atlas Library · Script File

`springer_table_A.at`

Springer correspondence from nilpotent orbits to representations of W in type A
Orbits and W-irreps both correspond to partitions of rank+1, which gives the
(here bijective) map from orbits to W-irreps for type A

Partition of $n$ to a ComplexNilpotent for a given type $A_{n-1}$ root datum

Source

atlas-scripts/springer_table_A.at (69 lines)

Definitions

Loads

<basic.at
<combinatorics.at { for |classic_permutation| }
<nilpotent_orbits.at { for the type |ComplexNilpotent|, |diagram_normalized| }
<springer_tables.at
<classical_character_tables.at { for |character_table_A| }

Loaded by

springer_tables_reductive.at

Definitions in source order

`nilpotent_orbit_GL`

L14nilpotent_orbit_GL ([int] parts) = ComplexNilpotent

`nilpotent_orbit_A`

L20nilpotent_orbit_A(RootDatum rd) = (Partition->ComplexNilpotent)

`diagram_A_to_partition`

L32diagram_A_to_partition ([int] diagram) = Partition

an inverse operation, starting from |stratified_diagram(orbit)|

Springer table construction.

For the mapping parts (dual, dual dual, and Springer) we shall impose on the
"simple" Springer tables the constraint that these maps will ignore any
radical component of |O.H|, in other words that vanishes on the roots of
|O.root_datum|. This will allow, for a root datum with disconnected Dynkin
diagram, the orbits to be passed to the Springer tables for sub-rootdata for
the individual Dynkin diagram components without having to project away the
part for the other components. This constraint is automatically met if we
handle orbits using only their diagram, which has this independence property.

L43

`dual_map_A`

L55dual_map_A(ComplexNilpotent O) = ComplexNilpotent

`springer_table_A`

L59springer_table_A (RootDatum rd) = SpringerTable

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