Script started on Mon 01 Jul 2013 06:37:32 PM EDT annegret.paul@atlas2:~/rx-scripts$ realex This is 'realex', version 0.8 (compiled on Jun 26 2013 at 03:49:20). It is the programmable interpreter interface to the library (version 0.5.4) of the Atlas of Lie Groups and Representations program. http://www.liegroups.org/ expr> 1+1 { one can do simple arithmetic } Value: 2 expr> expr> expr> expr> expr> +(1,1) { alternative syntax for the same } Value: 2 expr> =(1,1) Value: true expr> 1!=1 Value: false expr> 2/1 { division of integers produces a rational number } Value: 2/1 expr> =(2,1+1) { santiy check! } Value: true expr> set v=[3/2,4/5] { make a list } Identifier v: [rat] expr> v { the type [rat] is "list of rational", now look at its value } Value: [3/2,4/5] expr> for i:2 do v[i]=8/10 od { a simple loop produing a list of truth values } Value: [false,true] expr> showall { this command shows everything now defined, types and values } Overloaded operators and functions: +: (int,int->int): {+@(int,int)} +: (rat,rat->rat): {+@(rat,rat)} +: (ratvec,ratvec->ratvec): {+@(ratvec,ratvec)} +: (Split,Split->Split): {+@(Split,Split)} +: (ParamPol,Param->ParamPol): {+@(ParamPol,Param)} +: (ParamPol,(Split,Param)->ParamPol): {+@(ParamPol,(Split,Param))} +: (ParamPol,[(Split,Param)]->ParamPol): {+@(ParamPol,[(Split,Param)])} +: (ParamPol,ParamPol->ParamPol): {+@(ParamPol,ParamPol)} -: (int,int->int): {-@(int,int)} -: (int->int): {-@int} -: (rat,rat->rat): {-@(rat,rat)} -: (rat->rat): {-@rat} -: (ratvec,ratvec->ratvec): {-@(ratvec,ratvec)} -: (Split,Split->Split): {-@(Split,Split)} -: (Split->Split): {-@Split} *: (int,int->int): {*@(int,int)} *: (rat,rat->rat): {*@(rat,rat)} *: (vec,vec->int): {*@(vec,vec)} *: (mat,vec->vec): {*@(mat,vec)} *: (mat,ratvec->ratvec): {*@(mat,ratvec)} *: (mat,mat->mat): {*@(mat,mat)} *: (vec,mat->vec): {*@(vec,mat)} *: (Split,Split->Split): {*@(Split,Split)} *: (int,ParamPol->ParamPol): {*@(int,ParamPol)} *: (Split,ParamPol->ParamPol): {*@(Split,ParamPol)} \: (int,int->int): {\@(int,int)} %: (int,int->int): {%@(int,int)} %: (rat->int,int): {%@rat} %: (ratvec->vec,int): {%@ratvec} %: (LieType->[LieType]): {%@LieType} %: (Param->KGBElt,vec,ratvec): {%@Param} %: (Split->int,int): {%@Split} \%: (int,int->int,int): {\%@(int,int)} ^: (int,int->int): {^@(int,int)} ^: (rat,int->rat): {^@(rat,int)} ^: (vec->mat): {^@vec} ^: (mat->mat): {^@mat} /: (int,int->rat): {/@(int,int)} /: (rat,rat->rat): {/@(rat,rat)} /: (rat->rat): {/@rat} /: (vec,int->ratvec): {/@(vec,int)} =: (int,int->bool): {=@(int,int)} =: (rat,rat->bool): {=@(rat,rat)} =: (bool,bool->bool): {=@(bool,bool)} =: (string,string->bool): {=@(string,string)} =: (vec,vec->bool): {=@(vec,vec)} =: (mat,mat->bool): {=@(mat,mat)} =: (KGBElt,KGBElt->bool): {=@(KGBElt,KGBElt)} =: (Param,Param->bool): {=@(Param,Param)} !=: (int,int->bool): {!=@(int,int)} !=: (rat,rat->bool): {!=@(rat,rat)} !=: (bool,bool->bool): {!=@(bool,bool)} !=: (vec,vec->bool): {!=@(vec,vec)} !=: (mat,mat->bool): {!=@(mat,mat)} <: (int,int->bool): {<@(int,int)} <: (rat,rat->bool): {<@(rat,rat)} <=: (int,int->bool): {<=@(int,int)} <=: (rat,rat->bool): {<=@(rat,rat)} <=: (string,string->bool): {<=@(string,string)} >: (int,int->bool): {>@(int,int)} >: (rat,rat->bool): {>@(rat,rat)} >=: (int,int->bool): {>=@(int,int)} >=: (rat,rat->bool): {>=@(rat,rat)} #: (string,string->string): {#@(string,string)} #: (string->int): {#@string} #: (vec->int): {#@vec} #: (mat->int,int): {#@mat} #: (vec,int->vec): {#@(vec,int)} #: (int,vec->vec): {#@(int,vec)} #: (vec,vec->vec): {#@(vec,vec)} #: (LieType->int): {#@LieType} #: (ParamPol->int): {#@ParamPol} int_format: (int->string): {int_format@int} ascii: (string->int): {ascii@string} ascii: (int->string): {ascii@int} error: (string->*): {error@string} null: (int->vec): {null@int} null: (int,int->mat): {null@(int,int)} id_mat: (int->mat): {id_mat@int} diagonal: (vec->mat): {diagonal@vec} echelon: (mat->mat,[int]): {echelon@mat} diagonalize: (mat->vec,mat,mat): {diagonalize@mat} adapted_basis: (mat->mat,vec): {adapted_basis@mat} kernel: (mat->mat): {kernel@mat} eigen_lattice: (mat,int->mat): {eigen_lattice@(mat,int)} row_saturate: (mat->mat): {row_saturate@mat} inv_fact: (mat->vec): {inv_fact@mat} Smith_basis: (mat->mat): {Smith_basis@mat} Smith: (mat->mat,vec): {Smith@mat} invert: (mat->mat,int): {invert@mat} subspace_normal: ([vec]->[vec],[vec],[vec],[int]): {subspace_normal@[vec]} Lie_type: (string->LieType): {Lie_type@string} Lie_type: (RootDatum->LieType): {Lie_type@RootDatum} Cartan_matrix: (LieType->mat): {Cartan_matrix@LieType} Cartan_matrix: (RootDatum->mat): {Cartan_matrix@RootDatum} Cartan_matrix_type: (mat->LieType,vec): {Cartan_matrix_type@mat} rank: (LieType->int): {rank@LieType} rank: (RootDatum->int): {rank@RootDatum} semisimple_rank: (LieType->int): {semisimple_rank@LieType} semisimple_rank: (RootDatum->int): {semisimple_rank@RootDatum} str: (LieType->string): {str@LieType} Smith_Cartan: (LieType->mat,vec): {Smith_Cartan@LieType} filter_units: (mat,vec->mat,vec): {filter_units@(mat,vec)} ann_mod: (mat,int->mat): {ann_mod@(mat,int)} replace_gen: ((mat,vec),mat->mat): {replace_gen@((mat,vec),mat)} involution: (LieType,string->mat): {involution@(LieType,string)} involution: (LieType,mat,string->mat): {involution@(LieType,mat,string)} involution: (CartanClass->mat): {involution@CartanClass} involution: (KGBElt->mat): {involution@KGBElt} root_datum: (LieType,mat->RootDatum): {root_datum@(LieType,mat)} root_datum: ([vec],[vec],int->RootDatum): {root_datum@([vec],[vec],int)} root_datum: (LieType,[ratvec]->RootDatum): {root_datum@(LieType,[ratvec])} root_datum: (InnerClass->RootDatum): {root_datum@InnerClass} quotient_basis: (LieType,[ratvec]->mat): {quotient_basis@(LieType,[ratvec])} simply_connected: (LieType->RootDatum): {simply_connected@LieType} adjoint: (LieType->RootDatum): {adjoint@LieType} simple_roots: (RootDatum->mat): {simple_roots@RootDatum} simple_coroots: (RootDatum->mat): {simple_coroots@RootDatum} positive_roots: (RootDatum->mat): {positive_roots@RootDatum} positive_coroots: (RootDatum->mat): {positive_coroots@RootDatum} roots: (RootDatum->mat): {roots@RootDatum} coroots: (RootDatum->mat): {coroots@RootDatum} root_coradical: (RootDatum->mat): {root_coradical@RootDatum} coroot_radical: (RootDatum->mat): {coroot_radical@RootDatum} fundamental_weight: (RootDatum,int->ratvec): {fundamental_weight@(RootDatum,int)} fundamental_coweight: (RootDatum,int->ratvec): {fundamental_coweight@(RootDatum,int)} dual: (RootDatum->RootDatum): {dual@RootDatum} dual: (InnerClass->InnerClass): {dual@InnerClass} derived: (RootDatum->RootDatum,mat): {derived@RootDatum} integrality_datum: (RootDatum,ratvec->RootDatum): {integrality_datum@(RootDatum,ratvec)} integrality_points: (RootDatum,ratvec->[rat]): {integrality_points@(RootDatum,ratvec)} classify_involution: (mat->int,int,int): {classify_involution@mat} inner_class: (RootDatum,mat->InnerClass): {inner_class@(RootDatum,mat)} inner_class: (LieType,[ratvec],string->InnerClass): {inner_class@(LieType,[ratvec],string)} inner_class: (RootDatum,string->InnerClass): {inner_class@(RootDatum,string)} inner_class: (RealForm->InnerClass): {inner_class@RealForm} twisted_involution: (RootDatum,mat->InnerClass,vec): {twisted_involution@(RootDatum,mat)} distinguished_involution: (InnerClass->mat): {distinguished_involution@InnerClass} form_names: (InnerClass->[string]): {form_names@InnerClass} dual_form_names: (InnerClass->[string]): {dual_form_names@InnerClass} nr_of_real_forms: (InnerClass->int): {nr_of_real_forms@InnerClass} nr_of_dual_real_forms: (InnerClass->int): {nr_of_dual_real_forms@InnerClass} nr_of_Cartan_classes: (InnerClass->int): {nr_of_Cartan_classes@InnerClass} block_sizes: (InnerClass->mat): {block_sizes@InnerClass} occurrence_matrix: (InnerClass->mat): {occurrence_matrix@InnerClass} dual_occurrence_matrix: (InnerClass->mat): {dual_occurrence_matrix@InnerClass} real_form: (InnerClass,int->RealForm): {real_form@(InnerClass,int)} real_form: (DualRealForm->RealForm): {real_form@DualRealForm} real_form: (KGBElt->RealForm): {real_form@KGBElt} real_form: (Param->RealForm): {real_form@Param} real_form: (ParamPol->RealForm): {real_form@ParamPol} quasisplit_form: (InnerClass->RealForm): {quasisplit_form@InnerClass} components_rank: (RealForm->int): {components_rank@RealForm} count_Cartans: (RealForm->int): {count_Cartans@RealForm} KGB_size: (RealForm->int): {KGB_size@RealForm} Cartan_order: (RealForm->mat): {Cartan_order@RealForm} dual_real_form: (InnerClass,int->DualRealForm): {dual_real_form@(InnerClass,int)} dual_quasisplit_form: (InnerClass->DualRealForm): {dual_quasisplit_form@InnerClass} Cartan_class: (InnerClass,int->CartanClass): {Cartan_class@(InnerClass,int)} Cartan_class: (RealForm,int->CartanClass): {Cartan_class@(RealForm,int)} most_split_Cartan: (RealForm->CartanClass): {most_split_Cartan@RealForm} Cartan_info: (CartanClass->(int,int,int),vec,(int,int),(LieType,LieType,LieType)): {Cartan_info@CartanClass} real_forms: (CartanClass->[RealForm]): {real_forms@CartanClass} dual_real_forms: (CartanClass->[DualRealForm]): {dual_real_forms@CartanClass} fiber_part: (CartanClass,RealForm->[int]): {fiber_part@(CartanClass,RealForm)} KGB: (RealForm,int->KGBElt): {KGB@(RealForm,int)} cross: (int,KGBElt->KGBElt): {cross@(int,KGBElt)} Cayley: (int,KGBElt->KGBElt): {Cayley@(int,KGBElt)} status: (int,KGBElt->int): {status@(int,KGBElt)} twist: (KGBElt->KGBElt): {twist@KGBElt} torus_bits: (KGBElt->vec): {torus_bits@KGBElt} torus_factor: (KGBElt->ratvec): {torus_factor@KGBElt} param: (KGBElt,vec,ratvec->Param): {param@(KGBElt,vec,ratvec)} infinitesimal_character: (Param->ratvec): {infinitesimal_character@Param} is_standard: (Param->bool): {is_standard@Param} is_zero: (Param->bool): {is_zero@Param} is_final: (Param->bool): {is_final@Param} dominant: (Param->Param): {dominant@Param} orientation_nr: (Param->int): {orientation_nr@Param} reducibility_points: (Param->[rat]): {reducibility_points@Param} print_n_block: (Param->): {print_n_block@Param} n_block: (Param->[Param],int): {n_block@Param} KL_block: (Param->[Param],int,mat,[vec],vec,vec,mat): {KL_block@Param} partial_block: (Param->[Param],mat,[vec],vec,vec,mat): {partial_block@Param} null_module: (RealForm->ParamPol): {null_module@RealForm} deform: (Param->ParamPol): {deform@Param} full_deform: (Param->ParamPol): {full_deform@Param} KL_sum_at_s: (Param->ParamPol): {KL_sum_at_s@Param} raw_KL: (RealForm,DualRealForm->mat,[vec],vec): {raw_KL@(RealForm,DualRealForm)} dual_KL: (RealForm,DualRealForm->mat,[vec],vec): {dual_KL@(RealForm,DualRealForm)} print_gradings: (CartanClass,RealForm->): {print_gradings@(CartanClass,RealForm)} print_real_Weyl: (RealForm,CartanClass->): {print_real_Weyl@(RealForm,CartanClass)} print_strong_real: (CartanClass->): {print_strong_real@CartanClass} print_block: (RealForm,DualRealForm->): {print_block@(RealForm,DualRealForm)} print_blocku: (RealForm,DualRealForm->): {print_blocku@(RealForm,DualRealForm)} print_blockd: (RealForm,DualRealForm->): {print_blockd@(RealForm,DualRealForm)} print_blockstabilizer: (RealForm,DualRealForm,CartanClass->): {print_blockstabilizer@(RealForm,DualRealForm,CartanClass)} print_KGB: (RealForm->): {print_KGB@RealForm} print_X: (InnerClass->): {print_X@InnerClass} print_KL_basis: (RealForm,DualRealForm->): {print_KL_basis@(RealForm,DualRealForm)} print_prim_KL: (RealForm,DualRealForm->): {print_prim_KL@(RealForm,DualRealForm)} print_KL_list: (RealForm,DualRealForm->): {print_KL_list@(RealForm,DualRealForm)} print_W_cells: (RealForm,DualRealForm->): {print_W_cells@(RealForm,DualRealForm)} print_W_graph: (RealForm,DualRealForm->): {print_W_graph@(RealForm,DualRealForm)} Global variables: v: [rat]: [3/2,4/5] expr> [int]) Added definition [11] of #: (bool->int) Added definition [12] of #: (ratvec->int) Defined n_rows: (mat->int) Defined n_columns: (mat->int) Added definition [6] of !