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 LittlewoodRichardson rule developed by Ravi Vakil leads to
homotopy algorithms to solve a Schubert problem. LittlewoodRichardson
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 LittlewoodRichardson
homotopies are implemented using the path trackers of the software
package PHCpack.
This is work in progress joint with Frank Sottile and Ravi Vakil.
