Reference
Function Index
Every name defined in the 52 script files loaded by
all.at: functions, operators, values, types and type fields (1468 names). Each link goes to
the definition in its file; ×n counts the overloads in that file.
Symbols (operators)
!basic ×4!=basic · combinatorics ×3#basic ×5 · W_reps##basic ×4%basic ×2 · extParamPol*basic ×13 · extParamPol ×3 · complex · modules ×2 · hodge_tensor · tits_centralizer · stable ×3+basic ×8 · extParamPol ×3 · modules ×3-basic ×9 · extParamPol · modules ×5/basic ×2<basic ×3<=basic ×3 · combinatorics ×2=basic ×3 · combinatorics ×4 · modules ×3==basic ×2===basic>basic>=basic\basic ×3\%basic ×3^basic ×7~basic ×3
A
A_barlusztig_cells ×2a_column_withbasica_primegeck_generica_row_withbasicabsbasic ×2action_on_root_vectorstits_centralizer ×4addhodge_K_type_formulaadd_onesommersadjointbasic ×2adjoint_2rhobasicAffineSubspacetype, basicallbasic ×5all_0_1_vecsbasicall_0_1_vecs_with_sumbasicall_Levi_Springer_tablesgeck_genericall_levi_tablesgeck_genericall_levi_tables_plusgeck_genericall_parameters_x_gammaG2_unitary_dualall_posrootsbasicall_root_indextits_centralizer ×2all_rootsbasicall_simplesbasicall_wordsbasic ×3among_poscorootsbasicamong_posrootsbasicanalysehermitiananalyse_hermitian_form_irreduciblehermitiananisotropic_partfinite_dimensional_signatureanisotropic_part_stringfinite_dimensional_signatureanisotropic_part_tracefinite_dimensional_signatureanisotropic_part_trace_stringfinite_dimensional_signatureanybasic ×5appendbasicAq_longadams_johnson ×4Aq_reducible_longadams_johnson ×3Aq_rhoadams_johnson ×2Aq_rho_stableadams_johnson ×2Aq_rho_stable_testadams_johnson ×2Aq_stableadams_johnson ×2Aq_stable_longadams_johnson ×2Aq_stable_testadams_johnson ×2array_printsmodulesarthur_parametersarthur_parameters ×2arthur_parameters_by_orbitarthur_parametersarthur_parameters_by_yarthur_parametersas_bipartitioncombinatoricsas_columnbasicas_listbasicas_polbasic ×4 · modulesas_predictedtest_unitarityas_rowbasicas_signed_cyclescombinatoricsassertbasic ×6associated_variety_annassociated_variety_annihilator ×5associated_variety_ann_indassociated_variety_annihilator ×6associated_variety_ann_resassociated_variety_annihilator ×6associated_variety_ann_res_complexassociated_variety_annihilatorassociated_variety_ann_res_testassociated_variety_annihilatorassociated_variety_intassociated_variety_annihilator ×4avK_Nilpotent ×3av_annK_Nilpotent ×2av_ann_elemK_Nilpotent ×2av_ann_elem_longK_Nilpotent ×2av_elemK_Nilpotent ×2av_elem_longK_Nilpotent ×2av_from_av_annK_Nilpotent ×2av_from_av_ann_elemK_Nilpotent ×2av_from_av_ann_elem_longK_Nilpotent ×2av_maximal_idealassociated_variety_annihilator ×4av_maximal_ideal_0associated_variety_annihilator ×3av_verboseassociated_variety_annihilator
B
backward_differencesbasicbase_pointfield ofAffineSubspace, basicbasis_dual_latticetits_centralizer ×2basis_max_split_torusrestricted_rootsBezoutnumber_theorybig_blockbasicbinarybasicbinary_lookupbasic ×3binary_lookup_bybasicbinary_search_firstbasicbinary_search_getbasicbinary_search_inbasic ×2binomcombinatorics ×2BiPartitiontype, combinatoricsbirational_inductionsommers ×2bitwise_intersectionbasicbitwise_unionbasicblock_action_matrix_irrfourblock_action_matrix_stdfourblock_decomposeL_packetblock_matrixbasic ×2block_ofbasicblock_status_textbasicblocksbasic ×2bounded_listsall_finite_orderbranchbasic ×2 · modulesbranch_c_form_irreduciblehermitianbranch_function_stdhodge_tensorbruhatbruhat ×3bruhat_geqbruhat ×3bruhat_leqbruhat ×3Bruhat_orderbasicbruhat_Wbruhatbruhat_W_geqbruhatbruhat_W_leqbruhat
C
c_adjustbasicc_form_irreduciblehermitian ×2c_form_irreducible_bigextParamPol ×2c_form_irreducible_big_as_sum_of_standardsextParamPolc_form_irreducible_big_longextParamPolc_form_irreducible_longhermitianc_form_stdhermitianc_form_std_bigextParamPolcanbasicCartan_classbasic ×3Cartan_classesbasicCartan_matrixbasiccartansG2_unitary_dualCayleybasiccc_BCfinite_unipotentscc_Dfinite_unipotentsceilbasiccell_actionW_reps ×3cell_and_character_dataweak_packets_precomputed_cell_traces ×2cell_as_wcomplexcell_characterW_repscell_charactersW_repscell_representationW_repscell_representationsW_repscells_tableW_repscells_table_augmentedW_repscentral_invariantgalois ×2central_invariant_indexgalois ×2central_torus_rankbasic ×2centralizergood_W_representatives · nilpotent_centralizercentralizer_of_Kacall_finite_ordercentralizer_ordercombinatoricscentralizer_rootsnilpotent_centralizercentralizer_roots_and_corootsnilpotent_centralizer ×2certificatecertificatechainsommerschamberbasic ×6change_basisK_Nilpotentchange_nuextParamPolcharhodge_K_type_formulachar_indexbasicchar_polybasiccharactercombinatorics · modules · W_repscharacter_formulaextParamPol · modules ×2character_table_generic_degreesgeck_genericcheck_hermitianhermitiancheck_nilradical_dimensionsnilpotent_centralizer ×2choices_frombasicclass_sizecombinatoricsclassic_permutationcombinatoricsclassical_partsommersclassical_part_Bsommersclassical_part_CsommersClassicalOrbitSymboltype, sommersclosureK_Nilpotentclosure_of_real_orbitK_Nilpotentclosure_realK_Nilpotentcodimensionbasic ×2coherent_irrcoherent_irreducible ×7coherent_irr_newcoherent_irreducible ×6cokernelbasiccollapsespecial_repcollectbasiccolumnhodge_K_type_formula · stablecolumns_withbasic ×3combination_decodecombinatoricscombination_encodecombinatoricscombinegenericscompact_formbasiccompact_posrootsbasiccompact_rankbasic ×2compact_stringbasiccompact_torusbasiccomplementbasic ×2complex_nilpotent_from_partitionsommerscomplex_orbit_numberK_Nilpotent ×3complex_orbitsfield ofKNilpotentData, K_Nilpotentcomplex_posrootsbasic ×2complex_pseudo_Leviall_finite_ordercomposegenericscompose_permutationscombinatoricscomposition_seriesmodules ×2compressed_stringcombinatorics ×2compute_Y_matrixK_Nilpotentconcatbasicconcatenatecomplexconjugacy_classes_involutionsgaloisconjugacy_classes_reflectionsgeck_generic ×2conjugacy_classes_reflections_simplegeck_genericconjugatebasic ×2constantgenericscontainsbasic ×2convert_tobasiccoordsG2_unitary_dualcoradical_basisbasiccoresommerscore_lengthcombinatoricscore_numbercombinatoricscore_quotient_2combinatoricscoreflectbasic ×4coressommerscorootbasiccorootsbasiccoroots_all_positivebasiccosexp-generating-seriescountbasic ×3count_orbits_and_centralizersfinite_unipotentscount_permutations_with_cyclesexp-generating-series ×2count_unipotentsfinite_unipotentscoverdirac_indexcover_cdirac_indexcoxall_finite_ordercoxeter_dataweyltosemisimplecrossbasic ×5 · coherent_irreducible ×9cross_gradedcoherent_irreducible ×2cross_squarecoherent_irreducible ×4cross_square_testcoherent_irreducible ×2cumulate_backwardbasiccumulate_forwardbasiccurried_swapargsgenericscurrygenericscuthodge_K_type_formulacycle_centralizer_ordercombinatorics ×2cycle_class_sizecombinatorics ×2cycle_codecombinatoricscycle_powercombinatorics ×3cycle_productcombinatoricscycle_typecombinatoricscycle_type_ordercombinatorics ×3cyclic_permutationcombinatoricscyclotomic_polynomialnumber_theory
D
dfield ofgood_data, weyltosemisimpleD_classtype, combinatoricsD_classescombinatoricsD_irreduciblescombinatoricsD_irreptype, combinatoricsd_lambdabasicdefault_orbit_symbolsommersdefault_orbit_symbol_Bsommersdefault_orbit_symbol_Csommersdefault_weight_functiongeck_genericdeformextParamPol ×2deform_oldextParamPoldeform_to_heightbasicdeform_unreducedextParamPoldeletebasic ×3 · hodge_K_type_formuladelete_allsommers ×2deltafield ofgood_data, weyltosemisimpledenomfield ofAffineSubspace, basic · basic ×2derangement_numbersexp-generating-seriesderivedbasicderived_is_simplebasic ×2derived_is_simply_connectedbasicdescendbruhatdescribe_isogenynilpotent_centralizer ×2detbasicdiagcomplexdiag_Wcomplexdigits36basicdigraphsub_cellsdim_centralizernilpotent_centralizerdim_centralizer_nilradicalnilpotent_centralizerdim_K_types_std_uptoGK_dimensiondim_nilpotent_partitionsommersdim_nilradical_parabolic_Hnilpotent_centralizerdimensionbasic ×2 · field ofW_rep, W_repsdirac_index_dsdirac_index ×2dirac_index_irrdirac_index ×2dirac_index_stddirac_index ×2displayextParamPoldisplay_stringK_Nilpotentdistinguished_fiberbasicdivide_bybasic ×2divisorsnumber_theorydominance_leq_compositionscombinatoricsdominance_leq_partitionscombinatoricsdominantbasic ×6donefield ofmaybe_KTP, hermitiandrop_lowerbasic ×2drop_upperbasic ×2dualhodge_K_type_formula · hodge_tensordual_datasommersdual_integralbasicdual_KLbasicdual_orbitsommers ×3dual_parametersstabledual_parameters_matrixstable ×3dual_parameters_standard_basisstable ×2dual_parameters_standard_basis_poly_matstabledual_real_formsbasic ×2dual_symbolssommersduality_permutationstabledvdirac_index
E
E6_lusztig_map_ellipticexceptionalNilpotentDataE6_nilpotent_dataexceptionalNilpotentDataE7_lusztig_map_ellipticexceptionalNilpotentDataE7_nilpotent_dataexceptionalNilpotentDataE8_lusztig_map_ellipticexceptionalNilpotentDataE8_nilpotent_dataexceptionalNilpotentDataeasy_factorsnumber_theoryedge_sequencecombinatoricsEigenPairtype, good_W_representativeseigenpairsgood_W_representatives ×2eigenvaluesgood_W_representatives ×2eigenvectorgood_W_representativeseigenvectorsgood_W_representativesembed_leftcomplexempty_setfield ofLinearSolution, basicenumerate_classnew_conjugacyenumerate_classesnew_conjugacyequal_central_invariantgalois ×2equal_subgroupsgeck_genericevaluate_at_1stable ×2even_placescombinatoricsexact_dividebasicexceptional_lusztig_maps_ellipticexceptionalNilpotentDataexceptional_nilpotent_tablesexceptionalNilpotentDataexp_2basicexp_comp_inverseexp-generating-seriesexp_diffexp-generating-seriesexp_divideexp-generating-seriesexp_multexp-generating-seriesexp_multiplyexp-generating-seriesexp_of_exponential_seriesexp-generating-seriesexp_sbasicexp_substituteexp-generating-seriesexpandnumber_theoryexportsub_cellsextended_status_textsbasicExtParamPoltype, extParamPolextParamPolextParamPol ×2extract_nodessub_cells
F
