Experimentation and conjectures in the real Schubert calculus for flag manifolds

Details Experimentation and conjectures in the real Schubert calculus for flag manifolds Experimentation and conjectures in the real Schubert calculus for flag manifolds

2005-07-19

arxiv.org/abs/math/0507377v2

Experimental Mathematics, 15, No. 2 (2006), 199--221.

Mathematics

Algebraic Geometry

34 pages. Revised and one example removed. Expanded version of math.AG/0502040. Related WWW page http://www.math.tamu.edu/~s

Scientific paper

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.

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