Here, we record all of the code used in the paper Initial degenerations of spinor varieties by Daniel Corey.
We compute the tropicalizations of S4° and S5° using gfan version 0.6.2.
First, compute the starting cone by running
gfan_tropicalstartingcone < startingConeInput4.txt > outputFile.txt
To record the symmetries, append the lines to the outputFile.txt:
{{0,1,4,5,2,3,6,7}, {0,4,5,1,6,2,3,7}, {1,0,4,5,2,3,7,6}}
{{1,-1,1,1,1,1,1,-1}, {1,1,1,-1,1,-1,-1,-1}, {1,1,1,1,1,1,1,1}}
The result is stored in the file tropicalTraverseInput4.txt. To get the tropicalization of S4°, run
gfan_tropicaltraverse --symmetry --symsigns --nocones < tropicalTraverseInput4.txt
The result is contained in the file TS4.txt.
The computation is similar to the n=4 case above. First, compute the starting cone by running
gfan_tropicalstartingcone < startingConeInput5.txt > outputFile.txt
To record the symmetries, append the lines to the outputFile.txt:
{{1, 2, 3, 4, 0, 8, 11, 13, 14, 5, 6, 7, 9, 10, 12, 15}, {1, 0, 2, 3, 4, 5, 6, 8, 7, 9, 11, 10, 13, 12, 14, 15}, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {1, 0, 5, 6, 9, 2, 3, 8, 7, 4, 11, 10, 13, 12, 15, 14}, {2, 5, 0, 7, 10, 1, 8, 3, 6, 11, 4, 9, 14, 15, 12, 13}, {5, 2, 1, 8, 11, 0, 7, 6, 3, 10, 9, 4, 15, 14, 13, 12}, {3, 6, 7, 0, 12, 8, 1, 2, 5, 13, 14, 15, 4, 9, 10, 11}, {6, 3, 8, 1, 13, 7, 0, 5, 2, 12, 15, 14, 9, 4, 11, 10}, {7, 8, 3, 2, 14, 6, 5, 0, 1, 15, 12, 13, 10, 11, 4, 9}, {4, 9, 10, 12, 0, 11, 13, 14, 15, 1, 2, 5, 3, 6, 7, 8}, {9, 4, 11, 13, 1, 10, 12, 15, 14, 0, 5, 2, 6, 3, 8, 7}, {10, 11, 4, 14, 2, 9, 15, 12, 13, 5, 0, 1, 7, 8, 3, 6}, {12, 13, 14, 4, 3, 15, 9, 10, 11, 6, 7, 8, 0, 1, 2, 5}, {8, 7, 6, 5, 15, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 4}, {11, 10, 9, 15, 5, 4, 14, 13, 12, 2, 1, 0, 8, 7, 6, 3}, {13, 12, 15, 9, 6, 14, 4, 11, 10, 3, 8, 7, 1, 0, 5, 2}, {14, 15, 12, 10, 7, 13, 11, 4, 9, 8, 3, 6, 2, 5, 0, 1}, {15, 14, 13, 11, 8, 12, 10, 9, 4, 7, 6, 3, 5, 2, 1, 0}}
{{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}}
The result is stored in the file tropicalTraverseInput5.txt. To get the tropicalization of S5°, run
gfan_tropicaltraverse --symmetry --symsigns --nocones < tropicalTraverseInput5.txt
The result is contained in the file TS5.txt.
The rays of Trop S5° computed by gfan are complicated expressions, so in the paper we use different primitive generators modulo the lineality space. To verify that these are the same, see the Oscar notebook check_fan_oscar.ipynb.
The subdivision in section 6.3 was computed using Oscar. Furthermore, we verify all the claims in Proposition 6.15 (these can be done by hand as well). See the notebook prop-6-15.ipynb. (this works with Oscar 0.11.3).
To prove Lemma 7.10, we apply Lemma 7.3 to each nonmaximal cone τ of Σ5'. The collections of cones Aτ are listed in Table 7.1. To verify Equation 7.1 for these cones, see the Oscar notebook lem-7-11.ipynb (this works with Oscar 0.11.3).
This is done in polymake, and works with version 4.0. See notebooks subdivisionsS4.ipynb and subdivisionsS5.ipynb.
We use the following programs.
@Misc{gfan,
author = {Jensen, Anders N.},
title = {Gfan, a software system for {G}r{\"o}bner fans and tropical varieties},
howpublished = {{\tt http://home.imf.au.dk/jensen/software/gfan/gfan.html}}
}
@misc{OSCAR,
key = {OSCAR},
organization = {The OSCAR Team},
title = {{OSCAR} -- Open Source Computer Algebra Research system,
Version 0.10.1},
year = {2022},
url = {https://oscar.computeralgebra.de},
}
@Book{OSCAR-book,
editor = {Eder, Christian and Decker, Wolfram and Fieker, Claus and Horn, Max and Joswig, Michael},
title = {The {OSCAR} book},
year = {2024},
}
@incollection{polymake:2000,
AUTHOR = {Gawrilow, Ewgenij and Joswig, Michael},
TITLE = {{\tt polymake}: a framework for analyzing convex polytopes},
BOOKTITLE = {Polytopes---combinatorics and computation ({O}berwolfach, 1997)},
SERIES = {DMV Sem.},
VOLUME = {29},
PAGES = {43--73},
PUBLISHER = {Birkh\"auser, Basel},
YEAR = {2000},
MRCLASS = {52B55 (68U05)},
MRNUMBER = {1785292},
}
@InProceedings{polymakeJL,
author={Kaluba, Marek and Lorenz, Benjamin and Timme, Sascha},
editor={Bigatti, Anna Maria and Carette, Jacques and Davenport, James H. and Joswig, Michael and de Wolff, Timo},
title={Polymake.jl: A New Interface to polymake},
booktitle={Mathematical Software -- ICMS 2020},
year={2020},
publisher={Springer International Publishing},
address={Cham},
pages={377--385},
abstract={We present the Julia interfaceto polymake, a software for research in polyhedral geometry. We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art numerical software.},
isbn={978-3-030-52200-1}
}
In previous versions of this project, we used sage.
@manual{sagemath,
Key = {SageMath},
Author = {{The Sage Developers}},
Title = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 9.0)},
note = {{\tt https://www.sagemath.org}},
Year = {2020},
}