Given a simple Schubert problem (l,m) in Gr(k,n). Fix a set of flags F1,...,Fd and let S be the set of solutions of the intance of the Schubert problem given by the flags {Fi}. We compute a loop in the problem space based on the solution S by deforming one of the flags Fi using Homotopy continuation. This generates a loop in the problem space, which corresponds to a permutation in the Galois group.
i1 : l={1,1} o1 = {1, 1} o1 : List |
i2 : m={2,1} o2 = {2, 1} o2 : List |
i3 : (k,n) = (3,7) o3 = (3, 7) o3 : Sequence |
Generate a random set of flags to compute an instance of the problem
i4 : G = createRandomFlagsForSimpleSchubert((k,n),l,m); |
Solve the problem
i5 : S = solveSimpleSchubert((k,n),l,m,G); |
This is a problem with 77 solutions
i6 : #S o6 = 77 |
an element of the Galois group is:
i7 : findGaloisElement((l,m,k,n), G, S) o7 = {13, 15, 76, 8, 4, 58, 38, 31, 43, 44, 71, 67, 32, 54, 14, 57, 34, 22, ------------------------------------------------------------------------ 52, 59, 9, 20, 69, 75, 2, 21, 35, 33, 74, 17, 7, 49, 24, 28, 11, 55, 37, ------------------------------------------------------------------------ 25, 68, 63, 40, 66, 27, 73, 5, 51, 6, 45, 50, 30, 61, 36, 41, 62, 64, ------------------------------------------------------------------------ 48, 56, 26, 19, 53, 60, 42, 46, 39, 0, 47, 3, 23, 1, 29, 65, 70, 72, 12, ------------------------------------------------------------------------ 16, 18, 10} o7 : List |