The "kgporder" command prints the Hasse diagram for the closure
relation for orbits of K on the partial flag variety G/P. Here K
is the complexification of a maximal compact subgroup for a
particular (weak) real form. The software will prompt the user
for this if it has not already been chosen.
Once a real form is specified, the user is prompted to enter a
subset of the simple roots corresponding to the desired parabolic.
The input should be a list of integers in [1,rank], separated
by any whitespace or non-numeric character(s). Numbers outside this
range are ignored. An empty list corresponds to the case
B=P and regenerates K\B/G as a partially ordered set.
The output is as follows:
Individual orbits are listed by the "kgp" command, which numbers
them from 0 to n-1. Each row of the output corresponds to a single
orbit, and the first column is the orbit number followed by a colon.
Following the orbit number i is a comma-separated list of the immediate
predecessors of orbit #i in the closure order. Unlike the situation
for K\G/B, immediate predecessor orbits need not have codimension one
in the boundary.
For example:
empty: type
Lie type: B2 sc s
main: realform
(weak) real forms are:
0: so(5)
1: so(4,1)
2: so(3,2)
enter your choice: 2
real: kgporder
enter simple roots (1-2): 1
kgp size for roots {1}: 4
Name an output file (return for stdout, ? to abandon):
0:
1:
2: 0,1
3: 2
real:
Here the quasi-split form of type B2 is chosen, with P corresponding
to the root subset consisting of the unique long simple root. There
are 4 orbits of K on G/P. Orbits 0 and 1 are closed and orbit 2
contains orbits 0 and 1 as immediate predecessors. Orbit 3 is the
open orbit and thus contains every orbit in its closure, however
only orbit 2 is an immediate predecessor.