Atlas of Lie Groups and Representations

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atlas Library · Script File

springer_tables.at

tables encoding the Springer correspondence
Source
atlas-scripts/springer_tables.at (214 lines)
Definitions
32
Loads
Loaded by
special_rep.at cells.at springer_tables_reductive.at springer_table_A.at springer_table_BC.at springer_table_D.at springer_table_F.at springer_table_G.at springer_table_E6.at springer_table_E7.at springer_table_E8.at show_nilpotent_orbits.at

Definitions in source order

SpringerTable type

L7
set_type [SpringerTable=
  (CharacterTable ct,
  [ComplexNilpotent] orbits,
  (ComplexNilpotent->ComplexNilpotent) dual_map,
  (ComplexNilpotent->ComplexNilpotent) dual_map_i,
  (ComplexNilpotent->int) springer
  )]

Fields: ct, orbits, dual_map, dual_map_i, springer

character_table

L15character_table = ct@SpringerTable
longer name for the projector

also defined in character_tables.at, character_tables_reductive.at

root_datum

L16root_datum (SpringerTable st) = RootDatum

also defined in basic.at, W_reps.at, K_Nilpotent.at, sommers.at, parabolics.at, K.at, nilpotent_orbits.at, W_classes.at, extended_misc.at, character_tables.at, disconnected.at, cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

characters

L17characters (SpringerTable st) = [[int]]

also defined in character_tables.at

orbit

L18orbit (SpringerTable st) = (int i) ComplexNilpotent

also defined in W_reps_type_BC.at, exceptional.at

generic_degree

L19generic_degree (SpringerTable st) = (int->int)

also defined in character_tables.at

n_orbits

L21n_orbits (SpringerTable st) = int

equal

L24equal([[int]] a,[[int]] b) = bool
equality of normalized diagrams; |=@mat| must be avoided and cannot be used

also defined in poset.at

orbit_index

L26orbit_index(SpringerTable st,ComplexNilpotent orbit) = int
equivalently: first(st.n_orbits,(int i): orbit = st.orbits[i])
L29

special_closure 2 overloads

L31special_closure(SpringerTable st,ComplexNilpotent O) = ComplexNilpotent
L34special_closure (SpringerTable st) = (ComplexNilpotent->ComplexNilpotent)

is_special 2 overloads

L38is_special (SpringerTable st,ComplexNilpotent O) = bool
orbit is special if O=special_closure(O)
L42is_special(SpringerTable st) = (ComplexNilpotent->bool)

also defined in combinatorics.at, W_reps_type_BC.at, exceptional.at

special_orbits_indexed

L46special_orbits_indexed(SpringerTable st) = ([int],[ComplexNilpotent])
all special orbits, including numbering from list of all orbits

special_orbits

L50special_orbits(SpringerTable st) = [ComplexNilpotent]
st is the Springer table for RootDatum rd
O is an orbit for rd
special_piece(st,O)=list of orbits for rd:
all orbits satisfying d(orbit)=d(O)
L52

special_piece

L57special_piece(SpringerTable st, ComplexNilpotent O) = [ComplexNilpotent]

dual_special_piece

L71dual_special_piece (SpringerTable st, ComplexNilpotent O_check) = [ComplexNilpotent]
all orbits satisfying d_inverse(orbit^v)=d_inverse(O^v)

springer_character

L86springer_character(SpringerTable st) = (ComplexNilpotent->[int])

springer_inverse 2 overloads

L94springer_inverse(SpringerTable st,int special_character_number) = (bool,ComplexNilpotent)
inverse Springer map: try to find an orbit (not necessarily special) such
 that springer(orbit)=given character
if the character is special this succeeds and gives a special orbit
otherwise it might fail, in which case it returns (false,0-orbit)
L97springer_inverse(SpringerTable st) = (int->(bool,ComplexNilpotent))

springer_inverse_special 2 overloads

L100springer_inverse_special(SpringerTable st,int special_character_number) = ComplexNilpotent
case of special orbit don't need the boolean
L102springer_inverse_special(SpringerTable st) = (int->ComplexNilpotent)

special_star 2 overloads

L104special_star(SpringerTable st,ComplexNilpotent O) = string
L106special_star(SpringerTable st) = (ComplexNilpotent ->string)

also defined in W_reps_type_BC.at

special_representations

L110special_representations(SpringerTable st) = [int]
special representations defined in  character_tables.at
these versions for SpringerTable call those for the associated CharacterTable

also defined in character_tables.at

is_special_representation 2 overloads

L111is_special_representation(SpringerTable st,int i) = bool
L112is_special_representation(SpringerTable st) = (int->bool)

also defined in character_tables.at

special_representation_star

L113special_representation_star(SpringerTable st) = (int->string)

also defined in character_tables.at

integral_descent

L118integral_descent (SpringerTable st_int,ComplexNilpotent O_check) = (ComplexNilpotent,mat)
this version of integral_descent is used in weak_packets.at
this uses a permutation, which we return since it is needed later

also defined in nilpotent_orbits.at

show

L136show (SpringerTable st) = void

also defined in modules.at, K_Nilpotent.at, good_W_representatives.at, arthur_parameters.at, sub_cells.at, associated_variety_annihilator.at, geck_generic.at, L_packet.at, ratmat.at, Levi_subgroups.at, nilpotent_orbits.at, subgroups.at, poset.at, cyclotomic.at, polynomial.at, character_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, laurentPolynomial.at, weak_packets.at, cyclotomic_Lie_algebra.at, cyclotomicMat.at, bracket.at, exceptional.at, nilpotent_posets.at, lazy_lists.at

show_long

L162show_long(SpringerTable st) = void

also defined in character_tables.at, hodgeParamLaurentPol.at, hodgeParamPol.at, weak_packets.at, cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

show_reps

L195show_reps (SpringerTable st) = void

also defined in character_tables.at

Generated from atlas-scripts at commit 648506c (2026-10-13).