atlas Library · Script File

`springer_table_E8.at`

name,diagram,dim,special,dual,A/Abar

Source

atlas-scripts/springer_table_E8.at (291 lines)

Definitions

Loads

Loaded by

springer_tables_reductive.at

Definitions in source order

`E8_Springer_table_data`

L8!E8_Springer_table_data = [string,[int],int,bool,string,string]

`E8_names`

L81E8_names = [string]

`E8_diagrams`

L82E8_diagrams = [[int]]

`E8_dimensions`

L83E8_dimensions = [int]

`E8_special`

L84E8_special = [bool]

`E8_dual_names`

L85E8_dual_names = [string]

`name_to_diagram_E8`

L87name_to_diagram_E8 = (string->vec)

`E8_nilpotent_orbit_dual_map`

L100E8_nilpotent_orbit_dual_map = (vec->vec)

`diagram_to_name_E8`

L119diagram_to_name_E8 ([int] diagram) = string

`dual_diagram_E8`

L123dual_diagram_E8([int] diagram) = [int]

also in this file at line 141

`digit_to_int`

L129digit_to_int (string str)

`string_to_diagram`

L133string_to_diagram(string str) = [int]

`join`

L136join ([int] list) = string

also defined in basic.at

`dual_diagram_table_E8`

L138!dual_diagram_table_E8 = [[int],[int]]

`dual_diagram_E8`

L141dual_diagram_E8([int] diagram) = [int]

also in this file at line 123

`dual_E8`

L146dual_E8 (ComplexNilpotent O) = ComplexNilpotent

`springer_by_name_table_E8`

L153springer_by_name_table_E8 = [string,int]

The "Springer map" is given by the following mapping in terms of orbit names;
the image value are character indices according to the numbering of
|e8_gap_table| from e8_gap.at; this needs update if ordering there changes.

`E8_Springer_map`

L226E8_Springer_map = (vec->int)

orbit diagrams as |vec| values

`springer_table_E8` 2 overloads

L239springer_table_E8 () = SpringerTable

L257springer_table_E8 (RootDatum rd) = SpringerTable

In the general case we take into account the permutation of the simple root
indices returned by |test_simple_type| (it is a map from Bourbaki indices to
|rd| indices, which |stratified_diagram@(ComplexNilpotent,[int])| takes into
account to get into Bourbaki order, and which |permute| uses to transform the
dual diagram from |E8_nilpotent_orbit_dual_map| into the order of |dual(rd)|.

also in this file at line 285

`springer_by_name_E8`

L272springer_by_name_E8 (string name) = int

`springer_E8` 2 overloads

L277springer_E8 (ComplexNilpotent O) = int

L282springer_E8(CharacterTable ct,ComplexNilpotent O) = int