  
  [1X4 [33X[0;0YNilpotent Orbits[133X[101X
  
  [33X[0;0YThis  chapter  contains functions for dealing with the nilpotent orbits of a
  semisimple  Lie algebra [22XK[122X under its adjoint group [22XG[122X. We refer to the book by
  Collingwood and McGovern, [CM93] (and the references therein) for an account
  of the theory of nilpotent orbits.[133X
  
  [33X[0;0YA nilpotent orbit has two important attributes: the weighted Dynkin diagram,
  and  an [22Xsl_2[122X-triple. The weighted Dynkin diagram is represented by a list of
  integers in {0,1,2} of length equal to the rank of the Lie algebra. The i-th
  position  in this list corresponds to the i-th node of the Dynkin diagram of
  the  root  system. The Dynkin diagram of the root system is described by the
  Cartan  matrix  of  the  root  system.  Now in [5XGAP[105X this Cartan matrix can be
  somewhat  different  from the more usual forms. This holds most particularly
  for  type  F4, where the enumeration of the simple roots is rather different
  from  the one usually found. So when using the functions in this chapter one
  should keep this in mind.[133X
  
  [33X[0;0YEvery  nilpotent  orbit  has  an  [22Xsl_2[122X-triple,  that is, a triple [22X(y,h,x)[122X of
  elements  of  the  simple Lie algebra with [22X[x,y]=h[122X, [22X[h,x]=2x[122X, [22X[h,y]=-2y[122X. The
  nilpotent  orbit  corresponding  to this is the orbit of the element [22Xx[122X under
  the action of the adjoint group.[133X
  
  [33X[0;0YLet  [22XP[122X  be  a  parabolic  subalgebra  of  [22XK[122X  (i.e.,  generated by the Cartan
  subalgebra  of  [22XK[122X, all positive root vectors, along with the negative simple
  root  vectors corresponding to a given subset of the basis of simple roots),
  [22XL[122X  the  corresponding Levi subalgebra (i.e., the reductive part of [22XP[122X), and [22XN[122X
  the  nilradical  of  [22XP[122X.  Let  [22XO_L[122X  be a nilpotent orbit in [22XL[122X. There exists a
  unique  nilpotent orbit [22XO_K[122X in [22XK[122X such that the intersection of [22XO_K[122X and [22XO_L +
  N[122X  is  dense in the latter. In this situation [22XO_K[122X is said to be [13Xinduced[113X from
  [22XO_L[122X. Nilpotent orbits in [22XK[122X which are not induced are said to be [13Xrigid[113X.[133X
  
  [33X[0;0YNow  consider  the  variety of all [22XG[122X-orbits in [22XK[122X of a given dimension [22Xd[122X. The
  irreducible  components  of  this  variety are called the [13Xsheets[113X of [22XK[122X. Every
  sheet  has  a  unique  nilpotent  orbit.  Moreover,  this nilpotent orbit is
  induced  from  an  orbit  [22XO_L[122X,  and  [22XO_L[122X  is  rigid  in [22XL[122X. So the sheets are
  parametrised by pairs [22X(L,O_L)[122X, where [22XL[122X is a Levi subalgebra, and [22XO_L[122X a rigid
  nilpotent  orbit  in  it.  This  data  can  conveniently be given by a [13Xsheet
  diagram[113X:  this  is  the  Dynkin  diagram  of  [22XK[122X, where the nodes that do [13Xnot[113X
  correspond to simple roots of [22XL[122X have label 2. So, leaving out the nodes with
  label  2,  one  obtains the Dynkin diagram of [22XL[122X. The remaining labels in the
  sheet  diagram  then  correspond  to  the  weighted  Dynkin  diagram  of the
  nilpotent  orbit [22XO_L[122X. Since this orbit is rigid, its weighted Dynkin diagram
  has  labels 0 or 1. From that it follows that one can recover [22XL[122X and [22XO_L[122X from
  the  sheet  diagram.  The [13Xrank[113X of a sheet is defined as the dimension of the
  centre  of  [22XL[122X;  obviously  that  is  equal to the number of 2's in the sheet
  diagram.[133X
  
