Atlas of Lie Groups and Representations

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atlas Library · Script File

nilpotent_orbit_partitions.at

(complex) nilpotent orbits for simple classical groups,
classical groups: SL(n), GL(n), Sp(2n), SO(n),
orbits: partition-type parameterization
no attempt is made to handle I/II orbits in type D,
so these are really orbits for O(n,C) not SO(n,C)
semisimple_element: H\in X^* from Lie triple (H,X,Y),
standard (classical group) coordinates
(in type A_n lambda has n+1 coordinates, i.e. for GL(n))
Source
atlas-scripts/nilpotent_orbit_partitions.at (114 lines)
Definitions
17
Loads
Loaded by
representations.at nilpotent_orbits_exceptional.at springer_table_BC.at springer_table_D.at W_reps_type_BC.at

Definitions in source order

type_number 2 overloads

L17type_number (LieType t) = int
utility to convert type A-G of a simple root system to a number 0-6
we allow and ignore (trailing) torus factors; a pure torus classifies as 7
L24type_number (RootDatum rd) = int
same as previous applied to a RootDatum

nilpotent_orbit_partitions 2 overloads

L29nilpotent_orbit_partitions = …
(complex) nilpotent orbits, partition-type classification,
for simple group of type A-D
L43nilpotent_orbit_partitions(RootDatum rd) = [Partition]
insert conversion to LieType from (type converting to RootDatum)
(types A-D only; this only depends on complexification of real group)

dual_nilpotent_orbit_partitions 2 overloads

L47dual_nilpotent_orbit_partitions(RootDatum rd) = [Partition]
nilpotent orbits of dual group (types A-D only)
L49dual_nilpotent_orbit_partitions(RealForm G) = [Partition]

tworho

L53tworho (int n) = vec
tools for computing h(O^\vee): apply to rows of Young diagram
e.g., [2,0,-2] or [3,1,-1,-3]

rho

L54rho (int n) = [rat]

also defined in basic.at, nilpotent_orbits.at, extended_misc.at

String

L56String (int n) = Partition
e.g., [4,2] or [5,3,1]

semisimple_element 2 overloads

L62semisimple_element (LieType t,Partition P) = vec
compute 1/2 h(O^\vee) for G of type |t| (types A-D only)
O^\vee assumed to be an orbit of G^\vee, given by a partition |P|
only Lie type (of complexification) of G plays a role
a coweight
L73semisimple_element (RootDatum rd,Partition p) = vec
same as previous applied to (RootDatum,Partition p)
in this case apply dominant(.,rd) to resulting coweight; this only works if
RootDatum is a classical group with the usual classical coordinates
a dominant coweight
v,w are partitions of same n: sum(v)=sum(w)
v, w written in decreasing order (actually use reverse_sort to assume this)
v,w <-> nilpotent orbits in GL(n)
return true if orbit(v) contained in closure of orbit(w)
algorithm: v=(a_1,...,a_r), w=(b_1,...,b_s);
a_1+...+a_k \le b_1+...+b_k for all k\le min(r,s)sawtrset v
L76

nilpotent_orbit_semisimple_elements 2 overloads

L84nilpotent_orbit_semisimple_elements (LieType t) = [vec]
L86nilpotent_orbit_semisimple_elements (RootDatum rd) = [vec]
moved |lesseq_partition| (and renamed it |leq_partitions|) to combinatorics.at
L89

closures

L95closures([Partition] orbits) = [[int]]
orbits=[v_1,...,v_n]
closures(orbits)[i] is the set of j (not including i itself) such that
orbits[j] contained in the closure of orbits[i]

nilpotent_lambdas_classical

L107nilpotent_lambdas_classical (RootDatum rd) = [ratvec]
this only works for simple classical groups, fails for exceptional groups;
the more general version in nilpotent_orbits_exceptional handles the
simple exceptional case also

complex_nilpotent_from_partition

L110complex_nilpotent_from_partition(RootDatum rd, Partition p) = ComplexNilpotent

also defined in sommers.at

dim_nilpotent_partition

L113dim_nilpotent_partition(RootDatum rd,Partition p)

also defined in sommers.at

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