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%% define your title in the usual way
\title{Universal Pl\"ucker coordinates for the Wronski map and positivity in real Schubert calculus}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}
\author{Steven N.\ Karp\thanks{\href{mailto:skarp2@nd.edu}{skarp2@nd.edu}}\addressmark{1} \and Kevin Purbhoo\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Notre Dame \\ \addressmark{2}Combinatorics and Optimization Department, University of Waterloo}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{\noindent Given a $d$-dimensional vector space $V \subset \mathbb{C}[u]$ of polynomials, its Wronskian is the polynomial $(u + z_1) \cdots (u + z_n)$ whose zeros $-z_i$ are the points of $\mathbb{C}$ such that $V$ contains a nonzero polynomial with a zero of order at least $d$ at $-z_i$. Equivalently, $V$ is a solution to the Schubert problem defined by osculating planes to the moment curve at $z_1, \dots, z_n$. The \emph{inverse Wronski problem} involves finding all $V$ with a given Wronskian $(u + z_1) \cdots (u + z_n)$. We solve this problem by providing explicit formulas for the Grassmann--Pl\"ucker coordinates of the general solution $V$, as commuting operators in the group algebra $\mathbb{C}[\mathfrak{S}_n]$ of the symmetric group. The Pl\"ucker coordinates of individual solutions over $\mathbb{C}$ are obtained by restricting to an eigenspace and replacing each operator by its eigenvalue. This generalizes work of Mukhin, Tarasov, and Varchenko (2013) and of Purbhoo (2022), which give formulas in $\mathbb{C}[\mathfrak{S}_n]$ for the differential equation satisfied by $V$. Moreover, if $z_1, \dots, z_n$ are real and nonnegative, then our operators are positive semidefinite, implying that the Pl\"ucker coordinates of $V$ are all real and nonnegative. This verifies several outstanding conjectures in real Schubert calculus, including the positivity conjectures of Mukhin and Tarasov (2017) and of Karp (2021), the disconjugacy conjecture of Eremenko (2015), and the divisor form of the secant conjecture of Sottile (2003). The proofs involve the representation theory of $\mathfrak{S}_n$, symmetric functions, and $\tau$-functions of the KP hierarchy.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Wronskian, Schubert calculus, symmetric group, symmetric functions, KP hierarchy, total positivity}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}

For a system of real polynomial equations with finitely many solutions,
we normally expect that some --- but not all --- of the solutions are real,
while the remaining solutions come in complex-conjugate pairs.
The precise number of real solutions usually depends in a complicated way
on the coefficients of the equations.
However, in some rare cases, it is possible to obtain a better
understanding of the real solutions.
A remarkable example occurs in the Schubert calculus of
the Grassmannian $\Gr(d,m)$,
for Schubert problems defined by flags osculating
a rational normal curve.  In 1993, Boris and Michael Shapiro conjectured
that all such Schubert problems with real parameters have only real 
solutions.  The corresponding systems of equations arise in various
guises throughout mathematics, 
from algebraic curves \cite{eisenbud_harris83,kharlamov_sottile03} 
to differential equations \cite{mukhin_varchenko04} to pole-placement problems \cite{rosenthal_sottile98,eremenko_gabrielov02a}.
The conjecture was eventually proved by Mukhin, Tarasov, and Varchenko \cite{mukhin_tarasov_varchenko09a},
using a reformulation in terms of Wronski maps, 
and machinery from quantum integrable systems and representation
theory.

While the details of the Mukhin--Tarasov--Varchenko proof are rather
intricate, the basic idea is relatively straightforward.  
They consider a family of commuting linear operators arising from the Gaudin model,
and show that they satisfy algebraic equations defining a Schubert problem.
Hence, by considering the spectra of
these operators, they are able to infer some basic properties of
the solutions to the Schubert problem.  In this paper we extend these
results, making the connection between the commuting operators and the corresponding solutions more explicit and concrete.
Consequently, we obtain stronger results
in real algebraic geometry, including several generalizations of the Shapiro--Shapiro conjecture. Namely, we resolve the divisor form of the secant conjecture of Sottile (2003), the disconjugacy conjecture of Eremenko \cite{eremenko15}, and the positivity conjectures of Mukhin--Tarasov (2017) and Karp \cite{karp}. Proofs and further details appear in the paper \cite{karp_purbhoo}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The Wronski map and the Bethe algebra}
\label{sec:wronskiintro}