Fgood_W_representativesF4_lusztig_map_ellipticexceptionalNilpotentDataF4_nilpotent_dataexceptionalNilpotentDatafactorialbasicfactorizationnumber_theoryfactorsbasicfailfield ofOne_of, basicfailedK_Nilpotentfailed_avK_Nilpotentfailed_av_annK_Nilpotentfailsbasicfalling_powercombinatoricsfamiliesfamiliesfamilyfamilies ×2fd_reportfinite_dimensional_signaturefd_report_tracefinite_dimensional_signaturefgrepbasicfiberL_packet ×2fig_verbosespecial_repfillK_Nilpotent ×2fill_one_stepK_Nilpotentfill_outhodge_K_type_formulafill_out_recursivelyhodge_K_type_formulafilterbasicfindbasic ×7 · tits_centralizerfind_classgaloisfind_inbasicfind_induced_genericspecial_repfind_Levigeck_genericfirstbasic ×5 · sommers ×2first_K_typebasicfirst_parambasicfirst_set_bitbasicfirst_unset_bitbasicfive_inner_productsfour ×3five_inner_products_rfour ×2floorbasicfoldlgenericsfoldrgenericsfor_allbasicfor_dual_datumbasicfor_nonebasicfor_somebasicform_namebasicforward_differencesbasicfour_inner_productsfourfrequenciescombinatoricsFrobenius_Schurfinite_dimensional_signaturefrom_core_quotient_2combinatoricsfrom_dominantbasic ×6from_dominant_rootbasicfrom_stopsbasicfsfinite_dimensional_signaturefstfield ofPair, basicfull_bitsetbasicfull_deformbasic · extParamPolfunctions_on_real_orbitK_Nilpotentfundamental_Cartanbasicfundamental_coweightsbasicfundamental_weightsbasic
G
GG2_unitary_dualG2_lusztig_map_ellipticexceptionalNilpotentDataG2_nilpotent_dataexceptionalNilpotentDatag2_parameterG2_unitary_dualg_parametercomplexgamma_Lcomplexgamma_lG2_unitary_dualgamma_Rcomplexgamma_sG2_unitary_dualgcdbasic ×2gcd_Bezout_coefnumber_theorygcd_bigbasicgeneric_degreesgeck_generic ×2getfield ofIterator, basicget_dataall_finite_order ×2get_firstbasic ×2get_lastbasic ×2get_raw_dataall_finite_orderget_subsetsall_finite_orderget_subsets_given_orderall_finite_orderget_ystablegg_verbosegeck_genericGK_dimassociated_variety_annihilatorGK_dim_maximal_idealassociated_variety_annihilator ×2GL_partsommers ×2good_datatype, weyltosemisimplegood_infogood_W_representatives ×2good_reportgood_W_representatives ×2gp_alg_eltcomplexgp_alg_elt1complexgraded_composition_seriesjantzen ×3graded_multiplicityjantzen ×3graded_standardjantzen · coherent_irreduciblegraphfield ofWCell, W_repsgraph_actionW_reps ×3graph_Hecke_actioncoherent_irreducible ×2graph_Hecke_action_columncoherent_irreduciblegraph_Hecke_irr_actioncoherent_irreduciblegraph_Hecke_irr_action_columncoherent_irreduciblegroup_parameters_by_primitive_idealprojectors_using_character_tablesgroup_parameters_projectionprojectors_using_character_tablesgroup_vectors_up_to_scalarprojectors_using_character_tablesgroup_vectors_up_to_scalar_altprojectors_using_character_tablesgrowthGK_dimensiongrowth_irrGK_dimensiongrowth_stdGK_dimension
H
H1galois · G2_unitary_dualH2G2_unitary_dualh_initcoherent_irreducibleH_lG2_unitary_dualH_sG2_unitary_dualh_testcoherent_irreducible ×2halfbasic ×4has_all_generic_degreesgeck_generichas_connected_centerbasichas_generic_degreegeck_generichas_infinitesimal_charactermodules · stableheightbasic ×2height_splitbasic ×2hermitian_dualhermitianhermitian_form_irreduciblehermitian ×3hermitian_form_irreducible_bigextParamPolhermitian_form_irreducible_longhermitian ×2hermitian_form_is_purehermitianhermitian_form_stdhermitianhexadecimalbasichighest_coroot_indicesbasichighest_corootsbasichighest_rootbasichighest_root_indicesbasichighest_rootsbasichodge_branch_irrhodge_K_type_formulahodge_branch_stdhodge_K_type_formula ×3hodge_formula_irrhodge_K_type_formulahodge_function_stdhodge_K_type_formula · hodge_tensorhodge_K_type_formulahodge_K_type_formulahodge_K_type_formula_debugginghodge_K_type_formulahodge_K_type_formula_oldhodge_K_type_formulahodge_K_type_formula_verbosehodge_K_type_formulahodge_K_type_mathodge_K_type_formula ×2hodge_K_type_mat_longhodge_K_type_formula ×2hodge_K_type_mat_trivialhodge_K_type_formulahodge_tablehodge_K_type_formulahodge_tensorhodge_tensor ×2hodge_tensor_exterior_u_cap_shodge_tensorhodge_tensor_exterior_u_cap_s_stdhodge_tensor ×2hodge_tensor_stdhodge_tensor ×5hodge_tensor_wedge_k_u_cap_shodge_tensorhodge_tensor_wedge_k_u_cap_s_stdhodge_tensor ×2hodge_testhodge_test ×3hodge_test_verbosehodge_testhodgePrintPolyMatrixhodge_K_type_formula ×2hook_lengthscombinatorics ×2hyperoctahedral_charactercombinatorics
I
