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