=: (string,string->bool) Added definition [3] of >: (string,string->bool) Added definition [3] of >=: (string,string->bool) Added definition [3] of <: (string,string->bool) Added definition [11] of *: (string,int->string) Added definition [12] of *: (int,vec->vec) Added definition [13] of *: (int,mat->mat) Added definition [9] of +: (string,int->string) Added definition [10] of +: (int,string->string) Added definition [11] of +: (string,string->string) Added definition [5] of ^: (mat,int->mat) Added definition [8] of -: (vec->vec) Added definition [9] of -: (mat->mat) Added definition [12] of +: (vec,vec->vec) Added definition [10] of -: (vec,vec->vec) Added definition [7] of %: (vec,int->vec) Defined sum: (vec->int) Added definition [2] of sum: (mat->vec) Added definition [13] of +: (string,(int,int)->string) Defined floor: (rat->int) Defined ceil: (rat->int) Defined abs: (int->int) Defined sgn: (int->int) Defined numer: (rat->int) Defined denom: (rat->int) Added definition [2] of numer: (ratvec->vec) Added definition [2] of denom: (ratvec->int) Added definition [9] of =: (ratvec,ratvec->bool) Added definition [11] of -: (ratvec->ratvec) Added definition [14] of *: (int,ratvec->ratvec) Added definition [5] of /: (ratvec,int->ratvec) Added definition [15] of *: (rat,ratvec->ratvec) Added definition [16] of *: (ratvec,vec->rat) Added definition [2] of floor: ([rat]->[int]) Added definition [2] of ceil: ([rat]->[int]) Added definition [10] of =: ((int,int),(int,int)->bool) Added definition [7] of !=: ((int,int),(int,int)->bool) Added definition [11] of =: (Split,Split->bool) Added definition [8] of !=: (Split,Split->bool) Identifier s: Split Added definition [14] of +: (Split->int) Added definition [6] of ^: (Split->int) Defined split_power: (Split,int->Split) Added definition [7] of ^: (Split,int->Split) Added definition [12] of -: (ParamPol,(Split,Param)->ParamPol) Added definition [15] of +: (mat,mat->mat) Added definition [13] of -: (mat,mat->mat) Defined min: (int,int->int) Defined max: (int,int->int) Defined ones: (int->vec) Defined rho: (RootDatum->ratvec) Defined imaginary_roots: (RootDatum,mat->mat) Defined real_roots: (RootDatum,mat->mat) Defined imaginary_coroots: (RootDatum,mat->mat) Defined real_coroots: (RootDatum,mat->mat) Defined simple_reflection: (RootDatum,int->mat) Defined reflection: (RootDatum,int->mat) Added definition [5] of involution: (InnerClass,[int]->mat) Added definition [2] of imaginary_roots: (InnerClass,[int]->mat) Added definition [2] of real_roots: (InnerClass,[int]->mat) Added definition [2] of imaginary_coroots: (InnerClass,[int]->mat) Added definition [2] of real_coroots: (InnerClass,[int]->mat) Defined dual_integral: (InnerClass,ratvec->InnerClass) Added definition [2] of KGB: (RealForm->[KGBElt]) Defined is_complex: (int,KGBElt->bool) Defined is_real: (int,KGBElt->bool) Defined is_imaginary: (int,KGBElt->bool) Defined is_noncompact: (int,KGBElt->bool) Defined is_compact: (int,KGBElt->bool) Defined is_descent: (int,KGBElt->bool) Defined is_ascent: (int,KGBElt->bool) Defined status_code: (int,KGBElt->string) Added definition [2] of status_code: (KGBElt->[string]) Completely read file 'basic.rx'. Defined show: (string->string) Defined print_Cartan_info: (CartanClass->) Defined Cartan_dispatch: (int,int,int,int,bool->int) Defined Cartan_entry: (int,int,int,int->int) Added definition [3] of Cartan_matrix: (int,int->mat) Added definition [3] of Lie_type: (int,int->LieType) Defined block_repeat: (mat,int->mat) Defined complex: (RootDatum->RealForm) Defined complexification: (RealForm->RealForm) Defined GL_roots: (int->[vec]) Defined SL_coroots: (int->[vec]) Defined SL_roots: (int->[vec]) Defined GL: (int->RootDatum) Defined SL: (int->RootDatum) Defined PSL: (int->RootDatum) Defined type_B_roots: (int->[vec]) Defined type_C_roots: (int->[vec]) Defined type_D_roots: (int->[vec]) Defined Sp: (int->RootDatum) Defined PSp: (int->RootDatum) Defined GSp: (int->RootDatum) Defined SO: (int->RootDatum) Defined SO_lie_type: (int->string) Defined Spin: (int->RootDatum) Defined PSO: (int->RootDatum) Defined SU: (int,int,RootDatum->RealForm) Added definition [2] of SU: (int,int->RealForm) Defined PSU: (int,int->RealForm) Added definition [3] of SU: (int->RealForm) Added definition [2] of PSU: (int->RealForm) Defined U: (int,int->RealForm) Defined SL_R: (int->RealForm) Defined PSL_R: (int->RealForm) Defined PGL_R: (int->RealForm) Defined SL_C: (int->RealForm) Defined PSL_C: (int->RealForm) Defined PGL_C: (int->RealForm) Defined SL_H: (int->RealForm) Defined PSL_H: (int->RealForm) Defined PGL_H: (int->RealForm) Defined GL_R: (int->RealForm) Defined GL_C: (int->RealForm) Defined GL_H: (int->RealForm) Identifiers R: string, C: string, H: string Defined by_field: ((int->RealForm),(int->RealForm),(int->RealForm)->(string->(int->RealForm))) Added definition [2] of SL: (int,string->RealForm) Added definition [2] of GL: (int,string->RealForm) Added definition [2] of PSL: (int,string->RealForm) Defined PGL: (int,string->RealForm) Defined Sp_R: (int->RealForm) Defined PSp_R: (int->RealForm) Defined GSp_R: (int->RealForm) Added definition [2] of Sp: (int,int->RealForm) Added definition [2] of PSp: (int,int->RealForm) Defined Sp_C: (int->RealForm) Defined Sp_H: (int->RealForm) Defined PSp_C: (int->RealForm) Defined PSp_H: (int->RealForm) Defined GSp_C: (int->RealForm) Defined GSp_H: (int->RealForm) Added definition [3] of Sp: (int,string->RealForm) Added definition [3] of PSp: (int,string->RealForm) Added definition [2] of GSp: (int,string->RealForm) Defined SO_inner_class: (int,int->string) Defined SO_real_form_number: (int,int->int) Added definition [2] of SO: (int,int,RootDatum->RealForm) Added definition [2] of Spin: (int,int->RealForm) Added definition [3] of SO: (int,int->RealForm) Added definition [2] of PSO: (int,int->RealForm) Identifier E6_ic_e: InnerClass Identifier E6_c: RealForm Identifier E6_h: RealForm Identifier E6_D5T: RealForm Identifier E6_q: RealForm Identifier E6_ic_s: InnerClass Identifier E6_F4: RealForm Identifier E6_s: RealForm Identifier E6_C4: RealForm Identifier E7_ic_e: InnerClass Identifier E7_ic: InnerClass Identifier E7_c: RealForm Identifier E7_h: RealForm Identifier E7_E6T: RealForm Identifier E7_q: RealForm Identifier E7_D6A1: RealForm Identifier E7_s: RealForm Identifier E7_A7: RealForm Identifier E8_ic: (->InnerClass) Identifier E8_c: (->RealForm) Identifier E8_q: (->RealForm) Identifier E8_s: (->RealForm) Completely