  
  [1X3 [33X[0;0YSemisimple Lie Algebras and their Modules[133X[101X
  
  
  [1X3.1 [33X[0;0YSemisimple Lie algebras[133X[101X
  
  [1X3.1-1 IsomorphismOfSemisimpleLieAlgebras[101X
  
  [33X[1;0Y[29X[2XIsomorphismOfSemisimpleLieAlgebras[102X( [3XL1[103X, [3XL2[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL1[103X  and  [3XL2[103X  are  two  semisimple  Lie  algebras  that are known to be
  isomorphic  (i.e.,  they  have  the  same  type).  This  function returns an
  isomorphism.[133X
  
  [1X3.1-2 DisplayDynkinDiagram[101X
  
  [33X[1;0Y[29X[2XDisplayDynkinDiagram[102X( [3XL[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  semisimple  Lie  algebra.  This function displays its Dynkin
  diagram.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);[127X[104X
    [4X[28X<Lie algebra of dimension 52 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XDisplayDynkinDiagram(L);              [127X[104X
    [4X[28XF4:  2---4=>=3---1[128X[104X
  [4X[32X[104X
  
  [1X3.1-3 ApplyWeylPermToCartanElement[101X
  
  [33X[1;0Y[29X[2XApplyWeylPermToCartanElement[102X( [3XL[103X, [3Xw[103X, [3Xh[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is a semisimple Lie algebra, [3Xw[103X is a permutation which is an element
  of  [3XWeylGroupAsPermGroup( RootSystem(L) )[103X, and [3Xh[103X is an element of the Cartan
  subalgebra  [3XCartanSubalgebra(  L  )[103X.  The Weyl groups naturally acts on this
  Cartan subalgebra and this function returns the result of applying [3Xw[103X to [3Xh[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem(L);;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroupAsPermGroup(R);;[127X[104X
    [4X[25Xgap>[125X [27Xw:= Product( GeneratorsOfGroup(W) );[127X[104X
    [4X[28X(1,32,33,36,35,27,25,8,9,12,11,3)(2,30,34,44,37,39,26,6,10,20,13,15)(4,16,23,24,22,18,28,[128X[104X
    [4X[28X40,47,48,46,42)(5,31,41,43,45,38,29,7,17,19,21,14)[128X[104X
    [4X[25Xgap>[125X [27XH:= CartanSubalgebra(L);;[127X[104X
    [4X[25Xgap>[125X [27Xh:= Sum( Basis(H) );[127X[104X
    [4X[28Xv.49+v.50+v.51+v.52[128X[104X
    [4X[25Xgap>[125X [27XApplyWeylPermToCartanElement( L, w, h );[127X[104X
    [4X[28X(-1)*v.52[128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YRepresentations of semisimple Lie algebras[133X[101X
  
  [1X3.2-1 AdmissibleLattice[101X
  
  [33X[1;0Y[29X[2XAdmissibleLattice[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XV[103X  is  a  [13Xsimple[113X  module  over a semisimple Lie algebra. This function
  returns  a basis of [3XV[103X that spans an admissible lattice in [3XV[103X. This means that
  for  a  root  vector  [22Xx[122X  of  the  acting Lie algebra the matrix [22Xexp( mx )[122X is
  integral,  where  [22Xmx[122X  denotes  the  matrix  of  [22Xx[122X relative to the admissible
  lattice.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [2,0] );[127X[104X
    [4X[28X<27-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XB:=AdmissibleLattice(V);;[127X[104X
    [4X[25Xgap>[125X [27Xx:= L.1;[127X[104X
    [4X[28Xv.1[128X[104X
    [4X[25Xgap>[125X [27Xmx:= MatrixOfAction( B, x );;[127X[104X
    [4X[25Xgap>[125X [27XIsZero(mx^4); IsZero(mx^5);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xexp:=Sum( List( [0..4], i -> mx^i/Factorial(i) ) );;[127X[104X
    [4X[25Xgap>[125X [27XForAll( Flat(exp), IsInt );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.2-2 DirectSumDecomposition[101X
  
