Given a sequence of nested linear spaces (called flags) and prescribed dimensions for each flag, a Schubert problem asks for all planes that meet the given flags at the prescribed dimensions. A geometric Littlewood-Richardson rule developed by Ravi Vakil leads to homotopy algorithms to solve a Schubert problem. Littlewood-Richardson homotopies are the families of polynomial systems constructed by these homotopy algorithms. Symbolically, homotopy algorithms degenerate a moving flag, using polynomial equations to keep conditions imposed by other flags fixed. At the degenerate configuration of the flag, a linear system provides a start solution for a path to track by numerical continuation methods. The specialization of a flag follows a combinatorial checker game. For sufficiently generic Schubert problems, the number of paths to track is optimal. The Littlewood-Richardson homotopies are implemented using the path trackers of the software package PHCpack.

This is work in progress joint with Frank Sottile and Ravi Vakil.