| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/sla/gap/ (GAP Algebra Version 4.15.1©) Datei vom 12.6.2024 mit Größe 750 kB |
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Impressum nilporb.gi Sprache: unbekannt |
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Spracherkennung für: .gi vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] 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 weighte 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 ] ], [ [ 97, 2, 98, 2, 99, 2, 106, 2 ], [ 127, 4, 128, 6, 129, 8, 130, 12, 131, 10, 132, 8, 133, 4 ], [ 34, 1, 35, 1, 36, 1, 43, 1 ] ], [ [ 100, 1, 101, 1, 102, 2, 103, 2, 104, 1 ], [ 127, 4, 128, 7, 129, 8, 130, 12, 131, 9, 132, 6, 133, 3 ], [ 37, 1, 38, 1, 39, 1, 40, 1, 41, 1 ] ], [ [ 90, 3, 93, 4, 94, 1, 100, 3 ], [ 127, 6, 128, 7, 129, 10, 130, 14, 131, 11, 132, 8, 133, 5 ], [ 27, 1, 30, 1, 31, 1, 37, 1 ] ], [ [ 97, 2, 98, 2, 99, 2, 100, 1, 106, 2 ], [ 127, 5, 128, 7, 129, 10, 130, 14, 131, 11, 132, 8, 133, 4 ], [ 34, 1, 35, 1, 36, 1, 37, 1, 43, 1 ] ], [ [ 90, 3, 91, 4, 102, 3, 112, 1 ], [ 127, 6, 128, 8, 129, 11, 130, 16, 131, 12, 132, 8, 133, 4 ], [ 27, 1, 28, 1, 39, 1, 49, 1 ] ], [ [ 90, 1, 92, 3, 94, 3, 95, 4, 102, 1 ], [ 127, 6, 128, 8, 129, 12, 130, 16, 131, 12, 132, 8, 133, 4 ], [ 27, 1, 29, 1, 31, 1, 32, 1, 39, 1 ] ], [ [ 90, 3, 93, 4, 94, 1, 100, 3, 103, 1 ], [ 127, 6, 128, 8, 129, 11, 130, 16, 131, 13, 132, 9, 133, 5 ], [ 27, 1, 30, 1, 31, 1, 37, 1, 40, 1 ] ], [ [ 64, 10, 91, 6, 92, 6, 94, 6 ], [ 127, 10, 128, 12, 129, 18, 130, 24, 131, 18, 132, 12, 133, 6 ], [ 1, 1, 28, 1, 29, 1, 31, 1 ] ], [ [ 85, 1, 93, 1, 94, 4, 95, 3, 96, 3, 98, 1 ], [ 127, 6, 128, 9, 129, 12, 130, 17, 131, 13, 132, 9, 133, 5 ], [ 22, 1, 30, 1, 31, 1, 32, 1, 33, 1, 35, 1 ] ], [ [ 90, 2, 92, 3, 93, 2, 94, 3, 95, 4 ], [ 127, 6, 128, 9, 129, 12, 130, 18, 131, 14, 132, 10, 133, 5 ], [ 27, 1, 29, 1, 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155, 18, 157, 9, 158, 11, 159, -9 ], [ 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 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