Experimentation and conjectures in the real Schubert calculus for flag manifolds

James Ruffo, Yuval Sivan, Evgenia Soprunova, Frank Sottile

Published 2005-07-19, updated 2005-11-22Version 2

The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro conjecture for the flag manifold and present massive (15.76 gigahertz-years) computational experimentation in support of this refined conjecture. We also prove the conjecture in some special cases using discriminants and establish relationships between different cases of the conjecture.