Let $\Gr(d,m)$ denote the Grassmannian of all $d$-dimensional linear subspaces of $\CC^m$. We identify $\CC^m$ with $\CC_{m-1}[u]$, the $m$-dimensional vector space of univariate polynomials of degree at most $m-1$, via the isomorphism
\begin{equation}
\label{eq:isomorphism}
     (a_1, \dots, a_m)  \leftrightarrow  \sum_{j=1}^m a_j \frac{u^{j-1}}{(j-1)!}
\,.
\end{equation}
In particular, we also view $\Gr(d,m)$ as the space of $d$-dimensional subspaces of $\CC_{m-1}[u]$.

Now fix a nonnegative integer $n$, and let $\nu$ be a 
partition of $n$ with at most $d$ parts whose sizes are at most $m-d$. The \defn{Schubert cell}
$\scellnu \subseteq \Gr(d,m)$ is the space of all $d$-dimensional linear subspaces of $\CC[u]$
that have a basis $(f_1, \dots, f_d)$, with
$\deg(f_i) = \nu_i+d-i$.  As a scheme, $\scellnu$ is isomorphic to
$n$-dimensional affine space.

Let $\monics \subseteq \CC[u]$ denote the $n$-dimensional affine space of
monic polynomials of degree $n$.
Given $V \in \scellnu$, choose any basis $(f_1, \dots, f_d)$ for $V$.
We define $\Wr(V)$ to be the unique monic polynomial which is a scalar
multiple of the Wronskian 
\[ 
    \Wr(f_1, \dots, f_d) := 
   \begin{vmatrix}
    f_1 & f_1' & f_1'' & \dots & f_1^{(d-1)} \\
%    f_2 & f_2' & f_2'' & \dots & f_2^{(d-1)} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\\
    f_d & f_d' & f_d'' & \dots & f_d^{(d-1)} \\
   \end{vmatrix}
\,.
\]  
It is not hard to see that 
$\Wr(V) \in \monics$ is a polynomial of degree $n$, and 
is independent of the choice of basis.  Thus we
obtain a map $\Wr : \scellnu \to \monics$, called the \defn{Wronski map}
on $\scellnu$.  Abstractly, this is a finite morphism 
from $n$-dimensional affine
space to itself.

Suppose $g(u) = (u+z_1)\dotsm (u+z_n) \in \monics$,
where $z_1, \dots, z_n$ are complex numbers.   The 
\defn{inverse Wronski problem} is to compute the fibre 
$\Wr^{-1}(g) \subseteq \scellnu$.

In their study of the Gaudin model for $\gl_n$,
Mukhin, Tarasov, and 
Varchenko \cite{mukhin_varchenko04,mukhin_varchenko05,mukhin_tarasov_varchenko06,mukhin_tarasov_varchenko09a,mukhin_tarasov_varchenko09b} discovered a connection between the inverse Wronski problem,
and the problem of diagonalizing the Gaudin Hamiltonians \cite{gaudin76}. 
We will focus on the version of this story from \cite{mukhin_tarasov_varchenko13}, in which the Gaudin Hamiltonians generate the \emph{Bethe algebra (of Gaudin type)} $\bethe \subseteq \CC[\Sn]$, which is a commutative subalgebra of the group algebra of the symmetric group.

Let $\spechtnu$ be the Specht module (i.e.\ irreducible $\Sn$-representation) associated to the partition $\nu$. 
Then $\bethe$ acts on $\spechtnu$, and the image of 
this action defines a commutative subalgebra 
$\bethenu \subseteq \End(\spechtnu)$. We have the following correspondence:

\begin{theorem}[Mukhin, Tarasov, and Varchenko \cite{mukhin_tarasov_varchenko13}]
\label{thm:vague}
The eigenspaces $E \subseteq \spechtnu$ of the algebra 
$\bethenu$ are in one-to-one 
correspondence with the points $V_E \in \Wr^{-1}(g)$.
The eigenvalues of the generators of $\bethenu$ are
coordinates for $V_E$ in some coordinate system.
\end{theorem}

