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
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-.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
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-.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
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.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
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-.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
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-.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
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.989783-.142579ii .930733+.365699ii |), ({2, 4}, |
.990088-.140449ii .233835+.972276ii | |
-.999969-.00781125ii .565772+.824562ii | |
-.344326+.93885ii -.974074-.226228ii | |
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.276727+.960949ii -.166675+.986012ii .824149+.566373ii
.74405-.668124ii .936252-.351329ii .869663+.493645ii
-.191573+.981478ii -.987864+.155324ii .68453+.728985ii
.476077+.879404ii -.535925+.844266ii .415186-.909737ii
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-.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
|
The output consists of random numerical matrices that are assumed invertible. The code does not check invertibility.
The object randomSchubertProblemInstance is a method function with options.