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NumericalSchubertCalculus :: solveSchubertProblem

solveSchubertProblem -- uses Littlewood-Richardson homotopy to solve Schubert problems on Grassmannians

Synopsis

Description

Represent a Schubert variety in the Grassmannian Gr(k,n) by a condition c either a partition or a bracket (see partition2bracket for details) and a flag F (given as an n× n matrix). The codimention of the Schubert variety is |C|. A Schubert problem is a list of Schubert varieties, whose codimention add up to k(n-k), which is the dimension of the Grassmannian.

The function solves the Schubert problem by the Littlewood-Richardson homotopy. 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: Leykin, Martin del Campo, Sottile, Vakil, Verschelde "Numerical Schubert Calculus via the Littlewood-Richardson homotopy algorithm".

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

The Schubert conditions are either all partitions or all brackets.

See also

Ways to use solveSchubertProblem :