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NumericalSchubertCalculus :: isGaloisFullSymmetric

isGaloisFullSymmetric -- find Galois elements of a simple Schubert Problem until they generate the full symmetric group

Synopsis

Description

It runs a loop to find elements of the Galois group until it find a generating set or die after mx tries.

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 Schubert problem

i5 : S = solveSimpleSchubert((k,n),l,m,G);

Check if the Galois group is the symmetric group

i6 : isGaloisFullSymmetric((l,m,k,n), G, S, 5)
sh: gap: command not found
sh: gap: command not found
sh: gap: command not found
sh: gap: command not found
sh: gap: command not found

o6 = (false, {{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}, {40, 32, 23, 7, 70, 43, 69, 39, 59, 74, 71, 56, 53,
     ------------------------------------------------------------------------
     75, 14, 15, 66, 19, 28, 17, 6, 67, 73, 2, 37, 22, 27, 76, 54, 26, 5, 13,
     ------------------------------------------------------------------------
     21, 8, 0, 44, 24, 46, 38, 58, 34, 9, 61, 41, 45, 20, 29, 10, 30, 49, 31,
     ------------------------------------------------------------------------
     51, 55, 4, 11, 64, 68, 57, 52, 33, 60, 63, 18, 12, 48, 65, 42, 62, 16,
     ------------------------------------------------------------------------
     35, 25, 47, 72, 3, 50, 1, 36}, {56, 48, 2, 61, 0, 46, 74, 31, 52, 43,
     ------------------------------------------------------------------------
     45, 40, 5, 22, 14, 58, 11, 21, 67, 3, 8, 66, 10, 57, 39, 29, 38, 23, 37,
     ------------------------------------------------------------------------
     49, 30, 28, 55, 24, 42, 4, 7, 16, 64, 59, 60, 9, 12, 1, 13, 17, 35, 47,
     ------------------------------------------------------------------------
     76, 69, 73, 26, 62, 18, 54, 72, 71, 20, 65, 33, 51, 50, 36, 63, 70, 27,
     ------------------------------------------------------------------------
     15, 32, 68, 34, 19, 75, 6, 25, 44, 53, 41}, {0, 1, 2, 36, 23, 5, 16, 19,
     ------------------------------------------------------------------------
     8, 34, 10, 68, 29, 55, 76, 67, 74, 28, 50, 26, 41, 33, 47, 38, 56, 25,
     ------------------------------------------------------------------------
     32, 27, 37, 57, 46, 18, 22, 59, 9, 20, 64, 72, 58, 31, 14, 11, 42, 40,
     ------------------------------------------------------------------------
     66, 45, 12, 52, 48, 30, 4, 51, 21, 49, 62, 65, 13, 61, 15, 39, 60, 7,
     ------------------------------------------------------------------------
     71, 63, 17, 6, 44, 53, 54, 24, 69, 3, 43, 73, 70, 75, 35}, {56, 10, 40,
     ------------------------------------------------------------------------
     58, 28, 21, 33, 7, 8, 53, 44, 15, 22, 43, 27, 36, 11, 38, 1, 20, 17, 54,
     ------------------------------------------------------------------------
     62, 4, 52, 59, 63, 75, 6, 29, 50, 31, 30, 24, 34, 2, 3, 25, 35, 26, 18,
     ------------------------------------------------------------------------
     23, 42, 0, 55, 45, 12, 47, 48, 74, 61, 73, 67, 32, 13, 57, 51, 19, 16,
     ------------------------------------------------------------------------
     66, 60, 72, 5, 37, 64, 65, 46, 41, 68, 69, 70, 71, 39, 9, 49, 14, 76}})

o6 : Sequence

one permutation is not enough

i7 : isGaloisFullSymmetric((l,m,k,n), G, S, 1)
sh: gap: command not found

o7 = (false, {{36, 1, 20, 66, 65, 37, 28, 6, 52, 67, 10, 19, 35, 13, 14, 24,
     ------------------------------------------------------------------------
     11, 17, 18, 46, 57, 51, 64, 26, 45, 58, 7, 27, 22, 29, 32, 23, 2, 49,
     ------------------------------------------------------------------------
     73, 76, 56, 54, 38, 59, 4, 9, 42, 43, 44, 71, 55, 47, 5, 62, 3, 0, 16,
     ------------------------------------------------------------------------
     53, 39, 15, 30, 61, 12, 33, 60, 31, 8, 63, 68, 69, 25, 48, 70, 41, 40,
     ------------------------------------------------------------------------
     34, 72, 21, 74, 75, 50}})

o7 : Sequence

Caveat

This assumes that GAP runs when you type in the terminal gap and that we already know that the Galois group is the full symmetric group, otherwise it will output false after mx repetitions.

See also

Ways to use isGaloisFullSymmetric :