Gr38Schoen

Code used in the paper The Grassmannian of 3-planes in C^8 is schön by Daniel Corey and Dante Luber. This code works with OSCAR version 0.10.1. Note: it may be essential to use this version of OSCAR. To ensure that you are using this version of Oscar, do the following. First, run julia --project=. in the terminal from the root of this project. Next, open julia and run the following:

julia> using Pkg
julia> Pkg.instantiate()

Convention $TGr_{0}(3,8)$ is the support of a fan in $\wedge^{3}\mathbb{R}^{8} \cong \mathbb{R}^{\binom{[8]}{3}}$. We order the coordinates of $\mathbb{R}^{\binom{[8]}{3}} \cong \mathbb{R}$ lexicographically. In the paper, we use the MIN convention, however, the data of $TGr_{0}(3,8)$ is given to us with respect to the MAX convention. Thus, when forming the associated subdivision of $\Delta(3,8)$, one must negate the vector.

The seconday fan structure $\mathsf{TGr}_{0}(3,8)$: Raw data

The data of $\mathsf{TGr}_{0}(3,8)$ was shared with us by Benjamin Schröter, which was computed in the paper Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians arXiv:2003.13752 by Dominik Bendle, Janko Boehm, Yue Ren, and Benjamin Schröter. Their data is recorded at the following webpage.

https://www.mathematik.uni-kl.de/~boehm/singulargpispace/tropical/TGR38.htm

This is in the files group38, GrRays.data, and ConesDrOfGr.data.

group38 is a polymake data file containing an Array<Array<Int>>. Denote by $S_{n}$ the symmetric group on $[n]$. This file records the subgroup of $S_{56}$ isomorphic to $S_8$ induced by the action of $S_8$ on $\binom{[8]}{3}$ given by

$\sigma \cdot (i,j,k ) = (\sigma(i),\sigma(j),\sigma(k))$

Each of these permutation in $S_{56}$ is recorded in the standard ``one-line'' notation of a permutation, i.e., $\sigma = (\sigma(1), \sigma(2), \ldots, \sigma(56))$.

GrRays.data is a polymake data file containing a Matrix<Rational,NonSymmetric> recording the rays of $TGr_{0}(3,8)$. Each row of this matrix is a ray of $TGr_{0}(3,8)$ with its Gröbner fan structure; only the first 12 are needed for the secondary fan structure.

ConesDrOfGr.data is a text file recording the maximal cones of $\mathsf{TGr}_{0}(3,8)$ with its secondary fan structure. Each line represents a cone as a space separated list of symbols r#s. Here, r represents a row of GrRays.data, and s a row of group38. Thus, the symbol r#s mean ``the r-th ray whose coordinates are permuted by the s-th permutation in group38.''

The seconday fan structure $\mathsf{TGr}_{0}(3,8)$: The intermediate cones

See the notebook generateAllCones.ipynb for instructions on how to generate the remaining cones. The data most relevant for the verifications in Section 6 are the files codim_0.dat, codim_1.dat, codim_2.dat, codim_3.dat, codim_4.dat, codim_5.dat, codim_6.dat, codim_7.dat in the directory allRepsByCodim. The file codim_{j}.dat contains the data of the codimension j cones up to $S_8$--symmetry (replace {j} with 0 thru 7). Each line of codim_{j}.dat records a cone as a vector in its relative interior.

The (3,8)-matroids

Verifications necessary for Proposition 4.3 are contained in the notebook Matroids_3_8.ipynb.

Verifying propositions in Section 6

The cones in codim_0.dat, ..., codim_7.dat are reorganized into 6 groups G1.dat, ..., G6.dat in the directory groupsFinal; these groups are defined in Section 6 of the paper. These reorganization is done in the notebook annotated_grand_scheme.ipynb. The notebook G1.ipynb verifies that each representative w in G1.dat really does belong to the group G1. The notebooks G2.ipynb, G3.ipynb, G4.ipynb, G5.ipynb, G6.ipynb contain the code used in the proof of Propositions 6.7, 6.13, 6.14, 6.16, and 6.19, respectively. These rely on the functions in the files contained in the src directory. The documentation for these functions is in the notebook functionDocumentation.ipynb. Instructions on full verifications and examples are also provided in these notebooks.