S = solveSimpleSchubert(P,k,n)a Schubert variety in the Grassmannian $Gr(k,n)$ is represented by a partition $l$ (a weakly decreasing list of nonegative integers less than $n-k$) and a flag $F$ (given as an $n\times n$ matrix). A Schubert problem is a list of Schubert varieties $(l_1, F_1), \ldots, (l_m, F_m)$ such that $|l_1|+|l_2| + \cdots + |l_m| = k(n-k)$, where $|l_i|$ is the sum of the entries of $l_i$.
The function solves a Schubert problem by the Pieri homotopy algorithm. It assumes all partitions except possibly the first two are simple (e.g. equal to {1}). This algorithm uses homotopy continuation to track solutions of a simpler problem to a general problem according to the specializations of the geometric Pieri rule.
This algorithm is described in the paper: Huber, Sottile, and Sturmfels, "Numerical Schubert Calculus", J. Symb. Comp., 26 (1998), 767-788.
for instance, the Schubert problem {2,1}$^2$ {1}$^3$ in $Gr(3,6)$ has six solutions. Consider the following instance given by random flags
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Its solutions to this instance
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Need to input partitions together with flags.
The object solveSimpleSchubert is a method function.