  [33X[1;0Y[29X[2XDirectSumDecomposition[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XV[103X  is a module over a semisimple Lie algebra; this function computes a
  list of sub-modules such that [3XV[103X is their direct sum.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [1,0] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= TensorProductOfAlgebraModules( V, V );[127X[104X
    [4X[28X<49-dimensional left-module over <Lie algebra of dimension 14 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XDirectSumDecomposition( W );[127X[104X
    [4X[28X[ <left-module over <Lie algebra of dimension 14 over Rationals>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 14 over Rationals>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 14 over Rationals>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 14 over Rationals>> ][128X[104X
    [4X[25Xgap>[125X [27XList( last, Dimension );[127X[104X
    [4X[28X[ 27, 7, 14, 1 ][128X[104X
  [4X[32X[104X
  
  [1X3.2-3 IsIrreducibleHWModule[101X
  
  [33X[1;0Y[29X[2XIsIrreducibleHWModule[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YReturns  [3Xtrue[103X  if  [3XV[103X is an irreducible module over a semisimple Lie algebra,
  and [3Xfalse[103X otherwise[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);[127X[104X
    [4X[28X<Lie algebra of dimension 52 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [0,1,0,0] );[127X[104X
    [4X[28X<52-dimensional left-module over <Lie algebra of dimension 52 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleHWModule(V);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X3.2-4 HighestWeightVector[101X
  
  [33X[1;0Y[29X[2XHighestWeightVector[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere [3XV[103X is an irreducible module over a semisimple Lie algebra. This function
  returns  a  highest  weight  vector  [3Xv0[103X in [3XV[103X. This means that it is a weight
  vector for the Cartan subalgebra of the acting Lie algebra, and all positive
  root vectors send it to zero.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [1,0] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= TensorProductOfAlgebraModules( V, V );;[127X[104X
    [4X[25Xgap>[125X [27XdW:= DirectSumDecomposition( W );;[127X[104X
    [4X[25Xgap>[125X [27Xcg:= CanonicalGenerators( RootSystem(L) );;[127X[104X
    [4X[25Xgap>[125X [27Xv0:= HighestWeightVector( dW[3] );[127X[104X
    [4X[28X1*(1*v0<x>y1*v0)-1*(y1*v0<x>1*v0)[128X[104X
    [4X[25Xgap>[125X [27XList( cg[3], h -> h^v0 );[127X[104X
    [4X[28X[ <0-tensor>, 1*(1*v0<x>y1*v0)-1*(y1*v0<x>1*v0) ][128X[104X
    [4X[25Xgap>[125X [27XList( cg[1], h -> h^v0 );[127X[104X
    [4X[28X[ <0-tensor>, <0-tensor> ][128X[104X
  [4X[32X[104X
  
  [1X3.2-5 HighestWeight[101X
  
  [33X[1;0Y[29X[2XHighestWeight[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere [3XV[103X is an irreducible module over a semisimple Lie algebra. This function
  returns  the  highest  weight  of [3XV[103X. That is, the list of eigenvalues of the
  Cartan  elements  in  a  canonical  generating  set of the Lie algebra, when
  acting on a highest weight vector.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [1,0] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= TensorProductOfAlgebraModules( V, V );;[127X[104X
    [4X[25Xgap>[125X [27XdW:= DirectSumDecomposition( W );;[127X[104X
    [4X[25Xgap>[125X [27XList( dW, HighestWeight );[127X[104X
    [4X[28X[ [ 2, 0 ], [ 1, 0 ], [ 0, 1 ], [ 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.2-6 DisplayHighestWeight[101X
  
