In this file, we explain how to read the non-positivity data files.
These files give examples of Weyl group elements for each
non-distinguished rho of each exceptional group, for which the
associated score (zeta(w)_Delta in the article) is non-positive.
These files were created by the non-pos-weyl-elements.py code file,
which can be found at:
https://www.jleake.com/code/distinguished-rho-positivity/
Each file begins with the associated Dynkin diagram, with a labeling
of the nodes of the diagram 1 through n. After that, there is a list
of entries, one for each non-distinguished rho. In each entry, we
give a Weyl group element (and its score) which has non-positive
score with respect to the given rho. The Weyl group element is
represented in two ways: first as a product of simple reflections
(where the reflections are labeled via the Dynkin diagram labeling),
and second as a matrix (see the SageMath docs for the presentation
of the simple roots as vectors). The score is then given as a
weighted sum of simple roots (where the simple roots are labeled via
the Dynkin diagram labeling).