NumericalSchubertCalculus -- Numerical Algorithms for Schubert Calculus

Description

A Schubert problem on the Grassmannian $Gr(k,n)$ is a list of Schubert subvarieties (assumed general) of $Gr(k,n)$ whose codimensions add up to $k(n-k)$, the dimension of the Grassmannian. A Schubert variety is represented by a pair $(c,F)$, where $c$ is a Schubert condition (represented as a partition or a bracket) and $F$ is a flag, represented as an invertible $n\times n$-matrix. The Schubert variety for the pair $(c,F)$ consists of all $k$-planes that satisfy the incidence condition $c$ imposed by the flag $F$.

For a Schubert problem, we may fix one flag for each Schubert condition in the list --- i.e., describe an instance of a given Schubert problem --- and look for the solutions for this instance --- i.e., points in the intersection of the corresponding Schubert varieties. The methods of this package find approximations of these points.

General functions:

• randomSchubertProblemInstance -- returns a random instance of a given Schubert problem
• changeFlags -- continue solutions of a Schubert problem to another instance
• checkIncidenceSolution -- check if a solution satisfies an instance of a Schubert problem
• LRnumber -- returns the number of solutions to the given Schubert problem

Functions implementing homotopies specific to Schubert calculus:

• solveSchubertProblem -- uses Littlewood-Richardson homotopy algorithm to solve a Schubert problem
• solveSimpleSchubert -- uses Pieri homotopy algorithm to solve a simple Schubert problem

Service functions:

• setVerboseLevel -- set different levels of information printed on screen
• bracket2partition -- dictionary between different notations for Schubert conditions.
• partition2bracket -- dictionary between different notations for Schubert conditions.
• NSC2phc -- dictionary between different notations for Schubert problems.

Using PHCpack:

• LRrule -- computes the product of Schubert classes using geometric Littlewood-Richardson rule
• LRtriple -- uses Littlewood-Richardson homotopy algorithm to solve a Schubert problem
• parseTriplet -- Parses the output from LRtriple into Macaulay2 objects.
• LRcheater -- A cheater's homotopy to a real Schubert triple intersection problem
• PieriRootCount -- the number of solutions to a generic Pieri problem
• PieriHomotopies -- runs the Pieri homotopies to solve a general hypersurface problem

i1 : k = 3; n = 6;

i3 : SchPblm = {
     	 ({2,1}, random(CC^6,CC^6)),
     	 ({2,1}, random(CC^6,CC^6)),
     	 ({2,1}, random(CC^6,CC^6))
     	 };

i4 : solveSchubertProblem(SchPblm, k,n)

o4 = {| -.973546-.443329ii  -.193482-.957391ii -.460977-.267623ii  |, |
      | .153343-.291038ii   .339479-1.30008ii  -.577303-.171329ii  |  |
      | .326239-.770744ii   .120186-1.46205ii  -.0769985-.323186ii |  |
      | -.451322-.514227ii  -.144874-.987789ii -.589493-.209825ii  |  |
      | -.0525679-.692383ii .364061-.704785ii  -.325325-.185011ii  |  |
      | -.942864-.594173ii  -.826353-.203654ii -.236865+.691374ii  |  |
     ------------------------------------------------------------------------
     -5.47422+1.18674ii -.912706-1.64344ii -.590144+.126046ii |}
     -2.80395-.714508ii -.793467-2.20255ii -.016225+.340982ii |
     -2.29873-2.34724ii -.905617-2.59114ii .0951404+.355536ii |
     -3.36313+2.14592ii -.703202-1.58202ii -.375349+.205204ii |
     -3.91243-.845457ii -.528411-.900447ii -.187047+.545512ii |
     -4.19554+1.93172ii -1.11528-.480829ii .19859+1.24706ii   |

o4 : List

References:

• Vakil, "A geometric Littlewood-Richardson rule",Vakil, Ravi. A geometric Littlewood-Richardson rule. Ann. of Math. (2) 164 (2006), 371-421
• Huber, Sottile, Sturmfels, "Numerical Schubert calculus", J. Symb. Comp., 26 (1998), 767-788.
• Sottile, Vakil, Verschelde, "Solving Schubert Problems with Littlewood-Richardson Homotopies", ISSAC 2010, 179-186, ACM, New York, 2010.
• Leykin and Sottile, "Galois groups of Schubert problems via homotopy computation", Math. Comp. 78 (2009), 1749-1765.
• Leykin, Martin del Campo, Sottile, Vakil, Verschelde "Numerical Schubert Calculus via the Littlewood-Richardson homotopy algorithm", Math. Comp., 90 (2021), 1407-1433.