Wronskians, total positivity, and real Schubert calculus

TL;DR

This paper characterizes totally nonnegative and positive flags in real Grassmannians using Wronskian polynomials, linking total positivity to Chebyshev systems and Schubert calculus, and explores related conjectures about real solutions.

Contribution

It provides a new characterization of totally nonnegative flags via Wronskians and connects conjectures in real Schubert calculus to total positivity properties.

Findings

Totally nonnegative flags correspond to Wronskians nonzero on (0, ∞).

Totally positive flags correspond to Wronskians nonzero on [0, ∞].

A conjecture links zeros of Wronskians to positivity of Plücker coordinates.

Abstract

A complete flag in $\mathbb{R}^n$ is a sequence of nested subspaces $V_1 \subset \cdots \subset V_{n-1}$ such that each $V_k$ has dimension $k$. It is called totally nonnegative if all its Pl\"ucker coordinates are nonnegative. We may view each $V_k$ as a subspace of polynomials in $\mathbb{R}[x]$ of degree at most $n-1$, by associating a vector $(a_1, \dots, a_n)$ in $\mathbb{R}^n$ to the polynomial $a_1 + a_2x + \cdots + a_nx^{n-1}$. We show that a complete flag is totally nonnegative if and only if each of its Wronskian polynomials $\mathsf{Wr}(V_k)$ is nonzero on the interval $(0, \infty)$. In the language of Chebyshev systems, this means that the flag forms a Markov system or $ECT$-system on $(0, \infty)$. This gives a new characterization and membership test for the totally nonnegative flag variety. Similarly, we show that a complete flag is totally positive if and only if each $\mathsf{Wr}(V_k)$ is nonzero on $[0, \infty]$. We use these results to show that a conjecture of Eremenko (2015) in real Schubert calculus is equivalent to the following conjecture: if $V$ is a finite-dimensional subspace of polynomials such that all complex zeros of $\mathsf{Wr}(V)$ lie in the interval $(-\infty, 0)$, then all Pl\"ucker coordinates of $V$ are real and positive. This conjecture is a totally positive strengthening of a result of Mukhin, Tarasov, and Varchenko (2009), and can be reformulated as saying that all complex solutions to a certain family of Schubert problems in the Grassmannian are real and totally positive. We also show that our conjecture is equivalent to a totally positive version of the secant conjecture of Sottile (2003).