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We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.
We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is `robust in the sense that it is designed to prevent path jumping and in many cases, it can be used in (only) double precision arithmetic. It
The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians. It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has
We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus. A well-known web of conjectures and results in the real Schubert
In the recent paper [arXiv:1612.06893] P. Burgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $delta_{k,n}$ the average number of projective $k$-planes in $mathbb{R}textrm{P}^