read file 'groups.rx'. Starting to read from file 'hermitian.rx'. Starting to read from file 'misc.rx'. Defined is_even: (int->bool) Defined is_integer: (rat->bool) Added definition [2] of is_integer: (ratvec->bool) Defined rattoint: (rat->int) Defined ratvectovec: (ratvec->vec) Added definition [17] of *: (ratvec,ratvec->rat) Added definition [8] of %: (ParamPol->[(Split,Param)]) Defined split_form: (RootDatum->RealForm) Added definition [2] of split_form: (LieType->RealForm) Defined split_group: (string->RealForm) Defined Quasisplit_form: (RootDatum->RealForm) Added definition [2] of Quasisplit_form: (LieType->RealForm) Added definition [5] of root_datum: (Param->RootDatum) Defined print_real_forms: (RealForm->) Defined ss_rank: (RootDatum->int) Defined is_regular: (RootDatum,ratvec->bool) Added definition [2] of is_regular: (RealForm,ratvec->bool) Added definition [3] of is_regular: (Param->bool) Defined rhocheck: (RootDatum->ratvec) Added definition [2] of rhocheck: (RealForm->ratvec) Defined intrho: (RealForm->vec) Added definition [2] of intrho: (RootDatum->vec) Added definition [2] of integrality_datum: (Param->RootDatum) Defined x: (Param->KGBElt) Defined lambdaminusrho: (Param->vec) Defined lambda: (Param->ratvec) Defined nu: (Param->ratvec) Added definition [2] of param: (RealForm,int,vec,ratvec->Param) Defined normparam: (RealForm,int,ratvec,ratvec->Param) Defined finite_dimensional: (RealForm,vec->Param) Defined get_n_block: (Param->[Param]) Defined trivial: (RealForm->Param) Defined trivial_block: (RealForm->[Param]) Defined spherical_principal_series: (RealForm,ratvec->Param) Defined make_final: (Param->[Param]) Defined prettyPrint: (Split->string) Added definition [2] of prettyPrint: (ParamPol->) Added definition [13] of #: (Split,int->int) Added definition [14] of #: (Split,rat->rat) Added definition [15] of #: (ParamPol,int->ParamPol) Defined eval: (vec,Split->Split) Defined at_s: (vec->Split) Added definition [2] of eval: (vec,int->int) Completely read file 'misc.rx'. Starting to read from file 'kl.rx'. Defined length: (int,vec->int) Defined length_diff: (int,int,vec->int) Redefined +: (vec,vec->vec) Defined strip: (vec->vec) Defined mat_mult_int_poly: (mat,mat,[vec],int,int->vec) Defined KL: (int,mat,[vec]->[vec]) Added definition [2] of KL: (int,mat,[vec],mat->[vec]) Defined signedKL: (int,mat,[vec],vec->[vec]) Added definition [2] of signedKL: (int,mat,[vec],vec,vec,mat->[vec]) Defined KL_regular: (Param->[vec]) Defined signedKL_regular: (Param->[vec]) Defined KL_singular: (Param->[vec]) Defined signedKL_singular: (Param->[vec]) Added definition [3] of KL: (Param->[vec]) Added definition [3] of signedKL: (Param->[vec]) Defined KL_regular_matrix_at_one: (Param->mat) Defined signedKL_regular_matrix_at_one: (Param->mat) Defined KL_singular_matrix_at_one: (Param->mat) Defined signedKL_singular_matrix_at_one: (Param->mat) Defined KL_matrix_at_one: (Param->mat) Defined signedKL_matrix_at_one: (Param->mat) Defined extract_submatrix: (mat,vec,int->mat,int) Defined get_singular_index: (vec,int->int) Added definition [2] of get_singular_index: (Param->int) Defined KL_regular_inverse_matrix_at_one: (Param->mat) Defined KL_singular_inverse_matrix_at_one: (Param->mat) Defined KL_singular_inverse_matrix_at_one_alt: (Param->mat) Defined KL_inverse_matrix_at_one: (Param->mat) Defined regular_character_formula: (Param->ParamPol) Defined singular_character_formula: (Param->ParamPol) Defined character_formula: (Param->ParamPol) Defined regular_composition_series: (Param->ParamPol) Defined singular_composition_series: (Param->ParamPol) Defined composition_series: (Param->ParamPol) Defined stringPoly: (vec,string->string) Defined printPoly: (vec->) Defined print_KL: (Param->) Defined print_signedKL: (Param->) Defined print_regular_character_formula: (Param->) Defined print_singular_character_formula: (Param->) Defined print_character_formula: (Param->) Defined print_regular_composition_series: (Param->) Defined print_singular_composition_series: (Param->) Defined print_composition_series: (Param->) Completely read file 'kl.rx'. Identifier hermitian_debug: bool Added definition [14] of -: (ParamPol,ParamPol->ParamPol) Added definition [12] of =: (ParamPol,ParamPol->bool) Added definition [18] of *: (Param,rat->Param) Defined make_nu0: (Param->Param) Added definition [2] of make_nu0: (ParamPol->ParamPol) Defined orientation_nr_term: (int,Param->Split) Defined c_form_std: (Param->ParamPol) Defined c_form_irreducible_long: (Param->ParamPol,[(Param,Split,ParamPol)]) Defined KL_col: (Param->ParamPol) Defined c_form_irreducible: (Param->ParamPol) Defined print_c_form_irreducible_long: (Param->) Defined mixed: (Split->bool) Defined analyse: (ParamPol->) Defined lambda_plus_rho_res_T: (KGBElt,ratvec->ratvec) Defined mu: (Param->rat) Defined convert_cform_hermitian: (ParamPol->ParamPol) Defined hermitian_form_irreducible: (Param->ParamPol) Defined print_hermitian_form_irreducible: (Param->) Added definition [2] of print_hermitian_form_irreducible: ([Param]->) Defined is_unitary: (ParamPol->bool) Added definition [2] of is_unitary: (Param->bool) Defined test_spherical_unitary: (RealForm,[ratvec]->) Defined test_line: (Param->) Completely read file 'hermitian.rx'. Defined Cartan_list: (RealForm->[CartanClass]) Defined print_cartan_info_short: (CartanClass->) Defined list_cartans: (RealForm->[()]) Defined describe_Cartan: (KGBElt->) Identifier G2_ic: InnerClass Identifier G2_c: RealForm Identifier G2_s: RealForm Identifier F4_ic: InnerClass Identifier F4_c: RealForm Identifier F4_B4: RealForm Identifier F4_s: RealForm Completely read file 'workshop.rx'. expr> set G=SL(2,R) { give our favourite group the name 'G' } Identifier G: RealForm expr> list_cartans(G) { show its Cartan classes } Cartan Number 0: compact: 1, complex: 0, split: 0 Cartan Number 1: compact: 0, complex: 0, split: 1 Value: [(),()] expr> set H=E6_s { another group (the name E6_s was defined in a script } Identifier H (overriding previous): RealForm expr> list_cartans(H) { we could also have said list_cartans(E6_s) } Cartan Number 0: compact: 2, complex: 