Igenerics · modulesi_MG2_unitary_dualicrosscoherent_irreducible ×3icross_gradedcoherent_irreducibleid_Wbasicidentity_in_fund_domainall_finite_orderidentity_rowbasicimagebasicimaginary_corootsbasicimaginary_poscorootsbasic ×2imaginary_posrootsbasic ×2imaginary_rootsbasicimaginary_roots_and_corootsbasic ×2imaginary_sysbasic ×2imaginary_typebasic ×2in_distinguished_fiberbasicin_fppG2_unitary_dual ×3in_one_plus_delta_Zgaloisin_spanK_Nilpotent ×2in_string_listbasicin_taumodules · stable ×2 · sub_cellsin_tau_complementmodules · stable ×2include_Q_and_P_X_matricesfield ofKNilpotentData, K_Nilpotentincrfield ofIterator, basicindexfinite_dimensional_signatureindex_bipartitioncombinatoricsindex_D_classescombinatoricsindex_inbasic ×3index_partitioncombinatoricsindex_stringfinite_dimensional_signatureindex_tracefinite_dimensional_signatureindex_trace_stringfinite_dimensional_signatureindicesbasic · stableinducespecial_repinduce_characterW_repsinduce_orbittruncated_induction ×3infinitesimal_characterbasic ×2 · modules ×2 · sommersinfinitesimal_character_oldsommersinfoall_finite_order ×2info_reducedall_finite_orderinitialize_KNilpotentK_Nilpotent ×4inner_classbasic ×5inner_liftL_packet ×3inner_lift_stdL_packet ×4int_formatbasicint_partbasicintegrality_datumbasicintegrality_rankbasicintersectcomplexintersectionsub_cells ×2inv_factbasicinversebasic ×2 · combinatoricsinverse_make_coordstits_centralizerinverse_modnumber_theoryinversescomplexinvertibles_modulonumber_theoryinvolutionbasic ×3irrmodulesirreducibles_as_sums_of_standardsL_packetis_adjointbasicis_ascentbasic ×3is_birigidsommersis_compactbasic ×3is_compact_imaginarybasic ×4is_compatiblebasicis_complexbasic ×4 · complex · sub_cellsis_conjugateall_finite_orderis_conjugate_Kacall_finite_orderis_corootbasicis_descentbasic ×3is_disjointbasicis_disjoint_frombasicis_disjoint_from_sortedbasicis_distinguishedsommersis_dominantbasic ×2is_doubly_evencombinatoricsis_equal_rankbasicis_evenbasicis_hermitianhermitianis_imaginarybasic ×4is_integerbasic ×2is_integralbasic ×2is_integrally_dominantbasic ×2is_isomorphicW_repsis_memberbasic ×5is_member_bitsetbasicis_member_fastbasicis_member_sortedbasicis_multiplicative_generatornumber_theory ×2is_negative_corootbasic ×2is_negative_rootbasic ×2is_negcorootbasicis_negrootbasicis_noncompactbasic ×2is_noncompact_imaginarybasic ×4is_nonparitybasic ×2is_oddbasicis_orthogonalbasic ×3is_paritybasic ×2is_partitioncombinatoricsis_permutationcombinatoricsis_poscorootbasicis_positivecoherent_irreducibleis_positive_corootbasic ×2is_positive_rootbasic ×2is_posrootbasicis_primenumber_theoryis_purebasic ×4is_pure_1basic ×3is_pure_sbasicis_quasicompactbasicis_quasisplitbasicis_rational_multipleprojectors_using_character_tablesis_realbasic ×4is_regularbasic ×3is_rootbasicis_self_conjugatefinite_dimensional_signatureis_self_dualfinite_dimensional_signatureis_semisimplebasicis_simplebasic ×2is_simple_forbasicis_simply_connectedbasicis_solvable_for_torus_elementtits_centralizeris_specialcombinatoricsis_splitbasic ×2is_stableL_packetis_stable_irrL_packetis_stable_stdL_packetis_strict_descentbasicis_strictly_complexcomplex · sub_cellsis_strictly_dominantbasic ×2is_strongly_regularbasicis_subsetbasicis_subset_ofbasicis_subset_of_sortedbasicis_substringbasicis_unitaryhermitian ×2is_unitary_bigextParamPolis_validcombinatorics ×2 · L_packetis_very_evencombinatoricsis_weakly_unitaryhermitian ×2is_weakly_unitary_bigextParamPolisnt_memberbasic ×5isnt_member_bitsetbasicisnt_member_fastbasicisnt_member_sortedbasiciterate_H1galois ×3Iteratortype, basic
J
JmodulesJ_inducetruncated_induction ×2j_inducetruncated_induction ×2jJ_inducetruncated_inductionjJ_select_inducetruncated_inductionJM_tripletits_centralizer ×2JM_triple_onetits_centralizerJM_triple_strongtits_centralizer ×3JM_triplestits_centralizerjoinbasic
K
K_basisK_Nilpotent ×5K_basis_closureK_NilpotentK_basis_closure_of_real_orbitK_NilpotentK_intcomplexK_type_formulabasicK_type_lambdabasicK_type_polbasicK_typesmodulesKac_classes_given_orderall_finite_orderKac_classes_given_order_crudeall_finite_orderKac_diags_given_orderall_finite_orderKac_diags_of_identityall_finite_orderKac_diags_up_to_orderall_finite_orderKac_xall_finite_orderkeep_longrestricted_rootskeep_long_roots_short_corootsrestricted_rootskeep_shortrestricted_rootskeep_short_roots_long_corootsrestricted_rootskernel_vanishingstableKGBbasic ×2KGB_eltbasic ×3KGB_status_textbasickn_verboseK_NilpotentKNilpotentDatatype, K_Nilpotentkottwitz_invariantL_packetkottwitz_signL_packet
L
l_adjustbasicL_cap_K_basisK_Nilpotent ×3L_cap_K_basis_oppK_NilpotentL_packetL_packet ×2L_packet_representativeL_packetL_packet_representativesL_packet ×2L_packet_stable_sumL_packetL_packetsL_packetlabelsfield ofWCell, W_repslambdabasic ×2lambda_minus_rhobasic ×2lambda_rhobasiclastbasic ×5 · sommers ×2last_K_typebasiclast_parambasiclcmbasic ×2leftcomplex ×2left_cell_ofsub_cellsleft_cell_of_oldsub_cellsleft_cellssub_cellsleft_copysub_cellsleft_corootssub_cellsleft_factorssub_cellsleft_Gcomplex ×2left_kernelstableleft_reflectbasic ×2left_rhocomplexleft_root_indicessub_cellsleft_rootssub_cellsleft_Wcomplex ×2left_wcomplexlength_signcoherent_irreduciblelengthensbruhatlengths_signsstable ×2lengths_signs_matrixstable ×2less_eqcombinatoricslevigood_W_representativesLevi_AcombinatoricsLie_typebasicLinearSolutiontype, basiclinkssub_cellslistbasic · coherent_irreduciblelocate_sortedbasiclong_rat_as_integer_stringfinite_dimensional_signaturelookupbasic · hodge_K_type_formulalookup_character_tablegeck_genericlookup_character_table_oldgeck_genericlookup_columnbasiclookup_rowbasiclowerbasic ×2lowest_corootsbasiclowest_rootsbasiclusztig_celllusztig_cells ×2lusztig_cell_longlusztig_cellslusztig_cell_pluslusztig_cellslusztig_cellslusztig_cells ×3lusztig_cells_pluslusztig_cells
