The Wronski map and shifted tableau theory

TL;DR

This paper explores the relationship between the Wronski map, real roots of polynomials, and Young tableaux, specifically focusing on the orthogonal Grassmannian and shifted tableaux, providing new geometric insights into Schubert calculus.

Contribution

It extends the correspondence between the Wronski map and Young tableaux to the orthogonal Grassmannian, revealing symmetry conditions and offering a geometric proof of the Littlewood-Richardson rule.

Findings

Characterization of points on OG(n,2n+1) via tableau symmetry

Connection between real roots of polynomials and shifted tableaux

A new geometric proof of the Littlewood-Richardson rule for OG(n,2n+1)

Abstract

The Mukhin-Tarasov-Varchenko Theorem, conjectured by B. and M. Shapiro, has a number of interesting consequences. Among them is a well-behaved correspondence between certain points on a Grassmannian - those sent by the Wronski map to polynomials with only real roots - and (dual equivalence classes of) Young tableaux. In this paper, we restrict this correspondence to the orthogonal Grassmannian OG(n,2n+1) inside Gr(n,2n+1). We prove that a point lies on OG(n,2n+1) if and only if the corresponding tableau has a certain type of symmetry. From this we recover much of the theory of shifted tableaux for Schubert calculus on OG(n,2n+1), including a new, geometric proof of the Littlewood-Richardson rule for OG(n,2n+1).