Jonathan Leake · research resources

Distinguished Rho Positivity

Code and data for “Inequalities Characterizing Distinguished Unipotent Orbits.”

Project information

Article
Inequalities Characterizing Distinguished Unipotent Orbits
Authors
Alexander Bertoloni Meli, Teruhisa Koshikawa, and Jonathan Leake
arXiv
arXiv:2409.06006

This page collects the code and exceptional-group data for the computer checks in the paper. The non-positivity data give a Weyl group element witnessing failure of strict positivity for each non-distinguished {0,2}-valued rho. The rational-rho section provides the checker and input data for the general rational-weight computation in Section 8.1. Code and output for the cuspidal-weight computations are also included.

Resources

Positivity Checks

Original SageMath scripts for checking rho-positivity and producing the non-positivity data for non-distinguished {0,2}-valued weightings.

Download and instructions

Source archive and readme for the original positivity and non-positivity checks.

src.zipreadme.txt

Check positivity of a single rho

Parallel script for testing positivity for one rho.

pos-check-parallel.py

Compute non-positivity data

Script used to generate the exceptional-type non-positivity files.

non-pos-weyl-elements.py

Non-positivity data

The readme explains the format of the files. Each data file begins with the relevant Dynkin diagram and node labels, followed by entries for non-distinguished {0,2}-valued rho.

Exceptional types

Individual non-positivity data files by type.

G2F4E6E7E8

In the data files, the Weyl group element is represented both as a product of simple reflections and as a matrix; the score is given as a weighted sum of simple roots.

Rational Rho Vectors

Code and input data for the rational-weight computation in Section 8.1. The checker enumerates the nonnegative rational candidates from Proposition 8.1 and tests for additional non-degenerate weightings beyond the known distinguished lists.

Complete download and instructions

The checker, supporting Python modules, and precomputed positive-root and Weyl-row data for G2, F4, E6, E7, and E8.

rational-rho.zipreadme.txt

G2 input data

Positive-root coefficients and distinct Weyl-matrix rows in the simple-root basis.

positive rootsWeyl rows

F4 input data

Positive-root coefficients and distinct Weyl-matrix rows in the simple-root basis.

positive rootsWeyl rows

E6 input data

Positive-root coefficients and distinct Weyl-matrix rows in the simple-root basis.

positive rootsWeyl rows

E7 input data

Positive-root coefficients and distinct Weyl-matrix rows in the simple-root basis.

positive rootsWeyl rows

E8 input data

Positive-root coefficients and distinct Weyl-matrix rows in the simple-root basis.

positive rootsWeyl rows

The checker omits the known distinguished weightings. For each remaining candidate, it tests strict positivity of every score coordinate for every Weyl group element. The computation finds no additional non-degenerate weightings in any of the five exceptional types. See the readme for the commands, parallel arguments, and input-file layout.

Cuspidal Weights for Exceptional Groups

Code and output for the cuspidal-weight computations for the exceptional groups. The shared code constructs the relevant cuspidal polytopes and lattice points, while exceptional_cuspidal_output.py runs the exceptional-type calculations and prints the output collected below.

Complete download and instructions

All code, output files, and a readme for the exceptional-group cuspidal-weight computations.

cuspidal-weights.zipreadme.txt

For G2, F4, and E6, the output gives full cuspidal-polytope summaries and lattice points. For E7 and E8, it records the center calculation and whether that center is a lattice point.