  [33X[1;0Y[29X[2XDisplayHighestWeight[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere [3XV[103X is an irreducible module over a semisimple Lie algebra. This function
  displays  its  highest  weight,  that  is,  it  shows the coordinates of the
  highest weight on the Dynkin diagram of the Lie algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:= LieAlgebraAndSubalgebras( "E8" );;[127X[104X
    [4X[25Xgap>[125X [27XL:= r.liealg;;[127X[104X
    [4X[25Xgap>[125X [27XK:= r.subalgs[823];[127X[104X
    [4X[28X<Lie algebra of dimension 58 over CF(84)>[128X[104X
    [4X[25Xgap>[125X [27XDisplayDynkinDiagram(K);[127X[104X
    [4X[28XA1:  1[128X[104X
    [4X[28XB5:  2---3---4---5=>=6[128X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );[127X[104X
    [4X[28X<248-dimensional left-module over <Lie algebra of dimension 248 over CF(84)>>[128X[104X
    [4X[25Xgap>[125X [27XW:= ModuleByRestriction( V, K );[127X[104X
    [4X[28X<248-dimensional left-module over <Lie algebra of dimension 58 over CF(84)>>[128X[104X
    [4X[25Xgap>[125X [27XdW:= DirectSumDecomposition( W ); [127X[104X
    [4X[28X[ <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 58 over CF(84)>> ][128X[104X
    [4X[25Xgap>[125X [27XList( dW, Dimension );[127X[104X
    [4X[28X[ 33, 3, 3, 3, 64, 64, 11, 11, 55, 1 ][128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dW[5] );[127X[104X
    [4X[28XA1:  1[128X[104X
    [4X[28XB5:  0---0---0---0=>=1[128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dW[1] );[127X[104X
    [4X[28XA1:  2[128X[104X
    [4X[28XB5:  1---0---0---0=>=0[128X[104X
  [4X[32X[104X
  
  [1X3.2-7 IsomorphismOfIrreducibleHWModules[101X
  
  [33X[1;0Y[29X[2XIsomorphismOfIrreducibleHWModules[102X( [3XV1[103X, [3XV2[103X ) [32X operation[133X
  
  [33X[0;0YHere [3XV1[103X, [3XV2[103X are two irreducible modules over the same semisimple Lie algebra
  with  the same highest weights. This function returns an isomorphism between
  the two.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:= LieAlgebraAndSubalgebras( "E8" );;[127X[104X
    [4X[25Xgap>[125X [27XL:= r.liealg;;[127X[104X
    [4X[25Xgap>[125X [27XK:= r.subalgs[823];;[127X[104X
    [4X[25Xgap>[125X [27XDisplayDynkinDiagram(K);[127X[104X
    [4X[28XA1:  1[128X[104X
    [4X[28XB5:  2---3---4---5=>=6[128X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );;[127X[104X
    [4X[25Xgap>[125X [27XW:= ModuleByRestriction( V, K );;[127X[104X
    [4X[25Xgap>[125X [27XdW:= DirectSumDecomposition( W );;[127X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dW[5] );[127X[104X
    [4X[28XA1:  1[128X[104X
    [4X[28XB5:  0---0---0---0=>=1[128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dW[6] );[127X[104X
    [4X[28XA1:  1[128X[104X
    [4X[28XB5:  0---0---0---0=>=1[128X[104X
    [4X[25Xgap>[125X [27Xf:= IsomorphismOfIrreducibleHWModules( dW[5], dW[6] );;[127X[104X
    [4X[25Xgap>[125X [27XImage( f, HighestWeightVector( dW[5] ) );[127X[104X
    [4X[28Xv.205[128X[104X
    [4X[25Xgap>[125X [27XHighestWeightVector( dW[6] );[127X[104X
    [4X[28Xv.205[128X[104X
  [4X[32X[104X
  
  [1X3.2-8 DualAlgebraModule[101X
  
  [33X[1;0Y[29X[2XDualAlgebraModule[102X( [3XV[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XV[103X  is  a  module  over  a  Lie algebra. This function returns the dual
  module.[133X
  
