changeFlags -- continue solutions of a Schubert problem to another instance

Synopsis

Usage:

changeFlags(sols,CFG)
Inputs:
- sols, a list, solutions of a Schubert problem written as $n\times k$ matrices
- CFG, a sequence, a triple (C,F,G) where C is a list of m Schubert conditions, F is a list of m flags defining an instance with solution set sols, G is a list of m flags defining the instance whose solutions we want.
Optional inputs:
- OneHomotopy => ..., default value true, strategy for moving solutions to a Schubert problem
Outputs:
- a list, solutions of the problem with respect to flags G.

Description

If sols is a set of solutions to a Schubert problem $l_1,\ldots,l_m$ with respect to a set of flags $F_1,\ldots, F_m$, uses parameter homotopies to move $S$ to a solution set $S'$ for the same Schubert problem, but with respect to another set of flags $G_1,\ldots, G_m$.

For instance, consider the Schubert problem $(2,1)^3$ in $Gr(3,6)$.

i1 : k=3; n=6;

i3 : l1={2,1};

i4 : l2={2,1};

i5 : l3={2,1};

Generate flags $F$: standard flag, opposite flag, and one random.

i6 : F = {(l1, id_(CC^n)), (l2, rsort id_(CC^n)), (l3,random(CC^n,CC^n))}

o6 = {({2, 1}, | 1 0 0 0 0 0 |), ({2, 1}, | 0 0 0 0 0 1 |), ({2, 1}, |
               | 0 1 0 0 0 0 |            | 0 0 0 0 1 0 |            |
               | 0 0 1 0 0 0 |            | 0 0 0 1 0 0 |            |
               | 0 0 0 1 0 0 |            | 0 0 1 0 0 0 |            |
               | 0 0 0 0 1 0 |            | 0 1 0 0 0 0 |            |
               | 0 0 0 0 0 1 |            | 1 0 0 0 0 0 |            |
     ------------------------------------------------------------------------
     .892712+.673395ii  .0741835+.808694ii .830833+.538554ii 
     .29398+.632944ii   .362835+.706096ii  .873665+.415912ii 
     .0258884+.714827ii .127435+.254482ii  .606588+.848005ii 
     .89189+.231053ii   .741046+.108386ii  .191734+.403215ii 
     .461944+.775187ii  .348931+.562428ii  .615911+.0147867ii
     .909047+.314897ii  .246268+.153346ii  .223028+.388829ii 
     ------------------------------------------------------------------------
     .557119+.873708ii .0647412+.877846ii .444183+.644366ii |)}
     .7037+.681869ii   .0340514+.507989ii .194945+.518585ii |
     .276259+.605659ii .150294+.656391ii  .987173+.568273ii |
     .96518+.681683ii  .174853+.626892ii  .184779+.370833ii |
     .914199+.887381ii .350611+.379495ii  .339222+.062212ii |
     .169813+.965004ii .237252+.116721ii  .465736+.40273ii  |

o6 : List

Generate a random set of flags $G$.

i7 : G = randomSchubertProblemInstance({l1,l2,l3},k,n);

We solve with respect to $F$.

i8 : S = solveSchubertProblem(F,k,n);

i9 : FlagsF = F/last;

i10 : FlagsG = G/last;

and we transform the solutions to get solutions with respect to $G$

i11 : time S' = changeFlags(S,({l1,l2,l3},FlagsF,FlagsG))
     -- used 1.84162 seconds

o11 = {| .124063-.286284ii   .167867-.0639299ii  .295488-.366109ii   |, |
       | -.25181-.0448108ii  -.107521-.190275ii  -.0896067-.483543ii |  |
       | .219256+.214331ii   .0873511+.372915ii  -.213803+.410917ii  |  |
       | .24818-.0567624ii   .138074+.069527ii   -.446051+.0426311ii |  |
       | -.122857+.030327ii  -.0079778-.488031ii .214077+.374278ii   |  |
       | -.402855+.0597291ii -.43662-.200568ii   -.254234+.748617ii  |  |
      -----------------------------------------------------------------------
      -.00839714-.407126ii .205329-.387394ii  .424542-.396438ii  |}
      .192354-.0445916ii   .719274+.437354ii  .49019+.0178286ii  |
      -.396883+.032256ii   -.296361-.56774ii  -.362153+.372505ii |
      .153499+.396506ii    -.494939+.496139ii -.411614+.315486ii |
      -.282532+.248918ii   -.874085+.276867ii -.807461+.498361ii |
      .0284784+.374339ii   -.588673+.906394ii -.233277+.814607ii |

o11 : List

i12 : assert all(S', s-> checkIncidenceSolution(s,G))

We can also choose a different strategy

i13 : time S' = changeFlags(S,({l1,l2,l3},FlagsF,FlagsG), OneHomotopy=>false)
     -- used 1.08499 seconds

o13 = {| -.226702-.119043ii  -.326452-.119841ii  .0929953-.230085ii |, |
       | -.229738+.0833434ii -.445398+.0639826ii .150646-.436884ii  |  |
       | .279834-.0761677ii  .402182-.016251ii   -.178732+.299631ii |  |
       | -.119002+.197573ii  -.108802+.642985ii  -.820159-.281606ii |  |
       | -.134096+.0372165ii -.127528-.0349071ii .204901-.0516554ii |  |
       | .499185+.465779ii   1.13857+.442385ii   -.50217+1.18715ii  |  |
      -----------------------------------------------------------------------
      .132399-.579396ii   .78366-.359334ii   -.647254+1.41489ii  |}
      -.0614253+.257063ii -.356136+.210321ii .116836-.599086ii   |
      .127487-.876713ii   .739985-1.32758ii  .872113+1.6876ii    |
      -.103026+.750164ii  -.798533+.84669ii  -.0252049-1.72596ii |
      -.0428908-.187959ii .27551-.59017ii    .739685+.785045ii   |
      -.236198+.0362704ii -.341298-.374585ii .878579-.176735ii   |

o13 : List

i14 : assert all(S', s-> checkIncidenceSolution(s,G))

Caveat

There are two strategies: when OneHomotopy is set to true (default) it uses straight line homotopies to change flags, but assumes the initial flags are generic.

When OneHomotopy is set to false, it makes gradual changes in the flags by changing one column at a time, and using only linear homotopies, but in this setting, it generates polynomial equations using all minors of the incidence conditions (thus is not very effective).

Ways to use changeFlags :

changeFlags(List,Sequence)
changeFlags(Matrix,List,Sequence) -- recursive call of change flags

For the programmer

The object changeFlags is a method function with options.