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NumericalSchubertCalculus :: findGaloisElement

findGaloisElement -- computes a permutation from a loop of an instance of a simple Schubert problem.

Synopsis

Description

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

See also

Ways to use findGaloisElement :