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

README

This is the repository of the paper "The Chirotropical Grassmannian", by Dario Antolini and Nick Early.

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

README
本仓库对应Dario Antolini与Nick Early合著的论文《The Chirotropical Grassmannian》（《手性热带格拉斯曼流形》）。

以下为该数据集的使用指南：

### 格式文件说明
1. `.sobj`格式文件：此类文件为SageMath（SageMath）对象。若需加载文件`example.sobj`并以名称`example`调用其中内容，请在该文件所在目录下启动SageMath，执行如下命令：

example = load("example.sobj")

2. `.sage`格式文件：此类文件为SageMath脚本文件，内含SageMath对象（如函数、列表等）。若需加载`example.sage`中定义的所有函数与其他对象，请在该文件所在目录下启动SageMath，执行如下命令：

load("example.sage")

3. `.py`格式文件：此类文件为Python（Python）脚本文件。若需调用其中的对象，例如假设`example.py`中包含列表`L1`，若需加载该列表并以`L2`为名称调用，请在该文件所在目录下启动SageMath，执行如下命令：

from example.py import L1 as L2

该方法同样适用于`example.py`中定义的任意对象，如函数、字典或元组。若需导入`example.py`中的所有对象，可直接执行：

from example.py import *

### 仓库内容说明
本仓库包含如下内容：
1. 论文中算法2的实现，存储于文件`chirotropical_dressian.sage`中。
2. 用于在SageMath中生成所有普吕克关系（Plücker relations）的函数实现，存储于文件`pluecker_relations.sage`中。
3. 针对(3,6)、(3,7)与(3,8)三种情形的普吕克关系，存储于文件`pluecker_relations_3_6.sobj`、`pluecker_relations_3_7.sobj`与`pluecker_relations_3_8.sobj`中。
4. 针对(3,6)、(3,7)与(3,8)三种情形、用于生成德雷斯曼（Dressian）的3项式普吕克关系，存储于文件`3_term_pluecker_relations_3_6.sobj`、`3_term_pluecker_relations_3_7.sobj`与`3_term_pluecker_relations_3_8.sobj`中。
5. 针对(3,6)、(3,7)与(3,8)三种情形的全体手性胚（chirotopes）列表（按字典序排列），存储于文件`chirotopes.py`中。
6. 针对(3,6)、(3,7)与(3,8)三种情形的德雷斯曼流形Dr(k,n)模线性空间后的射线列表，存储于文件`R36.py`、`R37.py`与`R38.sobj`中。
7. 针对(3,6)、(3,7)与(3,8)三种情形的全体手性热带格拉斯曼流形（与对应手性热带德雷斯曼流形等价）模线性空间后的结果，存储于压缩文件`Trop_chi_3_6.zip`、`Trop_chi_3_7.zip`与`Trop_chi_3_8.zip`中。每个压缩包内均包含以Python字典形式存储的对象：
字典`d`满足`d[j]`为维度为`j`的扇锥列表。
每个压缩包中还包含一个专用于正部的特殊文件：`Trop_pos_3_6.sobj`、`Trop_pos_3_7.sobj`与`Trop_pos_3_8.sobj`。
8. 针对(3,8)情形，包含全体手性热带德雷斯曼流形/手性热带模空间`Trop^chi X(3,8)`的不同f-向量（f-vector）列表，存储于文件`f_vectors_trop_chi_3_8.sobj`中。该列表的第`i`个元素对应`chirotopes.py`文件中`chirotope_list_3_8`列表内第`i`个手性胚对应的手性热带化的f-向量。
9. 另有一个Python字典，其键为手性胚（以元组形式表示），当以某一手性胚作为键检索该字典时，将返回对应手性热带化的f-向量。该字典存储于文件`f_vectors_trop_chi_3_8_from_chi.sobj`中。
例如，执行如下代码：
python
f = load("f_vectors_trop_chi_3_8_from_chi.sobj")
f[tuple([1 for i in range(binomial(8,3))])]

将返回`[120, 2072, 14088, 48544, 93104, 100852, 57768, 13612]`，即正热带格拉斯曼流形`Trop^+ G(3,8)`的f-向量。

### 致谢
感谢Dominik Bendle、Janko Böhm、Yue Ren与Benjamin Schröter公开分享其关于德雷斯曼流形Dr(3,8)与热带格拉斯曼流形（tropical Grassmannian）Trop G(3,8)的数据集，本研究在(3,8)情形的计算中使用了其提供的射线数据。相关数据集可于以下网址获取：
https://agag-jboehm.math.rptu.de/~boehm/singulargpispace/tropical.htm