M
MG2_unitary_dualmnilpotent_centralizer · test_braidm_0nilpotent_centralizerm_2nilpotent_centralizerm_parametercomplexmain_diagonalbasicmake_coordstits_centralizermake_param_polstablemake_regularstablemake_specialcombinatorics ×3make_vectorfour ×2make_W_elementgood_W_representativesmapbasic ×7 · hermitianmap_onbasicmap_root_of_unitytits_centralizer ×2mat_as_poly_matstablematrixbasic ×2matrix_assignbasicmatrix_of_inner_productsW_repsmaxbasic ×7max_heightbasic ×2max_initbasicmax_Levi_subgroupsgeck_genericmax_listbasicmax_locbasic ×2max_standard_Levi_conjugacy_representativesgeck_genericmaxdexbasicMaybetype, basicmaybe_KTPtype, hermitianmerge_matricesK_Nilpotentmerge_P_X_matricesK_Nilpotentmerge_T_matricesK_Nilpotentmiddlesommers ×2millers_crossingcoherent_irreducible ×2millers_crossing_singcoherent_irreducible ×2minbasic ×7min_heightbasic ×4min_initbasicmin_listbasicmin_locbasic ×2mindexbasicminus_1basicmixedhermitianmixed_radix_nrbasicmixed_radix_wordbasicmixed_termshermitianmod_central_torusbasicmonomialbasic ×4monomialsbasic ×6most_split_Cartanbasicmucoherent_irreducible ×2mu_Ccomplexmulti_choosecombinatoricsmultinomcombinatoricsmultiplicitybasic · sommersmultiset_permutationscombinatoricsMurnaghan_Nakayamacombinatoricsmy_simple_rootsG2_unitary_dualmysortall_finite_order
N
Nfield ofKNilpotentData, K_Nilpotentnfield ofphi_data, number_theoryn_columnsbasicn_rowsbasicnamefield ofgood_data, weyltosemisimplenc_verbosenew_conjugacynegative_systembasicnegative_termscertificatenested_Levisgood_W_representativesnested_Levis_to_semisimplegood_W_representativesnew_linebasicnew_ordertits_centralizernice_classesall_finite_orderno_Cminus_rootsbasicno_Cplus_rootsbasicno_repsbasic ×2nodesfield ofWGraph, W_repsnon_dominant_simplesbasic ×2non_integrally_dominant_simplesbasicnoncompact_posrootsbasicnonefield ofMaybe, basic · basic ×5nonparity_poscorootsbasicnormalizecombinatoricsnubasicnu_Ccomplexnullall_finite_ordernull_ext_moduleextParamPolnull_K_modulebasic ×2null_modulebasic ×4null_poly_matstablenumberbasic ×2number_unipotentsfinite_unipotentsnumber_unipotents_BCfinite_unipotentsnumber_unipotents_Dfinite_unipotentsnumber_unipotents_simplefinite_unipotentsnumerbasic ×2
O
odd_part_partitionscombinatoricsodd_placescombinatoricsomega_Lgeck_genericOmega_S_inversehodge_tensorone_minus_sbasicOne_oftype, basicone_plus_sbasiconesbasicoperatorW_reps ×2 · new_conjugacy ×2operatorsfield ofW_rep, W_reps · field ofWCell, W_reps · new_conjugacy ×6orbit_symbolssommers ×2orbit_symbols_Bsommersorbit_symbols_Csommersorbits_and_unipotentsfinite_unipotentsorbits_and_unipotents_strictfinite_unipotentsOrbitSymboltype, sommersorderbasic ×2 · all_finite_order · field ofEigenPair, good_W_representativesorientation_nr_termbasicother_real_formsK_Nilpotentout_listfield ofWNode, W_reps
P
psommersP_indfield ofExtParamPol, extParamPolp_lG2_unitary_dualP_minusfield ofExtParamPol, extParamPolP_plusfield ofExtParamPol, extParamPolp_sG2_unitary_dualP_X_matricesK_Nilpotentpadbasic ×2Pairtype, basicpairsfield ofKNilpotentData, K_Nilpotent · field ofgood_data, weyltosemisimplepairs_of_total_sumcombinatoricsparamfield oftag_Param, modulesparam_polbasic ×2parameterbasic ×2 · L_packetparameter1_wcomplexparameter_gcomplexparameter_gammabasicparameter_mcomplexparameter_wcomplexparametersstableparameters_singularstableparameters_tau_contained_in_complementstableparameters_tau_containingstableparity_poscorootsbasicparity_restricted_partitionscombinatoricspart_containinggaloispart_restricted_partitionscombinatoricsPartitiontype, combinatoricspartition_pairscombinatoricspartition_sumsommerspartitionscombinatoricspartitions_all_even_partssommerspartitions_all_odd_partssommersparts_Bsommersparts_Csommerspassedflagtest_unitarity ×2Permutationtype, combinatoricspermutationcombinatoricspermutation_actcombinatoricspermutation_conjugationcombinatoricspermutation_cyclescombinatoricspermutation_decodecombinatoricspermutation_encodecombinatoricspermutation_inversecombinatoricspermutation_iteratorcombinatoricspermutation_matrixcombinatoricspermutation_matrix_sortstablepermutation_of_root_vectorstits_centralizerpermutation_productcombinatoricspermutation_right_actcombinatoricspermutationscombinatoricspermutecombinatorics ×2permute_characterfourpermute_columnscombinatoricspermute_rowscombinatoricspermuted_corootbasicpermuted_rootbasicPhiK_Nilpotent ×4phinumber_theoryphi_datatype, number_theoryphi_data_ofnumber_theoryphi_nfield ofphi_data, number_theoryPhi_Shodge_K_type_formula ×6Phi_S_inversehodge_tensorPhi_uptoK_Nilpotent ×2pi0galoispluralbasic ×2polfield oftag_ParamPol, modulespol_formatbasic ×2poscoroot_indexbasicposcoroot_sumbasicpositive_imaginary_roots_and_corootsbasic ×2positive_real_roots_and_corootsbasic ×2positive_to_negativebasicposroot_indexbasicposroot_sumbasicpower_modnumber_theorypower_setbasic ×2present_inbasicprime_candidats_afternumber_theoryprime_divisorsnumber_theoryprime_factors_phifield ofphi_data, number_theoryprime_phi_datanumber_theoryprimesnumber_theoryprint_blockbasicprint_blockdbasicprint_blockubasicprint_branch_irrprint_K_types ×4print_branch_irr_longprint_K_types ×4print_branch_K_irr_longprint_K_types ×2print_branch_K_std_longprint_K_types ×2print_branch_stdprint_K_types ×5print_branch_std_longprint_K_types ×4print_c_form_irreducible_big_longextParamPol ×2print_Cartan_infobasicprint_extended_blockbasicprint_extended_character_formulaextParamPol ×2print_extended_composition_seriesextParamPol ×2print_extended_indicesextParamPolprint_graded_composition_seriesjantzen ×5print_hermitian_form_irreduciblehermitian ×4print_hermitian_form_irreducible_longhermitian ×2print_hodge_irrhodge_K_type_formulaprint_hodge_irr_longhodge_K_type_formulaprint_hodge_K_type_mathodge_testprint_hodge_K_type_mat_trivialhodge_testprint_hodge_stdhodge_K_type_formulaprint_hodge_std_longhodge_K_type_formulaprint_is_unitaryhermitianprint_is_unitary_bigextParamPolprint_K_signature_irrprint_K_types ×2print_KGBbasicprint_KL_basisbasicprint_KL_listbasicprint_linesbasicprint_nilpotent_orbit_partitionsommersprint_nilpotent_orbit_partitionssommers ×2print_nilpotent_orbit_partitions_special_onlysommersprint_number_unipotentsfinite_unipotentsprint_prim_KLbasicprint_primitive_ideal_infoprojectors_using_character_tables ×2print_stablestableprint_stable_at_singularstable ×2print_stable_at_singular_unsortedstableprint_stable_sumsstable ×2print_strong_realgaloisprint_sumsstableprint_testtest_unitarity ×2print_test_all_real_induced_one_dimensionaltest_unitarityprint_test_Aqtest_unitarity ×2print_test_onetest_unitarityprint_test_spherical_unipotenttest_unitarityprint_test_Upq_Aqtest_unitarity ×2print_twisted_c_form_irreducible_longhermitianprint_W_cellsbasicprint_W_graphbasicprintParamPolstableprints_linesbasicproductbasic ×4project_to_dominant_conebasicprojectorK_Nilpotent · projectors_using_character_tables ×3projector_newnew_conjugacyprojector_new_plusnew_conjugacyprojector_verboseprojectors_using_character_tablesprojectorsprojectors_using_character_tables ×3projectors_newnew_conjugacyprojectors_new_plusnew_conjugacyprojectors_verboseprojectors_using_character_tables ×2promotetits_centralizer ×2prunerestricted_rootspsG2_unitary_dualPsi_irrstablepuritybasic
Q
qsommers · L_packetQ_and_P_X_matricesfield ofKNilpotentData, K_NilpotentQ_matricesK_Nilpotentquad_testcoherent_irreducible ×3quasicompact_formbasic
R
r_adjustbasicradical_basisbasicrangebasicrankbasic · combinatorics ×4 · sommersrat_as_intbasicratvec_as_vecbasicraw_KLbasicrdfield ofgood_data, weyltosemisimplerd_E6exceptionalNilpotentDatard_E7exceptionalNilpotentDatard_E8exceptionalNilpotentDatard_F4exceptionalNilpotentDatard_G2exceptionalNilpotentDatareal_complex_torusbasicreal_corootsbasicreal_formbasic · field ofKNilpotentData, K_Nilpotent · weak_packets_precomputed_cell_tracesreal_formsbasicreal_forms_given_central_invariantgaloisreal_forms_ofK_Nilpotentreal_orbit_numberK_Nilpotentreal_orbitsfield ofKNilpotentData, K_Nilpotentreal_poscorootsbasic ×2real_posrootsbasic ×2real_quaternionicfinite_dimensional_signaturereal_rootsbasicreal_roots_and_corootsbasic ×2real_sysbasic ×2real_typebasic ×2recursive_deformextParamPolreduceextParamPolrefine_raw_dataall_finite_orderreflectbasic ×4reflectionbasic ×2reflection_cobasicregroupsommersremove_duplicatesrestricted_rootsremove_zerosrestricted_rootsrepeat_partscombinatoricsreportweyltosemisimplerequired_solutionbasic ×3requisitionbasic ×2restrict_representation_isotypicprojectors_using_character_tablesrestricted_rootsrestricted_rootsrestricted_roots_longrestricted_rootsrestricted_roots_rawrestricted_rootsrestricted_roots_shortrestricted_rootsreversebasic · coherent_irreduciblereverse_pairsweyltosemisimplerhobasicrho_checkbasicrho_check_diffbasicrho_check_ibasic ×2rho_check_rbasic ×2rho_cibasicrho_coverdirac_indexrho_diffbasicrho_ibasic ×2rho_ncibasicrho_rbasic ×2rho_shiftsK_Nilpotentrightcomplex ×2right_cellssub_cellsright_copysub_cellsright_corootssub_cellsright_factorssub_cellsright_reflectbasic ×2right_root_indicessub_cellsright_rootssub_cellsrising_powercombinatoricsrlex_cmp_partitionscombinatoricsrlex_leq_partitionscombinatoricsrootbasicroot_datumbasic ×10 · field ofWGraph, W_reps · W_reps · K_Nilpotent · sommers ×2root_datum_from_positivebasicroot_indicesbasicroot_orbitscombinatoricsroot_span_projectorbasicrootsbasicroots_all_positivebasicroots_same_restrictionnilpotent_centralizerrounded_log_2GK_dimensionrows_withbasic ×3rqfinite_dimensional_signaturerun_hodge_testshodge_test ×2run_reportsgood_W_representatives
S