%(There are also scheme-theoretic analogues of \cref{thm:vague}, which we discuss in \cref{sec:schemeintro}.)
Unfortunately, \cref{thm:vague}
is poorly suited to studying certain properties of the Wronski map.  This is 
because the generators of $\bethe$ 
correspond to a somewhat unusual coordinate system for $\scellnu$.  
Namely, given $V \in \scellnu$, there is a unique \defn{fundamental differential operator} $D_V = \partial_u^d + \psi_1(u) \partial_u^{d-1} + \dots + \psi_d(u)$ with coefficients $\psi_j(u) \in \CC(u)$,
such that $V$ is the space of solutions to the differential equation 
$D_V f(u) = 0$.
The coefficients of $D_V$ can be 
regarded as a coordinate system on $\scellnu$.  In the precise formulation of \cref{thm:vague}, the point
$V_E\in\Wr^{-1}(g)$ is computed in these coordinates. In order to express $V_E$ in standard coordinates, we need to solve a differential equation, resulting in highly non-linear formulas.

Our main result is \cref{thm:main} below, which is a new version of \cref{thm:vague}. Rather than using the fundamental differential operator coordinates, it computes $V_E\in\Wr^{-1}(g)$ in the \defn{\Plucker coordinates}, which are the $d\times d$ minors of a $d\times m$ matrix whose rows form a basis for $V_E$.  We introduce (by explicit formulas)
a new set of generators $\beta^\lambda$ for $\bethe$, 
which are indexed by partitions $\lambda$.
For any eigenspace 
$E \subseteq \spechtnu$, the corresponding eigenvalues of the $\beta^\lambda$'s
are the \Plucker coordinates of $V_E$.

There are three major advantages of this formulation.  First, we obtain a more direct description of $V_E$ which does not require solving a differential equation; the implicit part of our construction lies entirely in understanding the representation theory of $\Sn$. Second, many
natural objects of interest are given by \emph{linear} 
functions of the \Plucker 
coordinates. For example, we readily obtain explicit bases for $V_E$; the Wronskian and
the fundamental differential operator coordinates are given as 
linear functions of the \Plucker coordinates; and Schubert varieties and 
Schubert intersections are defined by linear equations in the \Plucker 
coordinates.
Third, basic properties of the operators $\beta^\lambda$ imply positivity results about the \Plucker coordinates of $V_E$. This 
enables us to resolve several conjectures in real algebraic geometry, as we explain in \cref{sec:conjectures}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Universal \Plucker coordinates}
\label{sec:universalintro}

We now state our main theorem.
For every partition $\lambda$, define 
\begin{equation}
\label{eq:betadef}
  \beta^\lambda(t) := \sum_{\substack{X \subseteq [n], \\ |X| = |\lambda|}} \,
   \sum_{\sigma \in \symgrp{X}}
     \chi^\lambda(\sigma) \sigma \prod_{i \in [n] \setminus X} (z_i + t)
\,.
\end{equation}
Here $[n] = \{1, \dots, n\}$, $\symgrp{X} \subseteq \Sn$ is the group of 
permutations of $X$, and
$\chi^\lambda : \symgrp{X} \to \CC$ is the character of the Specht module $\specht{\lambda}$.  We note that $\chi^\lambda$ is integer-valued, so $\beta^\lambda(t)$ is in fact defined over $\ZZ$.
Also, $\beta^\lambda(t)$ is nonzero if and only if $|\lambda| \leq n$.
Set $\beta^\lambda := \beta^\lambda(0)$.
\begin{example}\label{ex:betadef}
If $\lambda = (1,1)$, then $\chi^\lambda$ is the sign character on $\symgrp{2}$. When $n=3$, we get
\[
\beta^{11} = (\Sidentity{3} - \trans{1}{2})z_3 + (\Sidentity{3} - \trans{1}{3})z_2 + (\Sidentity{3} - \trans{2}{3})z_1\,,
\]
where $\Sidentity{3}$ denotes the identity element of $\symgrp{3}$, 
and $\trans{i}{j} := (i \;\; j)$ is the transposition swapping $i$ and $j$.
\end{example}