  [33X[0;0YThe  basis  elements  of  this  module are printed as [3XF@v[103X where [3Xv[103X is a basis
  element  of  [3Xv[103X.  This  represents the function which takes the value 1 on te
  basis  element  [3Xv[103X  and 0 on all other basis elements. However, an element of
  the  module  is  a  module  element  and  not  a function. We can access the
  function  by  taking  the  [3XExtRepOfObj[103X  of  an  element  of  the  module, as
  illustrated by the example below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [0,0,1,0,0,0] );; Dimension(V);[127X[104X
    [4X[28X351[128X[104X
    [4X[25Xgap>[125X [27XVst:= DualAlgebraModule( V );[127X[104X
    [4X[28X<351-dimensional left-module over <Lie algebra of dimension 78 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( Vst );[127X[104X
    [4X[28X             0[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  0---0---0---1---0[128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( V );  [127X[104X
    [4X[28X             0[128X[104X
    [4X[28X             |[128X[104X
    [4X[28XE6:  0---1---0---0---0[128X[104X
    [4X[25Xgap>[125X [27Xv0:= HighestWeightVector( Vst );[127X[104X
    [4X[28X(1)*F@y15*y23*y36^(2)*v0[128X[104X
    [4X[25Xgap>[125X [27Xf:= ExtRepOfObj( v0 );         [127X[104X
    [4X[28X(1)*F@y15*y23*y36^(2)*v0[128X[104X
    [4X[25Xgap>[125X [27XImage(f, Basis(V)[10] );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [1X3.2-9 CharacteristicsOfStrata[101X
  
  [33X[1;0Y[29X[2XCharacteristicsOfStrata[102X( [3XL[103X, [3Xhw[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCharacteristicsOfStrata[102X( [3XL[103X, [3XB[103X, [3Xhw[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  semisimple  Lie  algebra  over  a field of characteristic 0.
  Secondly,  [3Xhw[103X  is  a  dominant weight, represented as a list of non-negative
  integers  (where  the  ordering  of  the fundamantal weights is given by the
  Cartan  matrix  of  the  root  system  of  [3XL[103X).  Let  [22XG[122X denote the semisimple
  algebraic group acting on the irreducible representation with highest weight
  [3Xhw[103X.  Alternatively,  [3Xhw[103X can also be a list of highest weights, in which case
  the representation is the direct sum of the irreducible representations with
  highest weights in the list. Hesselink ([Hes79]) defined a stratification of
  the  nullcone  relative  to the action of [22XG[122X. Popov and Vinberg ([VP89]) have
  described  this  stratification  in  terms  of  characteristics,  which  are
  elements  of  a  Cartan  subalgebra  of  [3XL[103X.  To  each  characteristic  there
  corresponds  a  stratum.  This function is an implementation of an algorithm
  due  to Popov ([Pop03]), for computing the characteristics of the strata. It
  returns  a  list  of two lists. The first list contains the characteristics.
  The  second list contains the dimensions of the corresponding strata. If the
  highest   weight   [3Xhw[103X   defines   the   adjoint   representation,  then  the
  characteristics  of  the  strata  are  exactly  the  characteristics  of the
  nilpotent  orbits in [3XL[103X. This means the following: let [22Xh[122X be a characteristic,
  then there are [22Xe,f[122X in [3XL[103X such that the triple [22Xh,e,f[122X satisfies the commutation
  relations  of  [22Xmathfraksl_2[122X,  and  the  elements  [22Xe[122X  thus  obtained  are the
  representatives of the nilpotent [22XG[122X-orbits in [3XL[103X.[133X
  
  [33X[0;0YWe  remark  that  the  characteristics  depend on the choice of an invariant
  bilinear  form.  This  form  is  unique  if [3XL[103X is simple. If we give just two
  arguments,  [3XL[103X,  [3Xhw[103X,  then  Killing  form  is chosen. It is possible to use a
  different form using the three argument variant of the function.[133X
  