Scoherent_irreducible ×7sbasicS_invcoherent_irreducible ×9S_inv_pluscoherent_irreducibles_one_nc_cx_restricted_rootsK_Nilpotents_one_rootsK_Nilpotents_one_roots_restrictedK_Nilpotents_partbasics_to_1basic ×3s_to_minus_1basic ×3same_ascombinatorics ×3same_complex_parabolicsadams_johnsonsaturated_spanbasicsearchK_Nilpotent ×3search_probable_generatornumber_theory ×2search_probable_primenumber_theoryseat_belt_onbasicselecttruncated_induction ×2select_both_degreestruncated_inductionselect_degreetruncated_inductionselect_generic_degreetruncated_inductionself_intersectcomplexsemisimplebasicsemisimple_rankbasicseparate_by_infinitesimal_characterbasicset_bit_positionsbasic ×2set_first_unset_bitbasicset_generic_degreegeck_generic ×2set_generic_degreesgeck_genericset_generic_degrees_2geck_genericset_generic_degrees_3geck_genericset_generic_degrees_geckgeck_genericset_generic_degrees_geck_longgeck_genericset_generic_degrees_plusgeck_genericshifthodge_tensorshift_by_minus_1weyltosemisimpleshort_formatG2_unitary_dualshort_stringsub_cellsshortensbruhatshowmodules ×4 · K_Nilpotent · good_W_representatives ×2 · arthur_parameters ×2 · sub_cells ×4 · associated_variety_annihilator · geck_generic · L_packetshow_A_barlusztig_cells ×2show_big_tablefourshow_big_table_rfour ×2show_bruhatbruhat ×2show_bruhat_geqbruhatshow_bruhat_leqbruhatshow_bruhat_Wbruhatshow_bruhat_W_geqbruhatshow_bruhat_W_leqbruhatshow_BXYI_tablesfour ×2show_centralizer_isogeniesnilpotent_centralizershow_centralizer_isogenynilpotent_centralizershow_count_orbits_and_centralizersfinite_unipotents ×2show_count_unipotentsfinite_unipotentsshow_degreesgeck_genericshow_dirac_indexdirac_index ×2show_dirac_index_param_poldirac_indexshow_lusztig_cellslusztig_cells ×3show_lusztig_cells_pluslusztig_cells ×3show_shortsub_cellsshow_signaturefinite_dimensional_signatureshow_stable_sums_irrL_packetshow_Xfoursigma_lusztiglusztig_cellssigma_lusztig_longlusztig_cellssigma_lusztig_verboselusztig_cellssignbasic ×3 · tits_centralizer · coherent_irreduciblesignaturefinite_dimensional_signaturesignature_stringfinite_dimensional_signaturesignature_tracefinite_dimensional_signaturesignature_trace_stringfinite_dimensional_signaturesigned_cycle_typecombinatoricssigned_cycle_type_codecombinatoricsSigned_cyclestype, combinatoricssigned_permutationcombinatoricssignssommerssimple_coroot_labelsbasicsimple_imaginary_reflectionsL_packetsimple_imaginary_rootsL_packetsimple_reflection_on_root_vectortits_centralizersimple_reflection_on_root_vectorstits_centralizer ×2simple_root_labelsbasicsimple_system_from_positivebasicsimply_connectedbasic ×2simply_connected_2rho_checkbasicsinexp-generating-seriessingular_blockbasicsingular_block_ofbasicsingular_root_datumbasicsingular_simple_indicesbasicslex_cmp_partitionscombinatoricsslex_leq_partitionscombinatoricssliceGK_dimensionsmaller_orbitsK_Nilpotentsmallest_degreeW_repsSmith_basisbasicSn_representation_dimensioncombinatoricssndfield ofPair, basicsolutionfield ofLinearSolution, basicsolvebasic ×3solve_for_tits_elementtits_centralizersolve_for_torus_elementtits_centralizersolve_for_torus_element_oldtits_centralizersomefield ofMaybe, basicsortsommerssort_by_dual_orbitsommerssort_by_lengthbruhatsort_by_length_descendingbruhatsort_K_types_by_log_2_heightGK_dimensionsort_special_firstfamiliessort_to_partitioncombinatoricssort_u_belowbasicsphodge_K_type_formulaspanning_subsettits_centralizerspecial_charactersfourspecial_geck_dataspecial_rep ×3special_projectorprojectors_using_character_tables ×2special_projectorsprojectors_using_character_tables ×2special_representation_of_cellprojectors_using_character_tables ×2SpehspehSpeh_0spehspeh_0spehspeh_longspehsplitbasic ×2split_1basicsplit_as_intbasicsplit_classfield ofD_class, combinatoricssplit_factor_formatbasicsplit_formbasic ×2split_formatbasicsplit_irrfield ofD_irrep, combinatoricssplit_linesbasicsplit_minus_1basicsplit_rankbasic ×2split_torusbasicSpringer_table_generic_degreesgeck_genericsquare_classesgaloisstablestablestable_at_regularstablestable_at_singularstable ×3stable_at_singular_unsortedstablestable_sumsstablestable_sums_irrL_packetstable_sums_irr_in_basis_of_standardsL_packetstable_sums_irr_in_basis_of_standards_as_matricesL_packetstable_sums_irr_matL_packetstable_sums_partialstablestable_sums_stdL_packetstable_sums_std_matricesL_packetstable_test_Aq_packetstable ×2standardizegood_W_representativesstarsommersstarting_set_of_K_typeshodge_K_type_formulastatusbasic ×3 · modulesstatus_textbasic ×4status_textsbasic ×2stdmodulesstrbasicstrict_partitionscombinatoricsstringifyK_Nilpotentstrip_to_partitioncombinatoricsstrong_componentssub_cells ×2strong_real_formsgaloisstrong_real_forms_central_invariant_egaloisstrong_real_forms_given_central_invariantgaloisstrong_real_forms_indicesgaloisstrong_real_forms_same_typegalois ×2strong_real_forms_typegaloisstrong_testhermitiansub_cellssub_cellssub_datumbasic ×2sub_digraphsub_cellssub_graphsub_cellssubsets_of_s_one_nc_cx_restricted_rootsK_Nilpotentsubspace_injection_matrixstablesubtract_onesommerssucceedfield ofOne_of, basicsucceedsbasicsumbasic ×10 · sommers ×2survivesbasicswapgenericsswapargsgenericsswapped_factorssub_cellsSymboltype, combinatoricssymbolcombinatoricssymbol_to_bipartitioncombinatorics
