香蕉视频

The real Wronski map and the Murnaghan-Nakayama rule.

The Wronski problem is a family of Schubert problems, defined with respect to flags osculating the rational normal curve. This family has attracted intensive interest in the last thirty years, thanks to a series of conjectures initiated by Boris and Michael Shapiro, later proven by Mukhin-Tarasov-Varchenko and generalized in many directions. Broadly, these results uncover rich topological structure, built out of Young tableaux and familiar combinatorial rules, in the Wronski map over the real numbers. Most prior work has focused on the case where the osculation points are all real. In this case, the Wronski map is described by standard tableaux. I will describe a recent extension (joint with Kevin Purbhoo) that allows the osculation points to occur in complex conjugate pairs. We describe certain fibers of the map using domino tableaux, and we relate its topological degree to the Murnaghan-Nakayama rule for symmetric group characters. Our result leads to a new, topological proof of the original Shapiro-Shapiro conjecture, and suggests some intriguing new topological-combinatorial questions to pursue.

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