Atlas of Lie Groups and Representations

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atlas Library · Script File

springer_table_F.at

Like for G2, the Springer table maps are given in tabular form, where
nilpotent orbits are analysed through their diagram. While for G2 many things
can be decided without knowing the numbering of the Dynkin diagram for a given
root datum (it only matters for the Springer correspondence proper, and it
sufficed to test for order reversal with respect to the Bourbaki numbering),
we need a stratified diagram for nearly everything in F4, so the functions
that deal with general F4 root data need to cater for arbitrary numberings.
A function that works with a fixed F4 root datum does not need such measures;
we shall first describe the maps for such a system and generalise it later.
Source
atlas-scripts/springer_table_F.at (119 lines)
Definitions
4
Loads
Loaded by
springer_tables_reductive.at

Definitions in source order

F4_nilpotent_orbit_dual_map

L21F4_nilpotent_orbit_dual_map = (vec->vec)
The "dual map" is given by the following mapping in terms of orbit diagrams;
the "source" diagram assumes the Bourbaki numbering for F4 (0--0=>0--0),
while the "image" diagram assumes the dual datum, numbered as (0--0<=0--0).
orbit diagrams as |vec| values

F4_Springer_map

L54F4_Springer_map = (vec->int)
The "Springer map" is given by the following mapping in terms of orbit
diagrams; the image value are character indices according to the numbering
of |character_table_F4_data| from character_table_F,at
orbit diagrams as |vec| values

springer_table_F 2 overloads

L83springer_table_F (int n) = SpringerTable
L105springer_table_F (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 |F4_nilpotent_orbit_dual_map| into the order of |dual(rd)|.
For the dual mapping in the opposite direction, both the input diagram and the
output diagram for |F4_nilpotent_orbit_dual_map| assume a reversal of the
diagram order with respect to the forward direction, so we prepare and use the
reverse |~map| of the |map| list on both occasions.

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