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NumericalSchubertCalculus :: solveSchubertProblem

solveSchubertProblem -- uses homotopies of geometric Littlewood-Richardson rule to solve Schubert problems on Grassmannians

Synopsis

Description

Represent a Schubert variety in the Grassmannian Gr(k,n) by a partition l (a weakly decreasing list of nonegative integers less than n-k) and a flag F (given as an n× n matrix). A Schubert problem is a list of Schubert varieties (l1, F1), ..., (lm, Fm) such that |l1|+|l2| + …+ |lm| = k(n-k), where |li| is the sum of the entries of li.

The function solves the Schubert problem by Littlewood-Richardson homotopies. This algorithm uses homotopy continuation to track solutions of a simpler problem to a general problem according to the specializations of the geometric Littlewood-Richardson.

This algorithm is described in the paper: Sottile,Vakil, and Verschelde, "Littlewood-Richardson homotopies"

i1 : k = 3;
i2 : 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 : stdio << "Schubert problem {2,1}^3 in Gr(3,6) with respect to random flags"<<endl;
Schubert problem {2,1}^3 in Gr(3,6) with respect to random flags
i5 : solveSchubertProblem(SchPblm, k,n)

o5 = {| -.973546-.443329i -.193482-.957391i -.460977-.267623i  |, |
      | .153343-.291038i  .339479-1.30008i  -.577303-.171329i  |  |
      | .326239-.770744i  .120186-1.46205i  -.0769985-.323186i |  |
      | -.451322-.514227i -.144874-.987789i -.589493-.209825i  |  |
      | -.052568-.692383i .364061-.704785i  -.325325-.185011i  |  |
      | -.942864-.594173i -.826353-.203654i -.236865+.691374i  |  |
     ------------------------------------------------------------------------
     -5.47422+1.18674i -.912706-1.64344i -.590144+.126046i |}
     -2.80395-.714508i -.793467-2.20255i -.016225+.340982i |
     -2.29873-2.34724i -.905617-2.59114i .0951404+.355536i |
     -3.36313+2.14592i -.703202-1.58202i -.375349+.205204i |
     -3.91243-.845457i -.528411-.900447i -.187047+.545512i |
     -4.19554+1.93172i -1.11528-.480829i .19859+1.24706i   |

o5 : List

Caveat

Need to input partitions together with flags. In the future, there will be an option for generating random flags and just input the partitions.

See also

Ways to use solveSchubertProblem :