\begin{theorem}
\label{thm:main}
Let $z_1, \dots, z_n \in \CC$, and set $g(u) := (u+z_1) \dotsm (u+z_n) \in \CC[u]$.
The operators $\beta^\lambda(t) \in \CC[\Sn]$ satisfy the following 
algebraic identities:
\begin{enumerate}[label=(\roman*)]
\item\label{main_commutativity} Commutativity relations:
\begin{equation}
\label{eq:commutativity}
      \beta^\lambda(s) \beta^\mu(t) = \beta^\mu(t) \beta^\lambda(s)
   \qquad \text{for all partitions $\lambda$ and $\mu$}\,.
\end{equation}
\item\label{main_translation} Translation identity:
\begin{equation}
\label{eq:translationidentity}
    \beta^\mu(s+t) = \sum_{\lambda \supseteq \mu} 
      \frac{\numsyt{\lambda/\mu}}{|\lambda/\mu|!} t^{|\lambda/\mu|} 
      \beta^\lambda(s) \qquad \text{for all partitions } \mu\,,
\end{equation}
where $\numsyt{\lambda/\mu}$ denotes the number of standard Young tableaux 
of shape $\lambda/\mu$.
\item\label{main_pluckers} The quadratic \Plucker relations.
\end{enumerate}
Furthermore:
\begin{enumerate}[label=(\roman*)] \setcounter{enumi}{3}
\item\label{main_generates} For every partition $\lambda$ and $t \in \CC$, 
      we have $\beta^\lambda(t) \in \bethe$.
      The set $\{\beta^\lambda \mid |\lambda| \leq n\}$ generates
      $\bethe$ as an algebra.
\item\label{main_eigenspace} If $E \subseteq \spechtnu$ is any eigenspace 
      of $\bethenu$, then the corresponding eigenvalues 
      of the operators $\beta^\lambda$ are the \Plucker coordinates of 
      a point $V_E \in \scellnu \subseteq \Gr(d,m)$ such that $\Wr(V_E) = g$.  
      Every point of $\Wr^{-1}(g)$ 
      corresponds to some eigenspace $E \subseteq \spechtnu$ 
      of $\bethenu$.  
\item\label{main_multiplicity} The multiplicity of $V_E$ as a point of 
      $\Wr^{-1}(g)$ is equal to $\dim \widehat{E}$, where 
      $\widehat{E} \subseteq \spechtnu$ is the generalized eigenspace of
      $\bethenu$ containing $E$.
\end{enumerate}
\end{theorem}

We note that while the translation identity in part \ref{main_translation} is linear, 
parts \ref{main_commutativity} and \ref{main_pluckers} both involve quadratic expressions in 
$\bethe$,
making them intractable to prove directly.  In both of these cases we proceed by reducing the problem to --- and then proving --- an easier identity, using a diverse set of algebraic tools. For part \ref{main_commutativity}, we use properties of $\bethe$ and combinatorial ideas which appeared in \cite{purbhoo}. For part \ref{main_pluckers}, we employ the translation identity, properties of the exterior algebra, new combinatorial identities of symmetric functions, and the theory of $\tau$-functions of the KP hierarchy.  Once
identities \ref{main_commutativity}--\ref{main_pluckers}
are established, parts \ref{main_generates}--\ref{main_multiplicity} are relatively straightforward
consequences. See \cite[Sections~3--4]{karp_purbhoo} for the details.

There is a precise scheme-theoretic formulation of \cref{thm:main}\ref{main_eigenspace}; see \cite[Section~5.1]{karp_purbhoo}. In \cite[Section~5.2]{karp_purbhoo}, we also use \cref{thm:main} to give two explicit bases for any element $V \in \Wr^{-1}(g)$, in terms of our operators $\beta^\lambda(t)$ acting on the associated eigenspace $E$.

%While the proof of \cref{thm:main} uses some of the same mathematical 
%constructions as the proof of \cref{thm:vague}/\ref{thm:precise} in \cite{mukhin_tarasov_varchenko13}, it does not use
%the result itself.  In fact, our proof of \cref{thm:main} also 
%provides a new proof of \cref{thm:precise} (see \cref{rmk:newproof}).
%The two arguments are interconnected, which plays a key role in
%establishing the \Plucker relations.

\begin{example}
\label{ex:main}
We illustrate \cref{thm:main} in the case $n=2$, for the Grassmannian $\Gr(2,4)$. Writing $\symgrp{2} = \{\Sidentity{2}, \trans{1}{2}\}$, we have
\[
\beta^0 = \Sidentity{2} \, z_1z_2\,, \qquad \beta^1 = \Sidentity{2} (z_1 +  z_2)\,, \qquad \beta^2 = \Sidentity{2} + \trans{1}{2}\,, \qquad \beta^{11} = \Sidentity{2} - \trans{1}{2}\,,
\]
and $\beta^\lambda = 0$ for all other partitions $\lambda$. Note that the $\beta^\lambda$'s satisfy the equation
\[
- \beta^0\beta^{22} + \beta^1\beta^{21} - \beta^{11}\beta^2 = 0\,,
\]
which is the first non-trivial \Plucker relation.

