randomSchubertProblemInstance -- returns a random instance of a given Schubert problem

Synopsis

Usage:

randomSchubertProblemInstance(conditions,k,n)
Inputs:
- conditions, a list, which is a list of Schubert conditions that are either all partitions or all brackets (see bracket2partition for details)
- k, an integer,
- n, an integer, $k$ and $n$ define the Grassmannian $Gr(k,n)$ of $k$-planes in $n$-space
Optional inputs:
- Strategy => ..., default value "unit circle", strategy for creating a random matrix representing a flag
Outputs:
- a list, random instance of the Schubert problem, which is a list of pairs of the form (condition,flag)

Description

This first verifies that the conditions are either all partitions or all brackets, and that they form a Schubert problem on $Gr(k,n)$.

Then it creates a list of random square invertible matrices that represent flags for the Schubert problem.

For instance, consider the problem of four lines, which is given by 4 partitions {1}$^4$ in $Gr(2,4)$

i1 : randomSchubertProblemInstance({{1},{1},{1},{1}},2,4)

o1 = {({1}, | .781262-.624203ii  -.462954-.886382ii -.272832+.962062ii
            | .9868+.161945ii    -.219207-.975678ii .778029-.628229ii 
            | -.971549+.236839ii .157598-.987503ii  .841103-.540875ii 
            | .893324+.449414ii  .360485-.932765ii  -.651048+.759036ii
     ------------------------------------------------------------------------
     -.670948-.741504ii |), ({1}, | .696205+.717843ii  -.0281556+.999604ii
     .118767+.992922ii  |         | -.58234+.812945ii  -.924052-.382267ii 
     -.396553+.918012ii |         | .486335-.873772ii  -.970803-.23988ii  
     -.272372-.962192ii |         | -.783999-.620762ii .577597-.816322ii  
     ------------------------------------------------------------------------
     -.0562287-.998418ii .776937+.629579ii  |), ({1}, | -.746312-.665596ii
     .023444+.999725ii   .570649+.821194ii  |         | -.936289-.351232ii
     .701149-.713015ii   -.863646+.504099ii |         | -.164243+.98642ii 
     .357971+.933733ii   -.820726+.571321ii |         | .85817-.513365ii  
     ------------------------------------------------------------------------
     .995687+.092774ii .168661+.985674ii  -.765803+.643076ii |), ({1}, |
     .701342-.712825ii -.286823-.957983ii -.415126-.909764ii |         |
     -.78761-.616174ii .976163-.21704ii   -.416188-.909278ii |         |
     .759918-.650019ii .482783+.87574ii   .975922-.218119ii  |         |
     ------------------------------------------------------------------------
     .918399+.395655ii  .719639-.694348ii  .9772+.212322ii   
     .58629+.810101ii   -.554833-.831962ii .454813+.890587ii 
     -.590887+.806754ii -.726792+.686857ii .0800129+.996794ii
     -.939129+.343564ii -.616052-.787706ii .339065+.940763ii 
     ------------------------------------------------------------------------
     -.998741-.0501734ii |)}
     -.698649-.715464ii  |
     .742918+.669382ii   |
     -.99319-.116506ii   |

o1 : List

the same problem but using brackets instead of partitions

i2 : randomSchubertProblemInstance({{2,4},{2,4},{2,4},{2,4}},2,4)

o2 = {({2, 4}, | .996754-.0805092ii -.909393-.415938ii .398424+.917201ii 
               | -.531694+.846937ii .92457+.381011ii   -.976915+.213627ii
               | .51095+.859611ii   -.227376-.973807ii .842765-.538281ii 
               | -.341238+.939977ii -.110622-.993863ii -.923063-.384649ii
     ------------------------------------------------------------------------
     -.688353+.725376ii |), ({2, 4}, | -.991977+.126418ii .398109-.917338ii 
     -.81898+.573823ii  |            | .91893+.394421ii   -.210222+.977654ii
     -.915156-.403101ii |            | -.834955-.550318ii .491643-.870797ii 
     -.684587-.728932ii |            | -.978842+.204616ii .0976385-.995222ii
     ------------------------------------------------------------------------
     .989783-.142579ii    .930733+.365699ii  |), ({2, 4}, |
     .990088-.140449ii    .233835+.972276ii  |            |
     -.999969-.00781125ii .565772+.824562ii  |            |
     -.344326+.93885ii    -.974074-.226228ii |            |
     ------------------------------------------------------------------------
     .276727+.960949ii  -.166675+.986012ii .824149+.566373ii
     .74405-.668124ii   .936252-.351329ii  .869663+.493645ii
     -.191573+.981478ii -.987864+.155324ii .68453+.728985ii 
     .476077+.879404ii  -.535925+.844266ii .415186-.909737ii
     ------------------------------------------------------------------------
     -.788618-.614883ii |), ({2, 4}, | .109319-.994007ii  .560285+.8283ii  
     .794218+.607632ii  |            | .984107-.177574ii  .0000532609+1ii  
     -.990213-.139564ii |            | -.764815-.64425ii  -.64519-.764022ii
     .194585-.980886ii  |            | -.129853+.991533ii .889567+.456804ii
     ------------------------------------------------------------------------
     .990856+.13492ii   -.211093+.977466ii |)}
     .978795+.20484ii   .241807+.970324ii  |
     .93104+.364917ii   .923802+.382872ii  |
     -.872433-.488734ii -.636591-.771201ii |

o2 : List

Caveat

The output consists of random numerical matrices that are assumed invertible. The code does not check invertibility.

Ways to use randomSchubertProblemInstance :

randomSchubertProblemInstance(List,ZZ,ZZ)

For the programmer

The object randomSchubertProblemInstance is a method function with options.