2, split: 0 Cartan Number 1: compact: 0, complex: 3, split: 0 Cartan Number 2: compact: 0, complex: 2, split: 2 Cartan Number 3: compact: 0, complex: 1, split: 4 Cartan Number 4: compact: 0, complex: 0, split: 6 Value: [(),(),(),(),()] expr> set cl=Cartan_list(H) { one can alternatively store a list of values } Identifier cl: [CartanClass] expr> cl { printing isn't very beautiful } Value: [Cartan class #0, occurring for 2 real forms and for 1 dual real form,Cartan class #1, occurring for 1 real form and for 1 dual real form,Cartan class #2, occurring for 1 real form and for 2 dual real forms,Cartan class #3, occurring for 1 real form and for 2 dual real forms,Cartan class #4, occurring for 1 real form and for 3 dual real forms] expr> print_Cartan_info(cl[2]) { here we select a Cartan class, show its info } compact: 0, complex: 2, split: 2 canonical twisted involution: 1,2,3,4,2,3,1,5,4,2,3,1,4,5,6,5,4,2,3,1,4,3,5,4,2,6,5,4,3,1 twisted involution orbit size: 270; fiber size: 1; strong inv: 270 imaginary root system: A1.A1 real root system: A3 complex factor: A1 expr> print_real_Weyl(H,cl[2]) { another one } real weyl group is W^C.((A.W_ic) x W^R), where: W^C is isomorphic to a Weyl group of type A1 A is an elementary abelian 2-group of rank 2 W_ic is trivial W^R is a Weyl group of type A3 generators for W^C: 3,5 generators for A 4 3,4,5,4,3 generators for W^R: 2,3,4,2,3 2,4,5,4,2 1,3,4,5,6,5,4,3,1 expr> print_real_Weyl(H,cl[5]) { the last of 5 is not number 5 } Runtime error: index 5 out of range (<5) in subscription cl[5] Evaluation aborted. expr> print_real_Weyl(H,cl[4]) { but rather number 4 } real weyl group is W^C.((A.W_ic) x W^R), where: W^C is trivial A is trivial W_ic is trivial W^R is a Weyl group of type E6 generators for W^R: 1 2 3 4 5 6 expr> set G=U(5,2) { nother example } Identifier G (overriding previous): RealForm expr> Lie_type(G) Value: Lie type 'A6.T1' expr> list_cartans(G) Cartan Number 0: compact: 7, complex: 0, split: 0 Cartan Number 1: compact: 5, complex: 1, split: 0 Cartan Number 2: compact: 3, complex: 2, split: 0 Value: [(),(),()] expr> print_KGB(G) { list the set K\G/B } kgbsize: 231 Base grading: [111111]. 0: 0 [c,c,n,n,n,n] 0 0 1 2 3 4 * * 36 31 26 21 (0,1,0,0,0,0,0)#0 e 1: 0 [c,n,n,c,n,n] 1 5 0 1 6 7 * 41 36 * 27 22 (0,1,1,1,0,0,0)#0 e 2: 0 [c,c,c,n,c,n] 2 2 2 0 2 8 * * * 31 * 23 (0,1,0,1,1,0,0)#0 e 3: 0 [c,c,n,c,n,c] 3 3 6 3 0 3 * * 37 * 26 * (0,1,0,0,1,1,0)#0 e 4: 0 [c,c,n,n,c,n] 4 4 7 8 4 0 * * 38 32 * 21 (0,1,0,0,0,1,1)#0 e 5: 0 [n,n,c,c,n,n] 9 1 5 5 10 11 46 41 * * 28 24 (0,0,0,1,0,0,0)#0 e 6: 0 [c,n,n,n,n,c] 6 10 3 12 1 6 * 42 37 33 27 * (0,1,1,1,1,1,0)#0 e 7: 0 [c,n,n,c,c,n] 7 11 4 7 7 1 * 43 38 * * 22 (0,1,1,1,0,1,1)#0 e 8: 0 [c,c,c,n,n,n] 8 8 8 4 13 2 * * * 32 29 23 (0,1,0,1,1,1,1)#0 e 9: 0 [n,c,c,c,n,n] 5 9 9 9 14 15 46 * * * 30 25 (1,1,0,1,0,0,0)#0 e 10: 0 [n,n,c,n,n,c] 14 6 10 16 5 10 47 42 * 34 28 * (0,0,0,1,1,1,0)#0 e 11: 0 [n,n,c,c,c,n] 15 7 11 11 11 5 48 43 * * * 24 (0,0,0,1,0,1,1)#0 e 12: 0 [c,n,c,n,c,c] 12 16 12 6 12 12 * 44 * 33 * * (0,1,1,0,0,1,0)#0 e 13: 0 [c,c,c,c,n,c] 13 13 13 13 8 13 * * * * 29 * (0,1,0,1,0,0,1)#0 e 14: 0 [n,c,c,n,n,c] 10 14 14 17 9 14 47 * * 35 30 * (1,1,0,1,1,1,0)#0 e 15: 0 [n,c,c,c,c,n] 11 15 15 15 15 9 48 * * * * 25 (1,1,0,1,0,1,1)#0 e 16: 0 [n,n,n,n,c,c] 17 12 18 10 16 16 49 44 39 34 * * (0,0,0,0,0,1,0)#0 e 17: 0 [n,c,n,n,c,c] 16 17 19 14 17 17 49 * 40 35 * * (1,1,0,0,0,1,0)#0 e 18: 0 [n,c,n,c,c,c] 19 18 16 18 18 18 50 * 39 * * * (0,0,1,1,0,1,0)#0 e 19: 0 [n,n,n,c,c,c] 18 20 17 19 19 19 50 45 40 * * * (1,1,1,1,0,1,0)#0 e 20: 0 [c,n,c,c,c,c] 20 19 20 20 20 20 * 45 * * * * (1,0,0,1,0,1,0)#0 e 21: 1 [c,c,n,n,C,r] 21 21 22 23 61 21 * * 52 51 * * (0,1,0,0,0,0,0) 1 6^e 22: 1 [c,n,n,c,C,r] 22 24 21 22 62 22 * 54 52 * * * (0,1,1,1,0,0,0) 1 6^e 23: 1 [c,c,c,n,C,r] 23 23 23 21 63 23 * * * 51 * * (0,1,0,1,1,0,0) 1 6^e 24: 1 [n,n,c,c,C,r] 25 22 24 24 64 24 57 54 * * * * (0,0,0,1,0,0,0) 1 6^e 25: 1 [n,c,c,c,C,r] 24 25 25 25 65 25 57 * * * * * (1,1,0,1,0,0,0) 1 6^e 26: 1 [c,c,n,C,r,C] 26 26 27 66 26 61 * * 53 * * * (0,1,0,0,0,0,0) 1 5^e 27: 1 [c,n,n,C,r,C] 27 28 26 67 27 62 * 55 53 * * * (0,1,1,1,0,0,0) 1 5^e 28: 1 [n,n,c,C,r,C] 30 27 28 69 28 64 58 55 * * * * (0,0,0,1,0,0,0) 1 5^e 29: 1 [c,c,c,C,r,C] 29 29 29 68 29 63 * * * * * * (0,1,0,1,0,0,1) 1 5^e 30: 1 [n,c,c,C,r,C] 28 30 30 70 30 65 58 * * * * * (1,1,0,1,0,0,0) 1 5^e 31: 1 [c,c,C,r,C,n] 31 31 71 31 66 32 * * * * * 51 (0,1,0,0,0,0,0) 1 4^e 32: 1 [c,c,C,r,C,n] 32 32 72 32 68 31 * * * * * 51 (0,1,0,0,0,1,1) 1 4^e 33: 1 [c,n,C,r,C,c] 33 34 73 33 67 33 * 56 * * * * (0,1,1,0,0,1,0) 1 4^e 34: 1 [n,n,C,r,C,c] 35 33 74 34 69 34 59 56 * * * * (0,0,0,0,0,1,0) 1 4^e 35: 1 [n,c,C,r,C,c] 34 35 75 35 70 35 59 * * * * * (1,1,0,0,0,1,0) 1 4^e 36: 1 [c,C,r,C,n,n] 36 76 36 71 37 38 * * * * 53 52 (0,1,0,0,0,0,0) 1 3^e 37: 1 [c,C,r,C,n,c] 37 77 37 73 36 37 * * * * 53 * (0,1,0,0,1,1,0) 1 3^e 38: 1 [c,C,r,C,c,n] 38 78 38 72 38 36 * * * * * 52 (0,1,0,0,0,1,1) 1 3^e 39: 1 [n,C,r,C,c,c] 40 79 39 74 39 39 60 * * * * * (0,0,0,0,0,1,0) 1 3^e 40: 1 [n,C,r,C,c,c] 39 80 40 75 40 40 60 * * * * * (1,1,0,0,0,1,0) 1 3^e 41: 1 [C,r,C,c,n,n] 81 41 76 41 42 43 * * * * 55 54 (0,0,0,1,0,0,0) 1 2^e 42: 1 [C,r,C,n,n,c] 82 42 77 44 41 42 * * * 56 55 * (0,0,0,1,1,1,0) 1 2^e 43: 1 [C,r,C,c,c,n] 83 43 78 43 43 41 * * * * * 54 (0,0,0,1,0,1,1) 1 2^e 44: 1 [C,r,C,n,c,c] 84 44 79 42 44 44 * * * 56 * * (0,0,0,0,0,1,0) 1 2^e 45: 1 [C,r,C,c,c,c] 85 45 80 45 45 45 * * * * * * (1,0,0,1,0,1,0) 1 2^e 46: 1 [r,C,c,c,n,n] 46 81 46 46 47 48 * * * * 58 57 (0,0,0,1,0,0,0) 1 1^e 47: 1 [r,C,c,n,n,c] 47 82 47 49 46 47 * * * 59 58 * (0,0,0,1,1,1,0) 1 1^e 48: 1 [r,C,c,c,c,n] 48 83 48 48 48 46 * * * * * 57 (0,0,0,1,0,1,1) 1 1^e 49: 1 [r,C,n,n,c,c] 49 84 50 47 49 49 * * 60 59 * * (0,0,0,0,0,1,0) 1 1^e 50: 1 [r,C,n,c,c,c] 50 85 49 50 50 50 * * 60 * * * (0,0,1,1,0,1,0) 1 1^e 51: 2 [c,c,C,r,C,r] 51 51 89 51 86 51 * * * * * * (0,1,0,0,0,0,0) 2 4^6^e 52: 2 [c,C,r,C,C,r] 52 93 52 89 88 52 * * * * * * (0,1,0,0,0,0,0) 2 3^6^e 53: 2 [c,C,r,C,r,C] 53 94 53 87 53 88 * * * * * * (0,1,0,0,0,0,0) 2 3^5^e 54: 2 [C,r,C,c,C,r] 99 54 93 54 91 54 * * * * * * (0,0,0,1,0,0,0) 2 2^6^e 55: 