  
  [1X4.1 [33X[0;0YThe functions[133X[101X
  
  [1X4.1-1 NilpotentOrbit[101X
  
  [33X[1;0Y[29X[2XNilpotentOrbit[102X( [3XL[103X, [3Xwd[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  simple Lie algebra and [3Xwd[103X a weighted Dynkin diagram (i.e., a
  list  containing  the  weights  of  the weighted Dynkin diagram, in the same
  order as the nodes of the Dynkin diagram of the root system of [3XL[103X; that order
  can  be  deduced  from  the  Cartan  matrix  of  the  same root system). The
  corresponding  nilpotent  orbit is returned. It is the responsibility of the
  user  to  make  sure  that  the  weighted  Dynkin  diagram  corresponds to a
  nilpotent orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xo:= NilpotentOrbit( L, [1,2,0,0,0,1] );[127X[104X
    [4X[28X<nilpotent orbit in Lie algebra of type E6>[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 NilpotentOrbits[101X
  
  [33X[1;0Y[29X[2XNilpotentOrbits[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a semisimple Lie algebra. This function returns the list of all
  nilpotent orbits of [3XL[103X.[133X
  
  [33X[0;0YIf  [3XL[103X  is  simple of classical type, then the nilpotent orbits correspond to
  partitions  (of  [22Xn+1[122X  for type [22XA_n[122X, of [22X2n+1[122X for type [22XB_n[122X, of [22X2n[122X for type [22XC_n[122X
  and  of [22X2n[122X for type [22XD_n[122X, see [CM93]). If [3XL[103X is of one of these types then the
  orbits  returned  by  this  function  have the attribute [3XOrbitPartition[103X set,
  which returns the corresponding partition.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs[10];[127X[104X
    [4X[28X<nilpotent orbit in Lie algebra of type E6>[128X[104X
    [4X[25Xgap>[125X [27XLength(orbs);[127X[104X
    [4X[28X20[128X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("B",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= NilpotentOrbits(L);;            [127X[104X
    [4X[25Xgap>[125X [27XOrbitPartition( orbs[10] );[127X[104X
    [4X[28X[ 5, 3, 1 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-3 WeightedDynkinDiagram[101X
  
  [33X[1;0Y[29X[2XWeightedDynkinDiagram[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xo[103X  is  a  nilpotent  orbit;  this function returns its weighted Dynkin
  diagram.[133X
  
  [1X4.1-4 WeightedDynkinDiagram[101X
  
  [33X[1;0Y[29X[2XWeightedDynkinDiagram[102X( [3XL[103X, [3Xx[103X ) [32X method[133X
  
  [33X[0;0YHere [3XL[103X is a semisimple Lie algebra, and [3Xx[103X a nilpotent element. This function
  returns the weighted Dynkin diagram of the orbit containing [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("B",3,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XWeightedDynkinDiagram( L, L.1+L.9 ); [127X[104X
    [4X[28X[ 2, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XWeightedDynkinDiagram(L, L.1+L.6+L.20+2*L.32 : table:= true );[127X[104X
    [4X[28X[ 0, 0, 0, 1, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-5 DisplayWeightedDynkinDiagram[101X
  
  [33X[1;0Y[29X[2XDisplayWeightedDynkinDiagram[102X( [3Xo[103X ) [32X method[133X
  
  [33X[0;0YThis  displays  the  weighted Dynkin diagram of the nilpotent orbit [3Xo[103X on the
  Dynkin diagram of the Lie algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XDisplayWeightedDynkinDiagram( no[10] );[127X[104X
    [4X[28X             1[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  0---1---0---1---0[128X[104X
  [4X[32X[104X
  
  [1X4.1-6 DisplayWeightedDynkinDiagram[101X
  
  [33X[1;0Y[29X[2XDisplayWeightedDynkinDiagram[102X( [3XL[103X, [3Xx[103X ) [32X method[133X
  
  [33X[0;0YHere  [3XL[103X  is a semisimple Lie algebra with nilpotent element [3Xx[103X. This displays
  the  weighted  Dynkin  diagram  of  the  nilpotent orbit containing [3Xx[103X on the
  Dynkin diagram of the Lie algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK1:= SimpleLieAlgebra("B",3,Rationals);;                                         [127X[104X
    [4X[25Xgap>[125X [27XK2:= SimpleLieAlgebra("F",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XL:= DirectSumOfAlgebras( K1, K2 );;[127X[104X
    [4X[25Xgap>[125X [27Xx:=L.1+L.3+L.17+L.33;[127X[104X
    [4X[28Xv.1+v.3+v.17+v.33[128X[104X
    [4X[25Xgap>[125X [27XDisplayWeightedDynkinDiagram( L, x );[127X[104X
    [4X[28XB3:  2---2=>=2[128X[104X
    [4X[28XF4:  0---0=>=0---1[128X[104X
  [4X[32X[104X
  
