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NumericalSchubertCalculus :: randomSchubertProblemInstance

randomSchubertProblemInstance -- Returns a random instance of a given Schubert problem by computing random matrices representing flags

Synopsis

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.

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

o1 = {({1}, | .781262-.624203i  -.462954-.886382i -.272832+.962062i
            | .9868+.161945i    -.219207-.975678i .778029-.628229i 
            | -.971549+.236839i .157598-.987503i  .841103-.540875i 
            | .893324+.449414i  .360485-.932765i  -.651048+.759036i
     ------------------------------------------------------------------------
     -.670948-.741504i |), ({1}, | .696205+.717843i  -.028156+.999604i
     .118767+.992922i  |         | -.58234+.812945i  -.924052-.382267i
     -.396553+.918012i |         | .486335-.873772i  -.970803-.23988i 
     -.272372-.962192i |         | -.783999-.620762i .577597-.816322i 
     ------------------------------------------------------------------------
     -.056229-.998418i .776937+.629579i  |), ({1}, | -.746312-.665596i
     .023444+.999725i  .570649+.821194i  |         | -.936289-.351232i
     .701149-.713015i  -.863646+.504099i |         | -.164243+.98642i 
     .357971+.933733i  -.820726+.571321i |         | .85817-.513365i  
     ------------------------------------------------------------------------
     .995687+.092774i .168661+.985674i  -.765803+.643076i |), ({1}, |
     .701342-.712825i -.286823-.957983i -.415126-.909764i |         |
     -.78761-.616174i .976163-.21704i   -.416188-.909278i |         |
     .759918-.650019i .482783+.87574i   .975922-.218119i  |         |
     ------------------------------------------------------------------------
     .918399+.395655i  .719639-.694348i  .9772+.212322i   -.998741-.050173i
     .58629+.810101i   -.554833-.831962i .454813+.890587i -.698649-.715464i
     -.590887+.806754i -.726792+.686857i .080013+.996794i .742918+.669382i 
     -.939129+.343564i -.616052-.787706i .339065+.940763i -.99319-.116506i 
     ------------------------------------------------------------------------
     |)}
     |
     |
     |

o1 : List
i2 : randomSchubertProblemInstance({{2,4},{2,4},{2,4},{2,4}},2,4)

o2 = {({2, 4}, | .996754-.080509i  -.909393-.415938i .398424+.917201i 
               | -.531694+.846937i .92457+.381011i   -.976915+.213627i
               | .51095+.859611i   -.227376-.973807i .842765-.538281i 
               | -.341238+.939977i -.110622-.993863i -.923063-.384649i
     ------------------------------------------------------------------------
     -.688353+.725376i |), ({2, 4}, | -.991977+.126418i .398109-.917338i 
     -.81898+.573823i  |            | .91893+.394421i   -.210222+.977654i
     -.915156-.403101i |            | -.834955-.550318i .491643-.870797i 
     -.684587-.728932i |            | -.978842+.204616i .097639-.995222i 
     ------------------------------------------------------------------------
     .989783-.142579i  .930733+.365699i  |), ({2, 4}, | .276727+.960949i 
     .990088-.140449i  .233835+.972276i  |            | .74405-.668124i  
     -.999969-.007811i .565772+.824562i  |            | -.191573+.981478i
     -.344326+.93885i  -.974074-.226228i |            | .476077+.879404i 
     ------------------------------------------------------------------------
     -.166675+.986012i .824149+.566373i -.788618-.614883i |), ({2, 4}, |
     .936252-.351329i  .869663+.493645i .794218+.607632i  |            |
     -.987864+.155324i .68453+.728985i  -.990213-.139564i |            |
     -.535925+.844266i .415186-.909737i .194585-.980886i  |            |
     ------------------------------------------------------------------------
     .109319-.994007i  .560285+.8283i   .990856+.13492i   -.211093+.977466i
     .984107-.177574i  .000053+i        .978795+.20484i   .241807+.970324i 
     -.764815-.64425i  -.64519-.764022i .93104+.364917i   .923802+.382872i 
     -.129853+.991533i .889567+.456804i -.872433-.488734i -.636591-.771201i
     ------------------------------------------------------------------------
     |)}
     |
     |
     |

o2 : List

Caveat

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

See also

Ways to use randomSchubertProblemInstance :