  [33X[0;0YIn  the  three  argument  variant  [3XL[103X is a reductive Lie algebra and [3XB[103X is the
  restriction  of  a  non-degenerate  invariant  bilinear  form  on the Cartan
  subalgebra of [3XL[103X. This bilinear form must be given with respect to a specific
  basis,  which  we  now  describe.  Let  [3XK[103X denote the derived subalgebra of [3XL[103X
  (which  is semisimple). Let [3Xh[103X be the list [3XCanonicalGenerators( RootSystem( K
  )  )[3][103X  (this  is  a  basis of a Cartan subalgebra of [3XK[103X). Let [3Xc[103X be the list
  [3XBasisVectors(  Basis(  LieCentre(K)  )  )[103X.  Then the basis we require is the
  concatenation  of  [3Xh[103X  and  [3Xc[103X. Again [3Xhw[103X can be a highest weight, or a list of
  highest  weights.  These  highest  weights  are  lists of eigenvalues of the
  elements  of  the  particular  basis  of  a Cartan subalgebra of [3XL[103X described
  above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XCharacteristicsOfStrata( L, [0,1] );[127X[104X
    [4X[28X[ [ v.13+(2)*v.14, (2)*v.13+(3)*v.14, (2)*v.13+(4)*v.14, (6)*v.13+(10)*v.14 ],[128X[104X
    [4X[28X  [ 6, 8, 10, 12 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn the next example we compute the strata of a representation of a reductive
  subalgebra  of  the  Lie  algebra  of type [22XE_6[122X, obtained as the set of fixed
  points   of   an   inner   automorphism.   We  compute  the  strata  of  the
  [22Xθ[122X-representation  corresponding  to the automorphism. For this we first need
  to  work  out  the  highest  weights of the module. The bilinear form is the
  restriction of the Killing form to the subalgebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms("E",6,3)[2];;[127X[104X
    [4X[25Xgap>[125X [27XM:= Source(f);;[127X[104X
    [4X[25Xgap>[125X [27Xgr:= Grading(f);;[127X[104X
    [4X[25Xgap>[125X [27XL:= Subalgebra(M,gr[1]);[127X[104X
    [4X[28X<Lie algebra over CF(3), with 28 generators>[128X[104X
    [4X[25Xgap>[125X [27XK:= LieDerivedSubalgebra( L );[127X[104X
    [4X[28X<Lie algebra of dimension 27 over CF(3)>[128X[104X
    [4X[25Xgap>[125X [27XV:= LeftAlgebraModuleByGenerators( K, function(x,v) return x*v; end, gr[2]); [127X[104X
    [4X[28X<left-module over <Lie algebra of dimension 27 over CF(3)>>[128X[104X
    [4X[25Xgap>[125X [27XDisplayDynkinDiagram( K ); [127X[104X
    [4X[28XA4:  1---4---3---2[128X[104X
    [4X[28XA1:  5[128X[104X
    [4X[25Xgap>[125X [27XdV:= DirectSumDecomposition(V);[127X[104X
    [4X[28X[ <left-module over <Lie algebra of dimension 27 over CF(3)>>, [128X[104X
    [4X[28X  <left-module over <Lie algebra of dimension 27 over CF(3)>> ][128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dV[1] );        [127X[104X
    [4X[28XA4:  0---0---0---1[128X[104X
    [4X[28XA1:  0[128X[104X
    [4X[25Xgap>[125X [27XDisplayHighestWeight( dV[2] );[127X[104X
    [4X[28XA4:  0---0---1---0[128X[104X
    [4X[28XA1:  1[128X[104X