  [1X4.1-7 AmbientLieAlgebra[101X
  
  [33X[1;0Y[29X[2XAmbientLieAlgebra[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xo[103X is a nilpotent orbit; this function returns the Lie algebra it lives
  in.[133X
  
  [1X4.1-8 SemiSimpleType[101X
  
  [33X[1;0Y[29X[2XSemiSimpleType[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xo[103X  is  a  nilpotent  orbit;  this function returns the type of the Lie
  algebra it lives in.[133X
  
  [1X4.1-9 SL2Triple[101X
  
  [33X[1;0Y[29X[2XSL2Triple[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xo[103X  is  a nilpotent orbit; this function returns an [22Xsl_2[122X-triple [22X(y,h,x)[122X
  corresponding  to  [3Xo[103X.  For  the  exceptional  types the [22Xx[122X is as in the paper
  [Gra08]. For the classical types the [22Xx[122X is computed on the fly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XSL2Triple( orbs[10] );[127X[104X
    [4X[28X[ (4)*v.51+(3)*v.53+(3)*v.56+v.59, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+([128X[104X
    [4X[28X    8)*v.77+(4)*v.78, v.15+v.17+v.20+v.23 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-10 RandomSL2Triple[101X
  
  [33X[1;0Y[29X[2XRandomSL2Triple[102X( [3Xo[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3Xo[103X  is  a  nilpotent  orbit; this function returns a random [22Xsl_2[122X-triple
  [22X(y,h,x)[122X corresponding to [3Xo[103X. This means that every call (potentially) returns
  a different [22Xsl_2[122X-triple.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XRandomSL2Triple( orbs[10] );[127X[104X
    [4X[28X[ (3)*v.49+(3)*v.50+v.51+(4)*v.59, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+([128X[104X
    [4X[28X    8)*v.77+(4)*v.78, v.13+v.14+v.15+v.23 ][128X[104X
    [4X[25Xgap>[125X [27XRandomSL2Triple( orbs[10] );[127X[104X
    [4X[28X[ (3)*v.49+(4)*v.54+(3)*v.56+v.57, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+([128X[104X
    [4X[28X    8)*v.77+(4)*v.78, v.13+v.18+v.20+v.21 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-11 SL2Grading[101X
  
  [33X[1;0Y[29X[2XSL2Grading[102X( [3XL[103X, [3Xh[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is  a Lie algebra, and [3Xh[103X is an element of it, such that there is an
  [22Xsl_2[122X-triple  of  which  it  is the Cartan element (the system does not check
  that). This function returns the grading of [3XL[103X in eigenspaces of [3Xh[103X.[133X
  
  [33X[0;0YA  list containing three lists is returned: the first list contains bases of
  the  components  with  degrees  1,2,3,...,  the second list has bases of the
  components  with degrees -1,-2,-3,..., the last list contains a basis of the
  zero component.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xorbs:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27Xsl2:= RandomSL2Triple( orbs[6] );[127X[104X
    [4X[28X[ (2)*v.37+(2)*v.39+v.41, (3)*v.49+(4)*v.50+(6)*v.51+(8)*v.52, v.13+v.15+v.17 [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XSL2Grading( L, sl2[2] );[127X[104X
    [4X[28X[ [ [ v.3, v.5, v.7, v.8, v.9, v.11 ], [128X[104X
    [4X[28X      [ v.10, v.12, v.13, v.14, v.15, v.16, v.17, v.18, v.20 ], [128X[104X
    [4X[28X      [ v.19, v.21 ], [ v.22, v.23, v.24 ] ], [128X[104X
    [4X[28X  [ [ v.27, v.29, v.31, v.32, v.33, v.35 ], [128X[104X
    [4X[28X      [ v.34, v.36, v.37, v.38, v.39, v.40, v.41, v.42, v.44 ], [128X[104X
    [4X[28X      [ v.43, v.45 ], [ v.46, v.47, v.48 ] ], [128X[104X
    [4X[28X  [ v.1, v.2, v.4, v.6, v.25, v.26, v.28, v.30, v.49, v.50, v.51, v.52 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.1-12 SL2Triple[101X
  