There are two Specht modules for $\symgrp{2}$, namely $\specht{2}$ and $\specht{11}$, which are both $1$-dimensional. In $\specht{2}$, both $\Sidentity{2}$ and $\trans{1}{2}$ act with eigenvalue $1$, and so
\begin{equation}
\label{eq:examplepluckers}
\beta^0 \rightsquigarrow z_1z_2\,, \qquad \beta^1 \rightsquigarrow z_1 + z_2\,, \qquad \beta^2 \rightsquigarrow 2\,, \qquad \beta^{1,1} \rightsquigarrow 0\,.
\end{equation}
These are the \Plucker coordinates of the element $V = \left\langle 1,\, z_1z_2u + \frac{z_1 + z_2}{2}u^2 + \frac{1}{3}u^3 \right\rangle \in \scell{2}$. That is, when we represent $V$ as the row span of the $2\times 4$ matrix
\[
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & z_1z_2 & z_1+z_2 & 2
\end{pmatrix}
\]
(where vectors correspond to polynomials as in \eqref{eq:isomorphism}), the maximal minors are precisely the $\beta^\lambda$'s, where we read off the column set of a minor from $\lambda$ as in \cref{fig:partition}.

On the other hand, in $\specht{11}$, the element $\Sidentity{2}$ acts with eigenvalue $1$ and $\trans{1}{2}$ acts with eigenvalue $-1$, giving the solution $V = \left\langle \frac{z_1 + z_2}{2} + u,\, -z_1z_2 + u^2 \right\rangle \in \scell{1,1}$.
We can check that both elements $V$ of $\Gr(2,4)$ have Wronskian $g(u) = (u+z_1)(u+z_2)$.
\end{example}
\begin{figure}[hbt]
\begin{center}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\tikzstyle{out1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=black]
\tikzstyle{in1}=[inner sep=0,minimum size=1.2mm,circle,draw=black,fill=white]
\pgfmathsetmacro{\unit}{0.922};
\pgfmathsetmacro{\s}{0.86}
\useasboundingbox(0,0*\unit)rectangle(2*\unit,-2*\unit);
\coordinate (vstep)at(0,-0.22);
\coordinate (hstep)at(0.16,0);
\coordinate (vepsilon)at(0,-0.02*\unit);
\coordinate (hepsilon)at(0.02*\unit,0);
\draw[thick](0,0)--(2*\unit,0)--(2*\unit,-2*\unit)--(0,-2*\unit)--cycle;
\node[inner sep=0]at(0,0){\scalebox{1.6}{\begin{ytableau}
\none \\
\none \\
\none & \none & *({black!20!}) & *({black!20!}) \\
\none & \none
\end{ytableau}}};
\node[inner sep=0]at($(0,-1.5*\unit)+(hstep)$){\scalebox{\s}{$1$}};
\node[inner sep=0]at($(2*\unit,-0.5*\unit)+(hstep)$){\scalebox{\s}{$4$}};
\node[inner sep=0]at($(0.5*\unit,-1*\unit)+(vstep)$){\scalebox{\s}{$2$}};
\node[inner sep=0]at($(1.5*\unit,-1*\unit)+(vstep)$){\scalebox{\s}{$3$}};
\node[inner sep=0]at(0,-1*\unit)[label={[left=2pt]$d=2$}]{};
\node[inner sep=0]at(1*\unit,0)[label={[above=2pt]$m-d=2$}]{};
\end{tikzpicture}
\caption{The partition $\lambda = (2)$ corresponds to the column set $\{1,4\}$, where $d = 2$ and $m = 4$. When we label the edges of the border of the diagram of $\lambda$ by $1, \dots, m$ from southwest to northeast, the elements of $I$ are the labels of the vertical edges.}\label{fig:partition}
\end{center}\vspace*{-24pt}
\end{figure}