    [4X[25Xgap>[125X [27Xt0:= Basis(LieCentre(L))[1];[127X[104X
    [4X[28Xv.73+(4/5)*v.75+(3/5)*v.76+(2/5)*v.77+(1/5)*v.78[128X[104X
    [4X[25Xgap>[125X [27XHighestWeightVector( dV[1] ); t0^last;[127X[104X
    [4X[28Xv.7[128X[104X
    [4X[28X(6/5)*v.7[128X[104X
    [4X[25Xgap>[125X [27XHighestWeightVector( dV[2] ); t0^last;[127X[104X
    [4X[28Xv.13[128X[104X
    [4X[28X(-3/5)*v.13[128X[104X
    [4X[25Xgap>[125X [27Xhw:= [ [0,1,0,0,0,6/5], [0,0,1,0,1,-3/5] ]; [127X[104X
    [4X[28X[ [ 0, 1, 0, 0, 0, 6/5 ], [ 0, 0, 1, 0, 1, -3/5 ] ][128X[104X
    [4X[25Xgap>[125X [27Xbas:= Concatenation( CanonicalGenerators( RootSystem(K) )[3],[127X[104X
    [4X[25X>[125X [27XBasis(LieCentre(L)) );;[127X[104X
    [4X[25Xgap>[125X [27XB:= List( bas, x -> [] );;[127X[104X
    [4X[25Xgap>[125X [27Xad:= List( bas, x -> AdjointMatrix( Basis(M), x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..Length(B)] do for j in [i..Length(B)] do[127X[104X
    [4X[25X>[125X [27XB[i][j]:= TraceMat( ad[i]*ad[j]); B[j][i]:= B[i][j];[127X[104X
    [4X[25X>[125X [27Xod; od;[127X[104X
    [4X[25Xgap>[125X [27XB;[127X[104X
    [4X[28X[ [ 48, 0, 0, -24, 0, 0 ], [ 0, 48, -24, 0, 0, 0 ], [ 0, -24, 48, -24, 0, 0 ], [128X[104X
    [4X[28X[ -24, 0, -24, 48, 0, 0 ], [ 0, 0, 0, 0, 48, 0 ], [ 0, 0, 0, 0, 0, 144/5 ] ][128X[104X
    [4X[25Xgap>[125X [27XCharacteristicsOfStrata( L, B, hw );[127X[104X
    [4X[28X[ [ v.74+v.75+v.76, v.73+v.75, (-2)*v.73, [128X[104X
    [4X[28X      (2)*v.74+(2)*v.75+(3)*v.76+(2)*v.77+v.78, (-1)*v.73+(-1)*v.76+(-1)*v.77,[128X[104X
    [4X[28X      v.73+v.74+(2)*v.75+v.76, (2)*v.73+(2)*v.74+(4)*v.75+(4)*v.76+(2)*v.77+([128X[104X
    [4X[28X        2)*v.78, (-2)*v.73+v.74, v.74+(4)*v.75+(2)*v.76+v.77+v.78, [128X[104X
    [4X[28X      (-1)*v.73+v.74+v.75+v.76, (2)*v.73+(3)*v.74+(5)*v.75+(5)*v.76+(3)*v.77+([128X[104X
    [4X[28X        2)*v.78, v.73+(4)*v.75+v.76, v.75+(-1)*v.76+(-1)*v.77, [128X[104X
    [4X[28X      v.73+v.74+(3)*v.75+(2)*v.76+v.78, (4)*v.73+(6)*v.74+(7)*v.75+(9)*v.76+([128X[104X
    [4X[28X        6)*v.77+(3)*v.78, (-3)*v.73+(-2)*v.75+(-2)*v.76+(-2)*v.77+(-1)*v.78, [128X[104X
    [4X[28X      (4)*v.75+(2)*v.76, (2)*v.73+(6)*v.74+(8)*v.75+(8)*v.76+(4)*v.77+(2)*v.78[128X[104X
    [4X[28X        , (2)*v.74+(4)*v.75+(2)*v.76+v.77+v.78, [128X[104X
    [4X[28X      (2)*v.74+(4)*v.75+(2)*v.76+(-2)*v.77, [128X[104X
    [4X[28X      v.73+v.74+(5)*v.75+(3)*v.76+v.77+v.78, [128X[104X
    [4X[28X      v.73+(2)*v.74+(4)*v.75+(3)*v.76+v.77+v.78, [128X[104X
    [4X[28X      (4)*v.73+(6)*v.74+(10)*v.75+(10)*v.76+(4)*v.77+(4)*v.78, [128X[104X
    [4X[28X      (3)*v.73+(6)*v.74+(10)*v.75+(10)*v.76+(5)*v.77+(5)*v.78, [128X[104X
    [4X[28X      (-1)*v.73+v.74+(3)*v.75+(-3)*v.77+(-1)*v.78, [128X[104X
    [4X[28X      (6)*v.74+(10)*v.75+(8)*v.76+(2)*v.77+(2)*v.78 ], [128X[104X
    [4X[28X  [ 8, 5, 16, 11, 12, 10, 13, 18, 18, 15, 15, 17, 13, 15, 16, 20, 20, 20, 19, [128X[104X
    [4X[28X      21, 19, 17, 20, 22, 22, 24 ] ][128X[104X
  [4X[32X[104X
  
