Wronskians, total positivity, and real Schubert calculus

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).