T
Tcoherent_irreducible ×5T_invcoherent_irreducible ×2T_matricesfield ofKNilpotentData, K_NilpotentT_matrixK_Nilpotenttableall_finite_order ×2table_reducedall_finite_ordertagfield oftag_Param, modules · field oftag_ParamPol, modulestag_Paramtype, modulestag_ParamPoltype, modulestakebasictanexp-generating-seriestangent_basisfield ofAffineSubspace, basictaubasic · modules · field ofWNode, W_reps · sub_cellstau_bitsetbasictau_complementbasic · modulestesttest_unitarity ×3 · all_finite_order · nilpotent_centralizer · sommers ×2test_all_orbitsK_Nilpotent ×2test_all_real_induced_one_dimensionaltest_unitaritytest_Aqtest_unitarity ×2test_av_maximal_idealassociated_variety_annihilator ×2test_braidtest_braid ×2test_elementary_hodge_grading_irr_at_1hodge_testtest_elementary_hodge_grading_irr_at_shodge_testtest_elementary_hodge_grading_std_at_1hodge_testtest_elementary_hodge_grading_std_at_shodge_testtest_enumerated_classnew_conjugacytest_Fermatnumber_theorytest_geck_generic_degreesgeck_generictest_generic_degreesgeck_generic ×2test_hodge_grading_std_at_1hodge_testtest_hodge_grading_std_at_shodge_testtest_hodge_K_type_mathodge_K_type_formula ×2 · hodge_test ×2test_induce_orbitstruncated_inductiontest_inv_conjfourtest_lG2_unitary_dualtest_linehermitiantest_minimal_orbitsK_Nilpotent ×2test_onetest_unitaritytest_one_unitarytest_unitaritytest_positivehodge_testtest_positive_hodge_K_type_mathodge_test ×2test_projectornew_conjugacytest_sG2_unitary_dualtest_simple_typebasictest_special_geckspecial_reptest_special_mapsspecial_rep ×3test_spherical_unipotenttest_unitaritytest_valid_onlysommerstheta_induce_irreducible_wgraphcoherent_irreducible ×2theta_orbit_repsK_Nilpotenttheta_plus_1_lambdabasic ×2ti_verbosetruncated_inductiontimebasictimed_outfield ofmaybe_KTP, hermitiantimes_sbasictits_actiontits_centralizer ×2tits_centralizer_weaktits_centralizer ×11tits_centralizes_weaktits_centralizer ×3tits_subgrouptits_centralizerto_basebasic · tits_centralizerto_base_fixed_lengthbasicto_base_polybasicto_cyclescombinatoricsto_D_classcombinatoricsto_listbasicto_partition_paircombinatoricsto_specialspecial_repto_special_geckspecial_rep ×3to_special_geck_longspecial_rep ×3to_strmodules ×4to_str_irrmodulesto_str_K_typesmodulesto_str_stdmodulestorusbasic · nilpotent_centralizertorus_actiontits_centralizertorus_datumbasictorus_elementweyltosemisimple ×2tracebasictrace_reportfinite_dimensional_signaturetransposebasic · combinatoricstrifinite_unipotentstrivialbasictrivial_WW_repstruncateGK_dimension · truncated_induction ×2truncate_by_degree_induce_charactertruncated_inductiontruncate_degree_induce_charactertruncated_inductiontruncate_induce_charactertruncated_induction ×5tTcoherent_irreducible ×5twistbasictwist_by_minus_2rho_u_cap_sK_Nilpotent ×2twist_orbitshermitiantwisted_c_form_irreduciblehermitian ×2twisted_c_form_irreducible_as_sum_of_standardshermitiantwisted_c_form_irreducible_contributionshermitiantwisted_c_form_irreducible_longhermitiantwisted_c_form_irreducible_unnormalizedhermitiantwisted_c_form_stdhermitiantwo_rhobasic ×2two_rho_checkbasic ×2
U
uflagtest_unitarity ×2uncurrygenericsunderlying_orbitsommersungroupsommersunionhodge_K_type_formula ×2 · weyltosemisimpleuniqueW_reps · hodge_K_type_formula ×2unset_bit_positionsbasic ×2unset_first_set_bitbasicunsplit_classfield ofD_class, combinatoricsunsplit_irrfield ofD_irrep, combinatoricsunzipbasicup_left_cornercomplexup_right_cornercomplexupdate_generic_degreesgeck_generic ×5update_generic_degrees_geckgeck_generic ×2update_generic_degrees_geck_longgeck_generic ×2update_generic_degrees_using_signgeck_generic ×2update_one_table_geckgeck_genericupdate_one_table_mixed_strategygeck_genericupdate_pairsK_Nilpotentupdate_Q_and_P_X_matricesK_Nilpotentupdate_T_matricesK_Nilpotentupdate_Y_matricesK_Nilpotentupperbasic ×2
V
vfield ofKNilpotentData, K_Nilpotentvalid_signssommersvalid_symbolssommersvalidatesommers ×2validate_signssommersvanishingstablevariants_Bsommers ×2variants_Csommers ×2vecsfield ofEigenPair, good_W_representativesVectorK_Nilpotent ×2vectorbasic · K_Nilpotent ×2verbose_fdsigfinite_dimensional_signaturevertex_and_W_cellsW_repsview_complexcomplex
W
wcomplexw0basicW_cells_ofW_repsW_crossbasicW_eltbasicW_elt_of_reflectionbasic ×2W_genbasicW_gensbasicW_graph_ofcoherent_irreduciblew_parametercomplexw_parameter1complexW_reptype, W_repswall_projectorbasicWCelltype, W_repsweak_packetweak_packets_precomputed_cell_tracesweak_packet_intweak_packets_precomputed_cell_traces ×2weak_packetsweak_packets_precomputed_cell_traces ×5weak_packets_preweak_packets_precomputed_cell_tracesweak_testhermitianWGraphtype, W_repswherebasicwidthbasicwith_decimalsbasicWNodetype, W_reps
X
xbasic ×2x_lG2_unitary_dualx_openbasicx_sG2_unitary_dualXORbasic
Y
y_genL_packetY_matricesfield ofKNilpotentData, K_Nilpotent
Z
zero_corootsgood_W_representativeszero_rootsall_finite_orderzipbasic
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