Schubert Galois groups in the Grassmannian G(4,9)

	Schubert Galois groups in the Grassmannian G(4,9) Abraham Martín del Campo, Frank Sottile, and Robert Williams.
	This page accompanies the article Classification of Schubert Galois groups in Gr(4,9), providing software and additional information on the computations.

Abstract (of paper) We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, only 149 have Galois group that does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and it begins to establish the inverse Galois problem for Schubert calculus.

Software and additional information

Maple code for computing all Schubert problems on a Grassmannian. GrassmannianProblems.maple. This needs the file Procedures, which contains procedures assisting the computation in the main file. Here is a sample output file for G(2,8). When this maple code is run, it generates all Schubert problems and for each, it computes the number of solutions. It writes these in text files in a host of subdirectories, before reading them and writing one large text file Gk.n.problems containing them, one per line and sorted by the number of solutions. The code is documented, and has the ability to filter by number of solutions, as well as whether or not a problem is essentially new for that Grassmannian.
Vakil wrote a Maple script that runs the algorithm using his criterion to test if a Schubert problem is at least alternating. His original script is available here. In it, Schubert problems are encoded by 0-1 sequences and multiplicities. It also did not run for me on G(4,9) (long lists), so I modified it. It is a list of procedures: Vakil_procedures.
This wrapper Vakil_Criterion.maple runs Vakil_procedures on all small Grassmannians. Here is its output. We additionally ran Vakil's code on G(3,10), G(3,11), and G(4,9) to produce Table 1 in our paper. We also used this to run Vakil_procedures on G(2,n), for n from 4 to 20, and here it its output. Here is a summary of these data.
We also have Python code in which we have implemented Vakil's algorithm. This is available in a gzipped .tar file. A guide and links to individual files is here.
The Schubert problems for which Vakil's algorithm was not able to determine if they were at least alternating are the input for further computation. The Frobenius algorithm is first run on them, sorting into full symmetric and unknown Galois group. We then computed many Frobenius elements and began to try and classify according to the cycle types found. All files and output are found here.
The output from running the Frobenius algorithm and computing Frobenius elements was used for the classification in Section 3 of the paper Classification of Schubert Galois groups in Gr(4,9). This classification is explained in a .html page: Enriched Schubert Galois groups in the Grassmannian G(4,9)
Here are data about the distribution of Frobenius elements for three enriched Schubert problems. These have Galois groups that are wreath products S₃WrS₂ (of order 48), S₂WrS₃ (of order 72), and S₂WrS₅ (of order 28800).
The proof of Corollary 22 relied on three symbolic computations in Maple.
For (31)*(4,1)*(111)*(211)*(211) : Here is the maple script, and Here is its (annotated) output.
For (31)*(4)*(211)*(211)*(211) : Here is the maple script, and Here is its (annotated) output.
For (3)*(41)*(211)*(211)*(211) : Here is the maple script, and Here is its (annotated) output.

Work of Sottile supported by the National Science Foundation under Grant DMS-1501370.

Last modified: Thu Dec 2 19:51:18 CST 2019