2 [C,r,C,C,r,C] 100 55 94 92 55 91 * * * * * * (0,0,0,1,0,0,0) 2 2^5^e 56: 2 [C,r,C,r,C,c] 101 56 90 56 92 56 * * * * * * (0,0,0,0,0,1,0) 2 2^4^e 57: 2 [r,C,c,c,C,r] 57 99 57 57 96 57 * * * * * * (0,0,0,1,0,0,0) 2 1^6^e 58: 2 [r,C,c,C,r,C] 58 100 58 97 58 96 * * * * * * (0,0,0,1,0,0,0) 2 1^5^e 59: 2 [r,C,C,r,C,c] 59 101 98 59 97 59 * * * * * * (0,0,0,0,0,1,0) 2 1^4^e 60: 2 [r,C,r,C,c,c] 60 95 60 98 60 60 * * * * * * (0,0,0,0,0,1,0) 2 1^3^e 61: 2 [c,c,n,C,C,C] 61 61 62 102 21 26 * * 88 * * * (0,1,0,0,0,0,0) 1 5x6^e 62: 2 [c,n,n,C,C,C] 62 64 61 103 22 27 * 91 88 * * * (0,1,1,1,0,0,0) 1 5x6^e 63: 2 [c,c,c,C,C,C] 63 63 63 104 23 29 * * * * * * (0,1,0,1,0,1,0) 1 5x6^e 64: 2 [n,n,c,C,C,C] 65 62 64 105 24 28 96 91 * * * * (0,0,0,1,0,0,0) 1 5x6^e 65: 2 [n,c,c,C,C,C] 64 65 65 106 25 30 96 * * * * * (1,1,0,1,0,0,0) 1 5x6^e 66: 2 [c,c,C,C,C,C] 66 66 107 26 31 102 * * * * * * (0,1,0,0,0,0,0) 1 4x5^e 67: 2 [c,n,C,C,C,C] 67 69 108 27 33 103 * 92 * * * * (0,1,1,0,1,0,0) 1 4x5^e 68: 2 [c,c,C,C,C,C] 68 68 109 29 32 104 * * * * * * (0,1,0,0,1,0,1) 1 4x5^e 69: 2 [n,n,C,C,C,C] 70 67 110 28 34 105 97 92 * * * * (0,0,0,0,1,0,0) 1 4x5^e 70: 2 [n,c,C,C,C,C] 69 70 111 30 35 106 97 * * * * * (1,1,0,0,1,0,0) 1 4x5^e 71: 2 [c,C,C,C,C,n] 71 112 31 36 107 72 * * * * * 89 (0,1,0,0,0,0,0) 1 3x4^e 72: 2 [c,C,C,C,C,n] 72 113 32 38 109 71 * * * * * 89 (0,1,0,0,0,1,1) 1 3x4^e 73: 2 [c,C,C,C,C,c] 73 114 33 37 108 73 * * * * * * (0,1,0,1,0,1,0) 1 3x4^e 74: 2 [n,C,C,C,C,c] 75 115 34 39 110 74 98 * * * * * (0,0,0,0,0,1,0) 1 3x4^e 75: 2 [n,C,C,C,C,c] 74 116 35 40 111 75 98 * * * * * (1,1,0,0,0,1,0) 1 3x4^e 76: 2 [C,C,C,C,n,n] 117 36 41 112 77 78 * * * * 94 93 (0,0,1,0,0,0,0) 1 2x3^e 77: 2 [C,C,C,C,n,c] 118 37 42 114 76 77 * * * * 94 * (0,0,1,0,1,1,0) 1 2x3^e 78: 2 [C,C,C,C,c,n] 119 38 43 113 78 76 * * * * * 93 (0,0,1,0,0,1,1) 1 2x3^e 79: 2 [C,C,C,C,c,c] 120 39 44 115 79 79 * * * * * * (0,0,0,0,0,1,0) 1 2x3^e 80: 2 [C,C,C,C,c,c] 121 40 45 116 80 80 * * * * * * (1,0,1,0,0,1,0) 1 2x3^e 81: 2 [C,C,C,c,n,n] 41 46 117 81 82 83 * * * * 100 99 (0,0,0,1,0,0,0) 1 1x2^e 82: 2 [C,C,C,n,n,c] 42 47 118 84 81 82 * * * 101 100 * (0,0,0,1,1,1,0) 1 1x2^e 83: 2 [C,C,C,c,c,n] 43 48 119 83 83 81 * * * * * 99 (0,0,0,1,0,1,1) 1 1x2^e 84: 2 [C,C,C,n,c,c] 44 49 120 82 84 84 * * * 101 * * (0,0,0,0,0,1,0) 1 1x2^e 85: 2 [C,C,C,c,c,c] 45 50 121 85 85 85 * * * * * * (0,1,0,1,0,1,0) 1 1x2^e 86: 3 [c,c,C,C,C,C] 86 86 124 122 51 122 * * * * * * (0,1,0,0,0,0,0) 2 5x4^6^e 87: 3 [c,C,C,C,C,C] 87 128 125 53 125 123 * * * * * * (0,1,0,0,0,0,0) 2 4x3^5^e 88: 3 [c,C,r,C,C,C] 88 130 88 123 52 53 * * * * * * (0,1,0,0,0,0,0) 2 3^5x6^e 89: 3 [c,C,C,C,C,r] 89 129 51 52 124 89 * * * * * * (0,1,0,0,0,0,0) 2 3x4^6^e 90: 3 [C,C,C,C,C,c] 135 131 56 131 126 90 * * * * * * (0,0,0,0,0,1,0) 2 3x2^4^e 91: 3 [C,r,C,C,C,C] 138 91 130 127 54 55 * * * * * * (0,0,0,1,0,0,0) 2 2^5x6^e 92: 3 [C,r,C,C,C,C] 139 92 126 55 56 127 * * * * * * (0,0,0,0,1,0,0) 2 2^4x5^e 93: 3 [C,C,C,C,C,r] 136 52 54 129 130 93 * * * * * * (0,0,1,0,0,0,0) 2 2x3^6^e 94: 3 [C,C,C,C,r,C] 137 53 55 128 94 130 * * * * * * (0,0,1,0,0,0,0) 2 2x3^5^e 95: 3 [C,C,C,C,c,c] 140 60 140 132 95 95 * * * * * * (0,0,0,0,0,1,0) 2 2x1^3^e 96: 3 [r,C,c,C,C,C] 96 138 96 133 57 58 * * * * * * (0,0,0,1,0,0,0) 2 1^5x6^e 97: 3 [r,C,C,C,C,C] 97 139 134 58 59 133 * * * * * * (0,0,0,0,1,0,0) 2 1^4x5^e 98: 3 [r,C,C,C,C,c] 98 132 59 60 134 98 * * * * * * (0,0,0,0,0,1,0) 2 1^3x4^e 99: 3 [C,C,C,c,C,r] 54 57 136 99 138 99 * * * * * * (0,0,0,1,0,0,0) 2 1x2^6^e 100: 3 [C,C,C,C,r,C] 55 58 137 139 100 138 * * * * * * (0,0,0,1,0,0,0) 2 1x2^5^e 101: 3 [C,C,C,r,C,c] 56 59 135 101 139 101 * * * * * * (0,0,0,0,0,1,0) 2 1x2^4^e 102: 3 [c,c,C,C,n,C] 102 102 141 61 104 66 * * * * 122 * (0,1,0,0,0,0,0) 1 4x5x6^e 103: 3 [c,n,C,C,c,C] 103 105 142 62 103 67 * 127 * * * * (0,1,1,0,1,0,0) 1 4x5x6^e 104: 3 [c,c,C,C,n,C] 104 104 143 63 102 68 * * * * 122 * (0,1,0,0,1,1,0) 1 4x5x6^e 105: 3 [n,n,C,C,c,C] 106 103 144 64 105 69 133 127 * * * * (0,0,0,0,1,0,0) 1 4x5x6^e 106: 3 [n,c,C,C,c,C] 105 106 145 65 106 70 133 * * * * * (1,1,0,0,1,0,0) 1 4x5x6^e 107: 3 [c,C,C,n,C,C] 107 146 66 108 71 141 * * * 125 * * (0,1,0,0,0,0,0) 1 3x4x5^e 108: 3 [c,C,C,n,C,C] 108 147 67 107 73 142 * * * 125 * * (0,1,0,1,1,0,0) 1 3x4x5^e 109: 3 [c,C,C,c,C,C] 109 148 68 109 72 143 * * * * * * (0,1,0,0,1,0,1) 1 3x4x5^e 110: 3 [n,C,C,c,C,C] 111 149 69 110 74 144 134 * * * * * (0,0,0,0,1,0,0) 1 3x4x5^e 111: 3 [n,C,C,c,C,C] 110 150 70 111 75 145 134 * * * * * (1,1,0,0,1,0,0) 1 3x4x5^e 112: 3 [C,C,c,C,C,n] 151 71 112 76 146 113 * * * * * 129 (0,0,1,0,0,0,0) 1 2x3x4^e 113: 3 [C,C,c,C,C,n] 152 72 113 78 148 112 * * * * * 129 (0,0,1,0,0,1,1) 1 2x3x4^e 114: 3 [C,C,n,C,C,c] 153 73 115 77 147 114 * * 131 * * * (0,0,1,1,0,1,0) 1 2x3x4^e 115: 3 [C,C,n,C,C,c] 154 74 114 79 149 115 * * 131 * * * (0,0,0,0,0,1,0) 1 2x3x4^e 116: 3 [C,C,c,C,C,c] 155 75 116 80 150 116 * * * * * * (1,0,1,0,0,1,0) 1 2x3x4^e 117: 3 [C,c,C,C,n,n] 76 117 81 151 118 119 * * * * 137 136 (0,0,1,0,0,0,0) 1 1x2x3^e 118: 3 [C,c,C,C,n,c] 77 118 82 153 117 118 * * * * 137 * (0,0,1,0,1,1,0) 1 1x2x3^e 119: 3 [C,c,C,C,c,n] 78 119 83 152 119 117 * * * * * 136 (0,0,1,0,0,1,1) 1 1x2x3^e 120: 3 [C,n,C,C,c,c] 79 121 84 154 120 120 * 140 * * * * (0,0,0,0,0,1,0) 1 1x2x3^e 121: 3 [C,n,C,C,c,c] 80 120 85 155 121 121 * 140 * * * * (0,1,1,0,0,1,0) 1 1x2x3^e 122: 4 [c,c,C,C,r,C] 122 122 157 86 122 86 * * * * * * (0,1,0,0,0,0,0) 2 4x5x4^6^e 123: 4 [c,C,C,C,C,C] 123 161 158 88 156 87 * * * * * * (0,1,0,0,0,0,0) 2 4x3^5x6^e 124: 4 [c,C,C,C,C,C] 124 162 86 156 89 157 * * * * * * (0,1,0,0,0,0,0) 2 3x5x4^6^e 125: 4 [c,C,C,r,C,C] 125 163 87 125 87 158 * * * * * * (0,1,0,0,0,0,0) 2 3x4x3^5^e 126: 4 [C,C,C,C,C,C] 168 164 92 160 90 159 * * * * * * (0,0,0,0,1,0,0) 2 3x2^4x5^e 127: 4 [C,r,C,C,c,C] 173 127 159 91 127 92 * * * * * * (0,0,0,0,1,0,0) 2 2^4x5x6^e 128: 4 [C,C,C,C,C,C] 169 87 160 94 163 161 * * * * * * (0,0,1,0,0,0,0) 2 2x4x3^5^e 129: 4 [C,C,c,C,C,r] 170 89 129 93 162 129 * * * * * * (0,0,1,0,0,0,0) 2 2x3x4^6^e 130: 4 [C,C,C,C,C,C] 172 88 91 161 93 94 * * * * * * (0,0,1,0,0,0,0) 2 2x3^5x6^e 131: 4 [C,C,r,C,C,c] 171 90 131 90 164 131 * * * * * * (0,0,0,0,0,1,0) 2 2x3x2^4^e 132: 4 [C,C,C,C,C,c] 174 98 166 95 165 132 * * * * * * (0,0,0,0,0,1,0) 2 2x1^3x4^e 133: 4 [r,C,C,C,c,C] 133 173 167 96 133 97 * * * * * * (0,0,0,0,1,0,0) 2 1^4x5x6^e 134: 4 [r,C,C,c,C,C] 134 165 97 134 98 167 * * * * * * (0,0,0,0,1,0,0) 2 1^3x4x5^e 135: 4 [C,C,C,C,C,c] 90 166 101 171 168 135 * * * * * * (0,0,0,0,0,1,0) 2 1x3x2^4^e 136: 4 [C,c,C,C,C,r] 93 136 99 170 172 136 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3^6^e 137: 4 [C,c,C,C,r,C] 94 137 100 169 137 172 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3^5^e 138: 4 [C,C,C,C,C,C] 91 96 172 173 99 100 * * * * * * (0,0,0,1,0,0,0) 2 1x2^5x6^e 139: 4 [C,C,C,C,C,C] 92 97 168 100 101 173 * * * * * * (0,0,0,0,1,0,0) 2 1x2^4x5^e 140: 4 [C,r,C,C,c,c] 95 140 95 174 140 140 * * * * * * (0,0,0,0,0,1,0) 2 1x2x1^3^e 141: 4 [c,C,C,n,n,C] 141 175 102 142 143 107 * * * 158 157 * (0,1,0,0,0,0,0) 1 3x4x5x6^e 142: 4 [c,C,C,n,c,C] 142 176 103 141 142 108 * * * 158 * * (0,1,0,1,1,0,0) 1 3x4x5x6^e 143: 4 [c,C,C,c,n,C] 143 177 104 143 141 109 * * * * 157 * (0,1,0,0,1,1,0) 1 3x4x5x6^e 144: 4 [n,C,C,c,c,C] 145 178 105 144 144 110 167 * * * * * (0,0,0,0,1,0,0) 1 3x4x5x6^e 145: 4 [n,C,C,c,c,C] 144 179 106 145 145 111 167 * * * * * (1,1,0,0,1,0,0) 1 3x4x5x6^e 146: 4 [C,C,c,n,C,C] 180 107 146 147 112 175 * * * 163 * * (0,0,1,0,0,0,0) 1 2x3x4x5^e 147: 4 [C,C,n,n,C,C] 181 108 149 146 114 176 * * 164 163 * * (0,0,1,1,1,0,0) 1 2x3x4x5^e 148: 4 [C,C,c,c,C,C] 182 109 148 148 113 177 * * * * * * (0,0,1,0,1,0,1) 1 2x3x4x5^e 149: 4 [C,C,n,c,C,C] 183 110 147 149 115 178 * * 164 * * * (0,0,0,0,1,0,0) 1 2x3x4x5^e 150: 4 [C,C,c,c,C,C] 184 111 150 150 116 179 * * * * * * (1,0,1,0,1,0,0) 1 2x3x4x5^e 151: 4 [C,c,c,C,C,n] 112 151 151 117 180 152 * * * * * 170 (0,0,1,0,0,0,0) 1 1x2x3x4^e 152: 4 [C,c,c,C,C,n] 113 152 152 119 182 151 * * * * * 170 (0,0,1,0,0,1,1) 1 1x2x3x4^e 153: 4 [C,c,n,C,C,c] 114 153 154 118 181 153 * * 171 * * * (0,0,1,1,0,1,0) 1 1x2x3x4^e 154: 4 [C,n,n,C,C,c] 115 155 153 120 183 154 * 174 171 * * * (0,0,0,0,0,1,0) 1 1x2x3x4^e 155: 4 [C,n,c,C,C,c] 116 154 155 121 184 155 * 174 * * * * (0,1,1,0,0,1,0) 1 1x2x3x4^e 156: 5 [c,C,C,C,C,C] 156 187 185 124 123 185 * * * * * * (0,1,0,0,0,0,0) 2 4x3x5x4^6^e 157: 5 [c,C,C,C,r,C] 157 188 122 185 157 124 * * * * * * (0,1,0,0,0,0,0) 2 3x4x5x4^6^e 158: 5 [c,C,C,r,C,C] 158 189 123 158 185 125 * * * * * * (0,1,0,0,0,0,0) 2 3x4x3^5x6^e 159: 5 [C,C,C,C,c,C] 195 190 127 186 159 126 * * * * * * (0,0,0,0,1,0,0) 2 3x2^4x5x6^e 160: 5 [C,C,C,C,C,C] 194 191 128 126 191 186 * * * * * * (0,0,0,1,0,0,0) 2 3x2x4x3^5^e 161: 5 [C,C,C,C,C,C] 196 123 186 130 187 128 * * * * * * (0,0,1,0,0,0,0) 2 2x4x3^5x6^e 162: 5 [C,C,c,C,C,C] 197 124 162 187 129 188 * * * * * * (0,0,1,0,0,0,0) 2 2x3x5x4^6^e 163: 5 [C,C,C,r,C,C] 198 125 191 163 128 189 * * * * * * (0,0,1,0,0,0,0) 2 2x3x4x3^5^e 164: 5 [C,C,r,C,C,C] 199 126 164 191 131 190 * * * * * * (0,0,0,0,1,0,0) 2 2x3x2^4x5^e 165: 5 [C,C,C,c,C,C] 200 134 193 165 132 192 * * * * * * (0,0,0,0,1,0,0) 2 2x1^3x4x5^e 166: 5 [C,C,C,C,C,c] 201 135 132 201 193 166 * * * * * * (0,0,0,0,0,1,0) 2 2x1x3x2^4^e 167: 5 [r,C,C,c,c,C] 167 192 133 167 167 134 * * * * * * (0,0,0,0,1,0,0) 2 1^3x4x5x6^e 168: 5 [C,C,C,C,C,C] 126 193 139 194 135 195 * * * * * * (0,0,0,0,1,0,0) 2 1x3x2^4x5^e 169: 5 [C,c,C,C,C,C] 128 169 194 137 198 196 * * * * * * (0,0,1,0,0,0,0) 2 1x2x4x3^5^e 170: 5 [C,c,c,C,C,r] 129 170 170 136 197 170 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x4^6^e 171: 5 [C,C,r,C,C,c] 131 201 171 135 199 171 * * * * * * (0,0,0,0,0,1,0) 2 1x2x3x2^4^e 172: 5 [C,c,C,C,C,C] 130 172 138 196 136 137 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3^5x6^e 173: 5 [C,C,C,C,c,C] 127 133 195 138 173 139 * * * * * * (0,0,0,0,1,0,0) 2 1x2^4x5x6^e 174: 5 [C,r,C,C,C,c] 132 174 201 140 200 174 * * * * * * (0,0,0,0,0,1,0) 2 1x2x1^3x4^e 175: 5 [C,C,c,n,n,C] 202 141 175 176 177 146 * * * 189 188 * (0,0,1,0,0,0,0) 1 2x3x4x5x6^e 176: 5 [C,C,n,n,c,C] 203 142 178 175 176 147 * * 190 189 * * (0,0,1,1,1,0,0) 1 2x3x4x5x6^e 177: 5 [C,C,c,c,n,C] 204 143 177 177 175 148 * * * * 188 * (0,0,1,0,1,1,0) 1 2x3x4x5x6^e 178: 5 [C,C,n,c,c,C] 205 144 176 178 178 149 * * 190 * * * (0,0,0,0,1,0,0) 1 2x3x4x5x6^e 179: 5 [C,C,c,c,c,C] 206 145 179 179 179 150 * * * * * * (1,0,1,0,1,0,0) 1 2x3x4x5x6^e 180: 5 [C,c,c,n,C,C] 146 180 180 181 151 202 * * * 198 * * (0,0,1,0,0,0,0) 1 1x2x3x4x5^e 181: 5 [C,c,n,n,C,C] 147 181 183 180 153 203 * * 199 198 * * (0,0,1,1,1,0,0) 1 1x2x3x4x5^e 182: 5 [C,c,c,c,C,C] 148 182 182 182 152 204 * * * * * * (0,0,1,0,1,0,1) 1 1x2x3x4x5^e 183: 5 [C,n,n,c,C,C] 149 184 181 183 154 205 * 200 199 * * * (0,0,0,0,1,0,0) 1 1x2x3x4x5^e 184: 5 [C,n,c,c,C,C] 150 183 184 184 155 206 * 200 * * * * (0,1,1,0,1,0,0) 1 1x2x3x4x5^e 185: 6 [c,C,C,C,C,C] 185 208 156 157 158 156 * * * * * * (0,1,0,0,0,0,0) 2 3x4x3x5x4^6^e 186: 6 [C,C,C,C,C,C] 212 209 161 159 207 160 * * * * * * (0,0,0,1,0,0,0) 2 3x2x4x3^5x6^e 187: 6 [C,C,C,C,C,C] 213 156 207 162 161 208 * * * * * * (0,0,1,0,0,0,0) 2 2x4x3x5x4^6^e 188: 6 [C,C,c,C,r,C] 214 157 188 208 188 162 * * * * * * (0,0,1,0,0,0,0) 2 2x3x4x5x4^6^e 189: 6 [C,C,C,r,C,C] 215 158 209 189 208 163 * * * * * * (0,0,1,0,0,0,0) 2 2x3x4x3^5x6^e 190: 6 [C,C,r,C,c,C] 216 159 190 209 190 164 * * * * * * (0,0,0,0,1,0,0) 2 2x3x2^4x5x6^e 191: 6 [C,C,C,C,C,C] 217 160 163 164 160 209 * * * * * * (0,0,0,1,0,0,0) 2 2x3x2x4x3^5^e 192: 6 [C,C,C,c,c,C] 218 167 211 192 192 165 * * * * * * (0,0,0,0,1,0,0) 2 2x1^3x4x5x6^e 193: 6 [C,C,C,C,C,C] 219 168 165 210 166 211 * * * * * * (0,0,0,0,1,0,0) 2 2x1x3x2^4x5^e 194: 6 [C,C,C,C,C,C] 160 210 169 168 217 212 * * * * * * (0,0,0,1,0,0,0) 2 1x3x2x4x3^5^e 195: 6 [C,C,C,C,c,C] 159 211 173 212 195 168 * * * * * * (0,0,0,0,1,0,0) 2 1x3x2^4x5x6^e 196: 6 [C,c,C,C,C,C] 161 196 212 172 213 169 * * * * * * (0,0,1,0,0,0,0) 2 1x2x4x3^5x6^e 197: 6 [C,c,c,C,C,C] 162 197 197 213 170 214 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x5x4^6^e 198: 6 [C,c,C,r,C,C] 163 198 217 198 169 215 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x4x3^5^e 199: 6 [C,C,r,C,C,C] 164 219 199 217 171 216 * * * * * * (0,0,0,0,1,0,0) 2 1x2x3x2^4x5^e 200: 6 [C,r,C,c,C,C] 165 200 219 200 174 218 * * * * * * (0,0,0,0,1,0,0) 2 1x2x1^3x4x5^e 201: 