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