  [33X[1;0Y[29X[2XSL2Triple[102X( [3XL[103X, [3Xx[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is a simple Lie algebra, and [3Xx[103X is a nilpotent element of it. A list
  of  three elements is returned, forming an [22Xsl_2[122X-triple, the last of which is
  equal to [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XSL2Triple( L, L.1+L.20 );[127X[104X
    [4X[28X[ v.16+v.25, v.49, v.1+v.20 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-13 Dimension[101X
  
  [33X[1;0Y[29X[2XDimension[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YReturns the dimension of the nilpotent orbit [3Xo[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XDimension( no[13] );                 [127X[104X
    [4X[28X60[128X[104X
  [4X[32X[104X
  
  [1X4.1-14 IsRegular[101X
  
  [33X[1;0Y[29X[2XIsRegular[102X( [3Xo[103X ) [32X operation[133X
  
  [33X[0;0YThe  regular  nilpotent orbit is the one of maximal dimension. This function
  returns [3Xtrue[103X if the nilpotent orbit [3Xo[103X is regular, [3Xfalse[103X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XIsRegular( no[13] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsRegular( no[20] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-15 RegularNilpotentOrbit[101X
  
  [33X[1;0Y[29X[2XRegularNilpotentOrbit[102X( [3XL[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  semisimple  Lie  algebra.  This function returns the regular
  nilpotent orbit of [3XL[103X.[133X
  
  [1X4.1-16 IsDistinguished[101X
  
  [33X[1;0Y[29X[2XIsDistinguished[102X( [3Xo[103X ) [32X operation[133X
  
  [33X[0;0YA  nilpotent  orbit is said to be distinguished if for a representative [3Xx[103X we
  have  that  the only Levi subalgebra containing [3Xx[103X is the ambient Lie algebra
  itself.   This   function   returns   [3Xtrue[103X  if  the  nilpotent  orbit  [3Xo[103X  is
  distinguished, [3Xfalse[103X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27XIsDistinguished( no[10] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsDistinguished( no[17] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-17 DistinguishedNilpotentOrbits[101X
  
  [33X[1;0Y[29X[2XDistinguishedNilpotentOrbits[102X( [3XL[103X ) [32X operation[133X
  
  [33X[0;0YReturns the list of distinguished nilpotent orbits of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xdis:= DistinguishedNilpotentOrbits( L );; Length(dis);[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XDisplayWeightedDynkinDiagram( dis[1] );[127X[104X
    [4X[28X             0[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  2---0---2---0---2[128X[104X
    [4X[25Xgap>[125X [27XDisplayWeightedDynkinDiagram( dis[2] );[127X[104X
    [4X[28X             2[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  2---2---0---2---2[128X[104X
    [4X[25Xgap>[125X [27XDisplayWeightedDynkinDiagram( dis[3] );[127X[104X
    [4X[28X             2[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  2---2---2---2---2[128X[104X
  [4X[32X[104X
  
  [1X4.1-18 ComponentGroup[101X
  
  [33X[1;0Y[29X[2XComponentGroup[102X( [3Xo[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xo[103X  is  a  nilpotent orbit in a [13Xsimple[113X Lie algebra [3XL[103X. Let [3X(f,h,e)[103X be an
  [22Xmathfraksl_2[122X-triple   as  returned  by  [3XSL2Triple(  o  )[103X.  We  consider  the
  stabilizer  [22XS[122X  of  [3X(f,h,e)[103X  in  the  adjoint  group  [22XG[122X of [3XL[103X. This stabilizer
  consists  of  the  elements  in  [22XG[122X  that map each of [3Xf, h, e[103X to itself. This
  function  returns  a subgroup of [22XG[122X that in all but four cases (see below) is
  isomorphic to the component group of [22XS[122X. The elements of this group are given
  by their matrices relative to the basis [3XBasis( L )[103X of [3XL[103X.[133X
  
  [33X[0;0YIn four cases (one in type [22XE_7[122X and three in type [22XE_8[122X) the group returned has
  order  4,  whereas the component group [22XS/S^∘[122X has order 2. In these cases the
  square of a generator of the returned group lies in [22XS^∘[122X.[133X
  
  [33X[0;0YWe  remark that it is known that the component group of [22XS[122X is the same as the
  component group of the stabilizer of [3Xe[103X.[133X
  
