initialDegenerationsFlagVarieties

This repository contains the code used in the paper Initial degenerations of flag varieties by Daniel Corey and Jorge Alberto Olarte, arXiv:2207.08094. We use polymake and OSCAR, and it works with version 0.10.1.

flagMatroids.ipynb is a jupyter notebook that generates all flag matroids of rank $(1,2,3)$ on $[4]$.

FansOnTFl4.ipynb is a jupyter notebook that contains the data of $\operatorname{TF\ell}^{\circ}((1,2,3),4)$. It records the fiber-fan structure (which coincides with the G"obner fan structure computed by gfan), together with all the matroid subdivisions, and the coarsest fan structure. It also contains a verification of Lemma 7.1.

We also have files used to compute $\operatorname{TF\ell}^{\circ}((1,2,3),4)$ with gfan. First, run

gfan_tropicalstartingcone < Fl_123_4_lex.txt > Fl_123_4_lex.cone

to get the starting cone. Then add to the end of the file Fl_123_4_lex.cone the lines

{(1,0,2,3,4,7,8,5,6,9,10,11,13,12),
(1,2,3,0,7,8,4,9,5,6,13,10,11,12),
(13,12,11,10,9,8,7,6,5,4,3,2,1,0)}

{(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)}

(these lines record the symmetry of the ideal.) Finally, run

gfan_tropicaltraverse --symmetry --symsigns < Fl_123_4_lex.cone > Fl_123_4_lex.trop

At the end of the notebook FansOnTFl4.ipynb, we verify that the presentation of this fan in the paper agrees with the output fron gfan.