Data on chirotropical Grassmannians Trop^\chi G(3,6), Trop^\chi G(3,7), Trop^\chi G(3,8), modulo lineality

<h1><strong>README</strong></h1> <p>This is the repository of the paper "The Chirotropical Grassmannian", by Dario Antolini and Nick Early.</p> <p>Here is some advice on how to use the data:</p> <ul> <li>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: <pre><code>example = load("example.sobj")</code></pre> </li> <li>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:&nbsp; <pre><code>load("example.sage")</code></pre> </li> <li>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: <pre><code>from example.py import L1 as L2</code></pre> <pre>The same can be done with any object defined in "example.py", for example a function or a dictionary or a tuple.<br><br>If you want to import every object of "example.py", you can simply type:</pre> <pre><code>from example.py import *</code></pre> </li> </ul> <h1>What does this repository contain?</h1> <ul> <li>The implementation of Algorithm 2 from the paper, stored in the file:&nbsp; <pre><code>chirotropical_dressian.sage</code></pre> </li> <li>the implementation of a function which generates all Pl&uuml;cker relations in SageMath, stored in the file:&nbsp; <pre><code>pluecker_relations.sage</code></pre> </li> <li>Pl&uuml;cker relations for the cases (3,6), (3,7) and (3,8), stored in the files:&nbsp; <pre><code>pluecker_relations_3_6.sobj</code> <code>pluecker_relations_3_7.sobj</code> <code>pluecker_relations_3_8.sobj</code></pre> </li> <li>3-term Pl&uuml;cker relations, generating the Dressian, for the cases (3,6), (3,7) and (3,8), stored in the files:&nbsp; <pre><code>3_term_pluecker_relations_3_6.sobj</code> <code>3_term_pluecker_relations_3_7.sobj</code> <code>3_term_pluecker_relations_3_8.sobj</code></pre> </li> <li>the list of all chirotopes (using the lexicographical order) for the cases (3,6), (3,7) and (3,8), stored in the file:&nbsp; <pre><code>chirotopes.py</code></pre> </li> <li>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:&nbsp; <pre><code>R36.py</code> <code>R37.py</code> <code>R38.sobj</code></pre> </li> <li>all chirotropical Grassmannians (equal to the corresponding chirotropical Dressians) modulo lineality for the cases (3,6), (3,7) and (3,8), stored in files:&nbsp; <pre><code>Trop_chi{i}_3_6.sobj</code> <code>Trop_chi{i}_3_7.sobj</code> <code>Trop_chi{i}_3_8.sobj</code></pre> &nbsp;as a Python dictionary: <pre><code>d</code> such that <code>d[j]</code> is the list of cones in the fan of dimension j<br><br>These SageMath objects are contained in the zipped files:<br><br><code>Trop_chi_3_6.zip</code> <code>Trop_chi_3_7.zip</code> <code>Trop_chi_3_8.zip</code><br><br>In each of these zip files, there is a special file dedicated to the positive part:<br><br><code>Trop_pos_3_6.sobj</code> <code>Trop_pos_3_7.sobj</code> <code>Trop_pos_3_8.sobj</code></pre> </li> <li>&nbsp;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:&nbsp; <pre><code>f_vectors_trop_chi_3_8.sobj</code><br><br>where the $i$-th element of the list corresponds to the f-vector of the <br>chirotropicalization corresponding to the $i$-th chirotope in the list: <br><br><code>chirotope_list_3_8</code> contained in the file <code>chirotopes.py</code> </pre> </li> <li>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:&nbsp; <pre><code>f_vectors_trop_chi_3_8_from_chi.sobj</code><br><br>For example, the following code:</pre> <pre><code>f = load("f_vectors_trop_chi_3_8_from_chi.sobj") f[tuple([1 for i in range(binomial(8,3))])]</code></pre> <p>returns:</p> <pre><code>[120, 2072, 14088, 48544, 93104, 100852, 57768, 13612]</code></pre> which is the f-vector of the positive tropical Grassmannian Trop^+ G(3,8).</li> </ul> <h1>Acknowledgements</h1> <p>We thank Dominik Bendle, Janko B&ouml;hm, Yue Ren, and Benjamin Schr&ouml;ter 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).&nbsp;Their data can be found at:</p> <pre><a href="https://agag-jboehm.math.rptu.de/~boehm/singulargpispace/tropical.htm">https://agag-jboehm.math.rptu.de/~boehm/singulargpispace/tropical.htm</a></pre>