, 1, 49, 1 ] ],
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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 ],
--> --------------------
--> maximum size reached
--> --------------------
[ Seitenstruktur0.80Drucken
etwas mehr zur Ethik
]
|
2026-04-02
|