\begin{example}
\label{ex:22}
We illustrate parts \ref{main_commutativity} and \ref{main_pluckers} of \cref{thm:main} in the case $n=4$. Consider the $2$-dimensional representation $\spechtnu$ of $\symgrp{4}$, $\nu = (2,2)$. Following the conventions used by \texttt{Sage} \cite{sagemath}, the simple transpositions $\trans{1}{2}$ and $\trans{3}{4}$ both act as $(\begin{smallmatrix}1 & 0 \\ 1 & -1\end{smallmatrix})$, and $\trans{2}{3}$ acts as $(\begin{smallmatrix}0 & -1 \\ -1 & 0\end{smallmatrix})$. Let $\beta^\lambda_\nu \in \End(\spechtnu)$ denote the operator $\beta^\lambda$ acting on $\spechtnu$, which we regard as a $2\times 2$ matrix. Then
\begin{gather*}
\beta^0_\nu = z_1z_2z_3z_4\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,,
\qquad
\beta^1_\nu = (z_1z_2z_3 + z_1z_2z_4 + z_1z_3z_4 + z_2z_3z_4)\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,,
\\[2pt]
\beta^2_\nu = \scalebox{0.85}{$\begin{pmatrix}
2z_1z_2 + z_1z_4 + z_2z_3 + 2z_3z_4 & z_1z_3 - z_1z_4 - z_2z_3 + z_2z_4 \\
z_1z_2 - z_1z_4 - z_2z_3 + z_3z_4 & 2z_1z_3 + z_1z_4 + z_2z_3 + 2z_2z_4
\end{pmatrix}$}\,,
\\[2pt]
\beta^{11}_\nu = \scalebox{0.85}{$\begin{pmatrix}
2z_1z_3 + z_1z_4 + z_2z_3 + 2z_2z_4 & -z_1z_3 + z_1z_4 + z_2z_3 - z_2z_4 \\
-z_1z_2 + z_1z_4 + z_2z_3 - z_3z_4 & 2z_1z_2 + z_1z_4 + z_2z_3 + 2z_3z_4
\end{pmatrix}$}\,,
\\[2pt]
\beta^{21}_\nu = 3(z_1 + z_2 + z_3 + z_4)\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,,
\qquad
\beta^{22}_\nu = 12\,\scalebox{0.85}{$\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$}\,,
\end{gather*}
and $\beta^\lambda_\nu = 0$ for all other partitions $\lambda$.
We can see that the $\beta^\lambda_\nu$'s pairwise commute and satisfy the \Plucker relation $- \beta^0_\nu\beta^{22}_\nu + \beta^1_\nu\beta^{21}_\nu - \beta^{11}_\nu\beta^2_\nu = 0\,.$
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conjectures in real algebraic geometry}
\label{sec:conjectures}

We continue to work with the Schubert cell $\scellnu \subseteq \Gr(d,m)$, where $\nu$ is a partition of $n$.
The Schubert variety $\svarnu \subseteq \Gr(d,m)$ is
the closure of $\scellnu$.
We write $\rectangle$ for the rectangular partition $(m-d)^d = (m-d, \dots, m-d)$.
In this case, $\svarrect = \Gr(d,m)$.

We will be mainly concerned with the following Schubert problem. 
Given $W_1, \dots, W_n$ in $\Gr(m-d,m)$, determine
all $d$-planes $V$ such that
\begin{equation}
\label{eq:schubertproblem}
V \in \svarnu \qquad \text{and} \qquad
  V \cap W_i \neq \{0\}\; \text{ for all $i =1, \dots, n$}
\,.
\end{equation}
When $W_1, \dots, W_n$ are sufficiently general, the number of distinct
solutions $V$ to the Schubert problem \eqref{eq:schubertproblem} is exactly $\numsyt{\nu} = \dim \spechtnu$.

We will be concerned with solving \eqref{eq:schubertproblem} over the real numbers when $W_1, \dots, W_n$ are real, and especially with instances for which \emph{all} the solutions are real. The interest in algebraic problems with only real solutions dates back at least to Fulton \cite[Section~7.2]{fulton84}, who wrote, ``The question of how many solutions of real equations can be real is still very much open, particularly for enumerative problems.'' Note that the property of having only real solutions is extremely rare; for example, for a `random' Schubert problem on $\Gr(d,m)$ defined over $\RR$, the number of real solutions is roughly the square root of the number of complex solutions \cite{burgisser_lerario20}. We refer to \cite{sottile11} for a detailed survey of real enumerative geometry.