6 [C,C,C,C,C,c] 166 171 174 166 219 201 * * * * * * (0,0,0,0,0,1,0) 2 1x2x1x3x2^4^e 202: 6 [C,c,c,n,n,C] 175 202 202 203 204 180 * * * 215 214 * (0,0,1,0,0,0,0)#1 1x2x3x4x5x6^e 203: 6 [C,c,n,n,c,C] 176 203 205 202 203 181 * * 216 215 * * (0,0,1,1,1,0,0)#1 1x2x3x4x5x6^e 204: 6 [C,c,c,c,n,C] 177 204 204 204 202 182 * * * * 214 * (0,0,1,0,1,1,0)#1 1x2x3x4x5x6^e 205: 6 [C,n,n,c,c,C] 178 206 203 205 205 183 * 218 216 * * * (0,0,0,0,1,0,0)#1 1x2x3x4x5x6^e 206: 6 [C,n,c,c,c,C] 179 205 206 206 206 184 * 218 * * * * (0,1,1,0,1,0,0)#1 1x2x3x4x5x6^e 207: 7 [C,C,C,c,C,C] 222 220 187 207 186 220 * * * * * * (0,0,0,1,0,0,0) 2 3x2x4x3x5x4^6^e 208: 7 [C,C,C,C,C,C] 223 185 220 188 189 187 * * * * * * (0,0,1,0,0,0,0) 2 2x3x4x3x5x4^6^e 209: 7 [C,C,C,C,C,C] 224 186 189 190 220 191 * * * * * * (0,0,0,1,0,0,0) 2 2x3x2x4x3^5x6^e 210: 7 [C,C,c,C,C,C] 225 194 210 193 225 221 * * * * * * (0,0,0,1,0,0,0) 2 2x1x3x2x4x3^5^e 211: 7 [C,C,C,C,c,C] 226 195 192 221 211 193 * * * * * * (0,0,0,0,1,0,0) 2 2x1x3x2^4x5x6^e 212: 7 [C,C,C,C,C,C] 186 221 196 195 222 194 * * * * * * (0,0,0,1,0,0,0) 2 1x3x2x4x3^5x6^e 213: 7 [C,c,C,C,C,C] 187 213 222 197 196 223 * * * * * * (0,0,1,0,0,0,0) 2 1x2x4x3x5x4^6^e 214: 7 [C,c,c,C,r,C] 188 214 214 223 214 197 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x4x5x4^6^e 215: 7 [C,c,C,r,C,C] 189 215 224 215 223 198 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x4x3^5x6^e 216: 7 [C,C,r,C,c,C] 190 226 216 224 216 199 * * * * * * (0,0,0,0,1,0,0) 2 1x2x3x2^4x5x6^e 217: 7 [C,C,C,C,C,C] 191 225 198 199 194 224 * * * * * * (0,0,0,1,0,0,0) 2 1x2x3x2x4x3^5^e 218: 7 [C,r,C,c,c,C] 192 218 226 218 218 200 * * * * * * (0,0,0,0,1,0,0) 2 1x2x1^3x4x5x6^e 219: 7 [C,C,C,C,C,C] 193 199 200 225 201 226 * * * * * * (0,0,0,0,1,0,0) 2 1x2x1x3x2^4x5^e 220: 8 [C,C,C,c,C,C] 228 207 208 220 209 207 * * * * * * (0,0,0,1,0,0,0) 2 2x3x2x4x3x5x4^6^e 221: 8 [C,C,c,C,C,C] 229 212 221 211 227 210 * * * * * * (0,0,0,1,0,0,0) 2 2x1x3x2x4x3^5x6^e 222: 8 [C,C,C,c,C,C] 207 227 213 222 212 228 * * * * * * (0,0,0,1,0,0,0) 2 1x3x2x4x3x5x4^6^e 223: 8 [C,c,C,C,C,C] 208 223 228 214 215 213 * * * * * * (0,0,1,0,0,0,0) 2 1x2x3x4x3x5x4^6^e 224: 8 [C,C,C,C,C,C] 209 229 215 216 228 217 * * * * * * (0,0,0,1,0,0,0) 2 1x2x3x2x4x3^5x6^e 225: 8 [C,C,c,C,C,C] 210 217 225 219 210 229 * * * * * * (0,0,0,1,0,0,0) 2 1x2x1x3x2x4x3^5^e 226: 8 [C,C,C,C,c,C] 211 216 218 229 226 219 * * * * * * (0,0,0,0,1,0,0) 2 1x2x1x3x2^4x5x6^e 227: 9 [C,C,c,c,C,C] 230 222 227 227 221 230 * * * * * * (0,0,0,1,0,0,0)#2 2x1x3x2x4x3x5x4^6^e 228: 9 [C,C,C,c,C,C] 220 230 223 228 224 222 * * * * * * (0,0,0,1,0,0,0) 2 1x2x3x2x4x3x5x4^6^e 229: 9 [C,C,c,C,C,C] 221 224 229 226 230 225 * * * * * * (0,0,0,1,0,0,0) 2 1x2x1x3x2x4x3^5x6^e 230: 10 [C,C,c,c,C,C] 227 228 230 230 229 227 * * * * * * (0,0,0,1,0,0,0) 2 1x2x1x3x2x4x3x5x4^6^e expr> set G=Sp(4,R) Identifier G (overriding previous): RealForm expr> print_KGB(G) { a smaller example is more easy to oversee } kgbsize: 11 Base grading: [11]. 0: 0 [n,n] 1 2 4 5 (0,0)#0 e 1: 0 [n,n] 0 3 4 6 (1,1)#0 e 2: 0 [c,n] 2 0 * 5 (0,1)#0 e 3: 0 [c,n] 3 1 * 6 (1,0)#0 e 4: 1 [r,C] 4 9 * * (0,0) 1 1^e 5: 1 [C,r] 7 5 * * (0,0) 2 2^e 6: 1 [C,r] 8 6 * * (1,0) 2 2^e 7: 2 [C,n] 5 8 * 10 (0,0)#2 1x2^e 8: 2 [C,n] 6 7 * 10 (0,1)#2 1x2^e 9: 2 [n,C] 9 4 10 * (0,0)#1 2x1^e 10: 3 [r,r] 10 10 * * (0,0)#3 1^2x1^e expr> list_cartans(G) Cartan Number 0: compact: 2, complex: 0, split: 0 Cartan Number 1: compact: 0, complex: 1, split: 0 Cartan Number 2: compact: 1, complex: 0, split: 1 Cartan Number 3: compact: 0, complex: 0, split: 2 Value: [(),(),(),()] expr> rho(G) Value: [ 2, 1 ]/1 expr> set G=E8_s() { this (huge) group is produced by a function call } Identifier G (overriding previous): RealForm expr> rho(G) { in the simply connected coordinates, rho is always [1,1...,1] } Value: [ 1, 1, 1, 1, 1, 1, 1, 1 ]/1 expr> rho_check(G) { but rho-check is more interesting } Value: [ 46, 68, 91, 135, 110, 84, 57, 29 ]/1 expr> set G=SL(2,R) Identifier G (overriding previous): RealForm expr> rho(G) Value: [ 1 ]/1 expr> set t=trivial(G) { here we produce the paramter for a representation } Identifier t: Param expr> t Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1) expr> list_cartans(G) Cartan Number 0: compact: 1, complex: 0, split: 0 Cartan Number 1: compact: 0, complex: 0, split: 1 Value: [(),()] expr> print_KGB(G) { see where that component x=2 lives } kgbsize: 3 Base grading: [1]. 0: 0 [n] 1 2 (0)#0 e 1: 0 [n] 0 2 (1)#0 e 2: 1 [r] 2 * (0)#1 1^e expr> print_n_block(t) { the block of the trivial gives more representations } Parameter defines element 2 of the following block: 0(0,1): 0 [i1] 1 (2,*) *(x=0, nu= [0]/1,lam=rho+ [0]) e 1(1,1): 0 [i1] 0 (2,*) *(x=1, nu= [0]/1,lam=rho+ [0]) e 2(2,0): 1 [r1] 2 (0,1) *(x=2, nu= [1]/1,lam=rho+ [0]) 1^e expr> composition_series(t) Value: (1+0s)*final parameter (x=0,lambda=[1]/1,nu=[0]/1) (1+0s)*final parameter (x=1,lambda=[1]/1,nu=[0]/1) (1+0s)*final parameter (x=2,lambda=[1]/1,nu=[1]/1) expr> infinitesimal_character (t) Value: [ 1 ]/1 expr> t Value: final parameter (x=2,lambda=[1]/1,nu=[1]/1) expr> set p=param(KGB(G)[2],[1],[1]/1) Identifier p: Param expr> p Value: final parameter (x=2,lambda=[2]/1,nu=[1]/1) expr> composition_series (p) Value: (1+0s)*final parameter (x=2,lambda=[2]/1,nu=[1]/1) expr> set p=param(KGB(G)[2],[1],[2]/1) Identifier p (overriding previous): Param expr> composition_series (p) Value: (1+0s)*final parameter (x=0,lambda=[2]/1,nu=[0]/1) (1+0s)*final parameter (x=1,lambda=[2]/1,nu=[0]/1) (1+0s)*final parameter (x=2,lambda=[2]/1,nu=[2]/1) expr> infinitesimal_character (p) Value: [ 2 ]/1 expr> is_unitary(p) { here is the ultimate (?) atlas algorithm! } Value: false expr> is_unitary(t) Value: true expr> set G=SL(3,R) Identifier G (overriding previous): RealForm expr> set t=trivial(G) Identifier t (overriding previous): Param expr> is_unitary(t) { this is too embassing } Value: false expr> list_cartans(G) Cartan Number 0: compact: 0, complex: 1, split: 0 Cartan Number 1: compact: 0, complex: 0, split: 2 Value: [(),()] expr> quit Bye. annegret.paul@atlas2:~/rx-scripts$ logout exit Script done on Mon 01 Jul 2013 07:37:40 PM EDT