  [33X[0;0YIn  the  next  example we construct the automorphisms that correspond to the
  generators of a component group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",8,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbits(L);; [127X[104X
    [4X[25Xgap>[125X [27XC:= ComponentGroup( no[41] ); [127X[104X
    [4X[28X<matrix group with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XLength( Elements(C) );[127X[104X
    [4X[28X120[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfGroup(C);;[127X[104X
    [4X[25Xgap>[125X [27Xf1:= function(x) return (gens[1]*Coefficients(Basis(L),x))*Basis(L); end;[127X[104X
    [4X[28Xfunction( x ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xf2:= function(x) return (gens[2]*Coefficients(Basis(L),x))*Basis(L); end;[127X[104X
    [4X[28Xfunction( x ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xsl2:= SL2Triple( no[41] );;[127X[104X
    [4X[25Xgap>[125X [27XList( sl2, f1 ) = sl2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XList( sl2, f2 ) = sl2;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-19 InducedNilpotentOrbits[101X
  
  [33X[1;0Y[29X[2XInducedNilpotentOrbits[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  simple  Lie  algebra.  This function returns the list of all
  induced  nilpotent  orbits  of  [3XL[103X.  An  induced  orbit  is given by a record
  containing  two  fields:  [3Xsheetdiag[103X,  which is a diagram describing the Levi
  subalgebra  and  the  rigid  nilpotent  orbit in it from which the nilpotent
  orbit is induced, and [3Xnorbit[103X, which is the induced nilpotent orbit in [3XL[103X.[133X
  
  [33X[0;0YThe sheet diagram is a labeled Dynkin diagram, and the labels are 0, 1 or 2.
  If we take the Dynkin diagram and erase the nodes which have label 2 then we
  obtain the Dynkin diagram of the Levi subalgebra. Moreover, the labels 0 and
  1  on  that  diagram  give the rigid nilpotent orbit in the Levi subalgebra.
  From this pair the nilpotent orbit [3Xnorbit[103X is induced.[133X
  
  [33X[0;0YIt  may  happen  that the same nilpotent orbit is induced from more than one
  pair  consisting  of a Levi subalgebra and a rigid nilpotent orbit in it. In
  that  case the same nilpotent orbit appears more than once in the list, each
  time with a different sheet diagram attached.[133X
  
  [33X[0;0YThis function works for the Lie algebras of exceptional type and for the Lie
  algebras  of type [22XA[122X regardless of the rank. It works for the Lie algebras of
  the other types up to rank 10.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xs:= InducedNilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27Xs[19];[127X[104X
    [4X[28Xrec( norbit := <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  sheetdiag := [ 2, 0, 0, 1, 0, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XWeightedDynkinDiagram( s[19].norbit );[127X[104X
    [4X[28X[ 0, 0, 0, 2, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-20 RigidNilpotentOrbits[101X
  
  [33X[1;0Y[29X[2XRigidNilpotentOrbits[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X is a simple Lie algebra. This function returns the list of all rigid
  nilpotent orbits of [3XL[103X, [13Xexcept[113X the zero orbit (which is always rigid).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XRigidNilpotentOrbits(L);[127X[104X
    [4X[28X[ <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6> ][128X[104X
    [4X[25Xgap>[125X [27XList( last, WeightedDynkinDiagram );[127X[104X
    [4X[28X[ [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 1, 0, 0, 1, 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.1-21 RichardsonOrbits[101X
  
  [33X[1;0Y[29X[2XRichardsonOrbits[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is a simple Lie algebra. A nilpotent orbit is said to be Richardson
  if  it  is  induced  from the zero orbit in a Levi subalgebra. This function
  returns the list of all Richardson nilpotent orbits of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XRichardsonOrbits(L);[127X[104X
    [4X[28X[ <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6>, [128X[104X
    [4X[28X  <nilpotent orbit in Lie algebra of type E6>, <nilpotent orbit in Lie algebra of type E6> ][128X[104X
    [4X[25Xgap>[125X [27XList( last, WeightedDynkinDiagram );    [127X[104X
    [4X[28X[ [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 0, 2, 2 ], [ 2, 2, 0, 2, 0, 2 ], [ 1, 2, 1, 0, 1, 1 ], [128X[104X
    [4X[28X  [ 1, 2, 0, 0, 0, 1 ], [ 1, 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 1, 0 ], [ 1, 1, 1, 0, 1, 1 ], [128X[104X
    [4X[28X  [ 0, 0, 0, 2, 0, 0 ], [ 2, 0, 0, 2, 0, 2 ], [ 2, 2, 0, 0, 0, 2 ], [ 0, 2, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 0, 2, 0, 2, 0, 0 ], [ 2, 0, 0, 0, 0, 2 ] ][128X[104X
  [4X[32X[104X
  
