DOI: 10.5281/zenodo.10732594
title: Data on chirotropical Grassmannians Trop^\chi G(3,6), Trop^\chi G(3,7), Trop^\chi G(3,8), modulo lineality

description: README

This is the repository of the paper

The Chirotropical Grassmannian, by Dario Antolini and Nick Early, Special volume on Positive Geometry, Le Matematiche 80 (1)  (2025), 123-142, doi: 10.4418/2025.80.1.4

Here is some advice on how to use the data:



files ".sobj": the files of this form are SageMath objects. In order to load the file "example.sobj" and use it with name "example", open SageMath in the folder where the file is contained and type the line:

example = load("example.sobj")


files ".sage": the files of this form are SageMath files. They contain SageMath objects, such as functions or lists. In order to load all the functions and the other objects defined in the file "example.sage", open SageMath in the folder where the file is contained and type the line: 

load("example.sage")


files ".py": the files of this form are Python files. In order to use the objects in them, do the following: suppose the file "example.py" contains a list "L1". In order to load the list and use it with the name "L2", open SageMath in the folder where the file is contained and type the line:

from example.py import L1 as L2

The same can be done with any object defined in "example.py", for example a function or a dictionary or a tuple.If you want to import every object of "example.py", you can simply type:

from example.py import *



What does this repository contain?



The implementation of Algorithm 2 from the paper, stored in the file: 

chirotropical_dressian.sage


the implementation of a function which generates all Plücker relations in SageMath, stored in the file: 

pluecker_relations.sage


Plücker relations for the cases (3,6), (3,7) and (3,8), stored in the files: 

pluecker_relations_3_6.sobj pluecker_relations_3_7.sobj pluecker_relations_3_8.sobj


3-term Plücker relations, generating the Dressian, for the cases (3,6), (3,7) and (3,8), stored in the files: 

3_term_pluecker_relations_3_6.sobj 3_term_pluecker_relations_3_7.sobj 3_term_pluecker_relations_3_8.sobj


the list of all chirotopes (using the lexicographical order) for the cases (3,6), (3,7) and (3,8), stored in the file: 

chirotopes.py


the list of the rays of the Dressian Dr(k,n) modulo lineality for the cases (3,6), (3,7) and (3,8), stored in the files: 

R36.py R37.py R38.sobj


all chirotropical Grassmannians (equal to the corresponding chirotropical Dressians) modulo lineality for the cases (3,6), (3,7) and (3,8), stored in files: 

Trop_chi{i}_3_6.sobj Trop_chi{i}_3_7.sobj Trop_chi{i}_3_8.sobj
 as a Python dictionary:

d such that d[j] is the list of cones in the fan of dimension jThese SageMath objects are contained in the zipped files:Trop_chi_3_6.zip Trop_chi_3_7.zip Trop_chi_3_8.zipIn each of these zip files, there is a special file dedicated to the positive part:Trop_pos_3_6.sobj Trop_pos_3_7.sobj Trop_pos_3_8.sobj


 for the case (3,8), a list with all the f-vectors of the different chirotropical Dressians/chirotropical moduli spaces Trop^chi X(3,8). This is organized in the list: 

f_vectors_trop_chi_3_8.sobjwhere the $i$-th element of the list corresponds to the f-vector of the chirotropicalization corresponding to the $i$-th chirotope in the list: chirotope_list_3_8 contained in the file chirotopes.py 


for the same purpose, there is also a Python dictionary whose keys are chirotopes (written as tuples), such that, when evaluating the dictionary at the chirotope, it gives the f-vector of the corresponding chirotropicalization as an output. This is stored in the file: 

f_vectors_trop_chi_3_8_from_chi.sobjFor example, the following code:

f = load("f_vectors_trop_chi_3_8_from_chi.sobj")
f[tuple([1 for i in range(binomial(8,3))])]

returns:

[120, 2072, 14088, 48544, 93104, 100852, 57768, 13612]
which is the f-vector of the positive tropical Grassmannian Trop^+ G(3,8).


Acknowledgements

We thank Dominik Bendle, Janko Böhm, Yue Ren, and Benjamin Schröter for making their data on the Dressian Dr(3,8) and the tropical Grassmannian Trop G(3,8) publicly available; we used the ray data for the computations in the case (3,8). Their data can be found at:

https://agag-jboehm.math.rptu.de/~boehm/singulargpispace/tropical.htm