POSITIVITY CHECKS FOR EXCEPTIONAL ROOT SYSTEMS

Companion code for "Inequalities Characterizing Distinguished Unipotent
Orbits" by Alexander Bertoloni Meli, Teruhisa Koshikawa, and Jonathan Leake.

REQUIREMENTS AND SETUP
SageMath is required. The scripts construct the root systems and Weyl
groups in SageMath; no separate input-data files are needed. The parallel
scripts use Python multiprocessing and expect fork-based process creation.

Extract src.zip and run the commands from its src directory:
  cd src

Rho coordinates follow SageMath's simple-root numbering. The numbering is
shown in the Dynkin diagrams printed by non-pos-weyl-elements.py and in
the non-positivity data files on the resource page.

CHECK ONE RHO
  sage -python pos-check-parallel.py G2 "(0,2)"
The first argument is the root-system type; the second is a quoted Python
tuple with one entry per simple root. The script tests strict positivity
of every score coordinate for every Weyl group element. It prints the
rho followed by YES or NO, then the elapsed time.

CHECK ALL ELEVEN DISTINGUISHED E8 WEIGHTINGS
  sage -python pos-check-parallel-E8.py

This script takes no arguments. It prints progress, then eleven YES lines
on success, or a failing rho with NO. The exhaustive E8 run can take a
long time; wait for the final result and total elapsed time.

GENERATE NON-POSITIVITY WITNESSES
  sage -python non-pos-weyl-elements.py E6 > E6.txt
Use G2, F4, E6, E7, or E8. The script enumerates {0,2}-valued rho, skipping
the known distinguished list. For each remaining rho it prints a Weyl
element (as reflections and a matrix) and a score with a nonpositive
coordinate. An unexpected positive rho is labelled IS POSITIVE.
The output begins with the labelled Dynkin diagram; progress goes to
standard error. These files concern {0,2}-valued rho, not general rational
weightings; the latter have a separate checker on the resource page.