CUSPIDAL WEIGHTS FOR THE EXCEPTIONAL TYPES

This download contains SageMath code and saved output for the distinguished
rho vectors of types G2, F4, E6, E7, and E8.

CONTENTS
  exceptional_cuspidal_output.py  Driver for all five exceptional types.
  weyl_utils_sage.py             Root systems, polytopes, and lattice routines.
  weyl_utils.py                  Lists of distinguished rho vectors.
  X.txt and X_w.txt              Saved output, with X = G2, F4, E6, E7, or E8.

RUNNING THE CODE
Install SageMath and keep the three Python files in the same directory.
From that directory, run:

  sage -python exceptional_cuspidal_output.py

The driver takes no command-line arguments and prints all results to standard
output. To save the combined output in one file, run:

  sage -python exceptional_cuspidal_output.py > exceptional_cuspidal_output.txt

This command creates a combined log; it does not create the individual X.txt
and X_w.txt files automatically. Full polytope computations can take time.

READING THE OUTPUT
The entries of each rho tuple are its values on the simple roots, in Sage's
numbering. Changing output coordinates does not change this rho tuple.

X.txt uses simple-root coordinates; X_w.txt uses fundamental-weight coordinates.
Both versions refer to the weight lattice. In particular, weight-lattice points
can have fractional coordinates in the simple-root basis.

For G2, F4, and E6, each summary gives the dimension, number of facets, number
of vertices, and all weight-lattice points in the closed cuspidal polytope.
The marker "<- interior" identifies points in the interior. The vertices field
is a count, not a list of vertex coordinates.

For E7 and E8, the files give only the center and whether it belongs to the
weight lattice. The center is half the sum of the roots beta with rho(beta)=2.
These files do not enumerate their polytopes or all of their lattice points.