Atlas of Lie Groups and Representations

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atlas Library · Script File

character_tables.at

Source
atlas-scripts/character_tables.at (342 lines)
Definitions
72
Loads
Loaded by
projectors_using_character_tables.at geck_generic.at special_rep.at families.at cells.at character_table_reps.at springer_tables.at isomorphism_W.at classical_character_tables.at character_table_F.at character_table_G.at character_table_E6.at character_table_E7.at character_table_E8.at springer_table_E6.at springer_table_E7.at springer_table_E8.at generic_degrees.at

Definitions in source order

sum

L5sum ([int] v,[int] w) = [int]
free function on characters; avoid |vec| to allow very large representations

also in this file at line 120; also defined in basic.at, sommers.at, cyclotomic_Lie_algebra.at, lazy_lists.at

tensor_product

L8tensor_product ([int] v,[int] w) = [int]

also defined in finite_dimensional.at, tensor_product.at

tensor

L10tensor = tensor_product@([int],[int])
convenience

also in this file at line 160

Cartesian_power

L12Cartesian_power([int] x,int n) = [int]

also in this file at line 165

tensor_power

L14tensor_power([int] v,int power) = [int]

also in this file at line 162

Character table structure for Weyl groups (integer valued characters)
L16
The structure below was inspired by the way GAP represents character tables
L18

CharacterTable type

L20
set_type
[ CharacterTable =
  ( WeylClassTable class_table { conjugacy classes }
  , (int->string) class_label { used to identify classes on output }
{ irreducible characters }
  , (int->string) irreducible_label
  , mat table { by row=character }
  , [int,int] deg_spec { degree, and associated special irreducible }
  )
]

Fields: class_table, class_label, irreducible_label, table, deg_spec

character_table 3 overloads

L32character_table ( WeylClassTable !Wct , [string] class_names , [[int],string] chars , (int->int) to_special ) = CharacterTable
a constructor that keeps the class ordering of |Wct|, attaches names in order
L53character_table ( WeylClassTable Wct , [WeylElt,string] class_reps , [[int],string] chars , (int->int) to_special ) = CharacterTable
L80character_table ( RootDatum rd , [WeylElt,string] class_reps , [[int],string] chars , (int->int) to_special ) = CharacterTable

also defined in springer_tables.at, character_tables_reductive.at

trivial_character_table

L88!trivial_character_table = CharacterTable
A valid character table that can be used to initialise local variables

root_datum

L93root_datum (CharacterTable ct) = RootDatum
attributes inherited from the |class_table| component

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, springer_tables.at, disconnected.at, cyclotomic_Lie_algebra.at, bracket.at, exceptional.at

rank

L94rank (CharacterTable ct)

also defined in basic.at, combinatorics.at, sommers.at, affine.at

order_W

L95order_W (CharacterTable ct)

also defined in Weylgroup.at

n_classes

L96n_classes (CharacterTable ct)

also defined in W_classes.at

n_irreps

L97n_irreps = n_classes@CharacterTable
alias, for readability

class_representatives

L99class_representatives (CharacterTable ct) = [WeylElt]

class_representative

L101class_representative (CharacterTable ct) = (int->WeylElt)

also defined in W_classes.at

class_of

L103class_of(CharacterTable ct)

class_order

L105class_order (CharacterTable ct) = (int i) int
element order

class_sizes

L107class_sizes (CharacterTable ct)

class_size

L108class_size (CharacterTable ct) = (int i) int

also defined in combinatorics.at, W_classes.at

centralizer_order

L109centralizer_order(CharacterTable ct) = (int i) int

also defined in combinatorics.at, W_classes.at

class_power

L111class_power(CharacterTable ct) = (int,int->int)

id_class

L113id_class(CharacterTable ct)

dimension 2 overloads

L114dimension (CharacterTable ct) = ([int] chi) int
L115dimension (CharacterTable ct,int i) = int

also defined in basic.at, finite_dimensional.at, K_highest_weights.at, nilpotent_orbits.at, W_reps_type_BC.at, tensor_product.at, exceptional.at

inner

L117inner(CharacterTable ct) = ([int],[int]->int)

norm2

L118norm2(CharacterTable ct) = ([int] x)int

sum

L120sum(CharacterTable ct) = ([int],[int]->[int])

also in this file at line 5; also defined in basic.at, sommers.at, cyclotomic_Lie_algebra.at, lazy_lists.at

product

L121product(CharacterTable ct) = (([int],[int])->[int])

also defined in basic.at

power_map

L123power_map(CharacterTable ct) = (int->[int])

sym_power_refl

L128sym_power_refl(CharacterTable ct)

trivial

L130trivial(CharacterTable ct) = [int]

also defined in basic.at

sign

L131sign(CharacterTable ct) = [int]

also defined in basic.at, tits_centralizer.at, coherent_irreducible.at, extended.at

reflection

L132reflection (CharacterTable ct) = [int]

also defined in basic.at

sym_power

L135sym_power (CharacterTable tab)
symmetric power of a character

also defined in W_classes.at

ext_power

L136ext_power (CharacterTable tab)

also defined in W_classes.at

character

L139character(CharacterTable t) = (int i) [int]

also defined in combinatorics.at, modules.at, W_reps.at, character_table_reps.at

characters

L140characters(CharacterTable ct) = [[int]]

also defined in springer_tables.at

character_index

L142character_index (CharacterTable ct) = ([int] char) int

index

L144index (CharacterTable ct,vec char) = int

also defined in finite_dimensional_signature.at

degree

L146degree (CharacterTable ct) = (int i)int

also defined in cyclotomic.at, polynomial.at, laurentPolynomial.at

special

L147special (CharacterTable ct) = (int i)int

generic_degree

L148generic_degree (CharacterTable ct) = (int i)int

also defined in springer_tables.at

tensor_sign

L151tensor_sign (CharacterTable ct) = ([int]->[int])
tensor_sign: defined on any character: [int] -> [int]
"free" |tensor| above

tensor_sign_index

L155tensor_sign_index (CharacterTable ct) = (int->int)
tensor_sign_index: involution on set of irreducible numbers

tensor

L160tensor (CharacterTable t) = (int i,int j)[int]
tensor product of two irreducibles by index; it this really useful?
use "free" function above

also in this file at line 10

tensor_power

L162tensor_power (CharacterTable t) = (int i,int k)[int]
"free" |tensor_power| above

also in this file at line 14

Cartesian_power

L165Cartesian_power(CharacterTable ct) = ([int] x,int n)[int]

also in this file at line 12

scalar_product

L168scalar_product (CharacterTable ct) = (int i0, int i1) int

inner_product

L172inner_product = (CharacterTable->([int],[int]->int))
different name for use with [int],[int] arguments
we already had that function

check_orthogonality

L175check_orthogonality (CharacterTable ct) = bool
The following few functions assume |ct| has a simple root datum
L178

index_long_reflection

L181index_long_reflection(CharacterTable ct) = int
find index j of the conjugacy class of a long root reflection, or -1 if none

index_short_reflection

L188index_short_reflection(CharacterTable ct) = int

signature 2 overloads

L195signature(CharacterTable ct,[int] char) = [int]
character values at two or three special elements:
identity, long root reflection, and if different short root reflection
L201signature(CharacterTable ct) = ([int] char)[int]

also defined in finite_dimensional_signature.at

decompose

L203decompose(CharacterTable ct) = (vec v)[int]
list of irrep multiplicities

also defined in lattice.at

memo_1_rd

L206memo_1_rd ((RootDatum->CharacterTable) compute, RootDatum special) = (RootDatum->CharacterTable)

smallest_degree_summands

L220smallest_degree_summands (CharacterTable ct,[int] mults) = [int]
|mults| is a vector of multiplicities of the irreducible characters;
return a list (without multiplicities) of the irreducible characters
of smallest degree occurring
convenience functions
L228

show_classes

L230show_classes (CharacterTable ct) = void

show_classes_long

L253show_classes_long(CharacterTable ct) = void

show_reps

L278show_reps (CharacterTable ct) = void

also defined in springer_tables.at

show

L299show (CharacterTable ct) = 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, springer_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

L303show_long (CharacterTable ct) = void

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

show_very_long

L307show_very_long (CharacterTable ct) = void

view_multiplicities

L312view_multiplicities (CharacterTable ct,vec v ) = void
multiplicities

view_char 2 overloads

L328view_char(CharacterTable ct,vec character) = void
L331view_char(CharacterTable ct) = (vec v)void
stuff about special representations moved from springer_tables.at
L333

is_special_representation 2 overloads

L335is_special_representation(CharacterTable ct, int i) = bool
L336is_special_representation(CharacterTable ct) = (int ->bool)

also defined in springer_tables.at

special_representations

L337special_representations(CharacterTable ct) = [int]

also defined in springer_tables.at

special_representation_star

L339special_representation_star(CharacterTable ct) = (int ->string)

also defined in springer_tables.at

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