\subsection{The Shapiro--Shapiro conjecture}
\label{sec:shapiroshapiro}
The \defn{moment curve}
$\gamma : \CC \to \CC_{m-1}[u]$ is the parametric curve
\begin{equation}
\label{eq:moment}
   \gamma(t) := \frac{(u+t)^{m-1}}{(m-1)!}
\,.
\end{equation}
The closure of the image of $\gamma$ in $\PP^{m-1}$ is 
a rational normal curve.
A $d$-plane $V \in \Gr(d,m)$ \defn{osculates} $\gamma$ at 
$w \in \CC$ if 
$(\gamma(w), \gamma'(w), \gamma''(w), \dots, \gamma^{(d-1)}(w))$
is a basis for $V$.
The \defn{Shapiro--Shapiro conjecture} can be stated as follows:
\begin{theorem}[Mukhin, Tarasov, and Varchenko \cite{mukhin_tarasov_varchenko09a}]
\label{thm:ssc}
Let $z_1, \dots, z_n$ be distinct real numbers.
For $i =1, \dots, n$, let $W_i \in \Gr(m-d,m)$ be the osculating 
$(m-d)$-plane to $\gamma$ at $z_i$.
Then there are exactly $\numsyt{\nu}$ distinct solutions to the Schubert
problem \eqref{eq:schubertproblem}, and all solutions are real.
\end{theorem}

\cref{thm:ssc} was conjectured by Boris and Michael Shapiro in 1993, and extensively tested and popularized by Sottile \cite{sottile00}. It was proved in the cases $d \le 2$ and $m-d \le 2$ by Eremenko and Gabrielov \cite{eremenko_gabrielov02}, and in general by Mukhin, Tarasov, and Varchenko \cite{mukhin_tarasov_varchenko09a}. Their proof was later restructured and simplified in \cite{purbhoo}.  A very different proof, based on geometric and topological arguments, is given in \cite{levinson_purbhoo21}.

%A notable feature of \cref{thm:ssc} is that the Schubert problem
%has $\numsyt{\nu}$ distinct solutions, despite the fact that $W_1, \dots, W_n$ are
%explicitly specified, and hence not assumed to be general.  
%In fact, this follows from the claim that all 
%solutions are real (see \cite[Theorem 3.2]{sottile11}).
%A more general form of \cref{thm:ssc} describes how the story changes 
%in the limit as $z_1, \dots, z_n$ become non-distinct,
%as we discuss in \cref{sec:dimensions,sec:generalssc}.

Using \cref{thm:main}, we obtain a number of generalizations of \cref{thm:ssc}:

\subsection{The divisor form of the secant conjecture}
\label{sec:secantconjecture}
Let $I \subseteq \RR$ be an interval.  An $(m-d)$-plane $W \in \Gr(m-d,m)$ 
is a \defn{secant} to $\gamma$ along $I$ if there exist distinct points 
$w_1, \dots, w_{m-d} \in I$ such that $(\gamma(w_1), \dots, \gamma(w_{m-d}))$
is a basis for $W$.
More generally, $W$ is a \defn{generalized secant} to $\gamma$ along $I$ 
if there
exist distinct points $w_1, \dots, w_k \in I$ and positive integers 
$m_1, \dots, m_k$, such that $m_1+ \dots + m_k = m-d$ and $\big(\gamma(w_1), \gamma'(w_1), \dots, \gamma^{(m_1-1)}(w_1),\ \dots\ , \gamma(w_k), \gamma'(w_k), \dots, \gamma^{(m_k-1)}(w_k)\big)$ is a basis for $W$.
%Working projectively, these definitions naturally
%extend to cyclic intervals of $\RP^1 = \RR \cup \{\infty\}$, 
%where the interval is allowed to wrap around infinity.  When
%one of the points $w_i\in I$ is $\infty$, 
%$\gamma^{(j)}(w_i)$ is replaced by $u^j$.

Around 2003, Frank Sottile formulated the \defn{secant conjecture}, which asserts in particular that \cref{thm:ssc} remains true when $W_1, \dots, W_n$ are generalized secants to $\gamma$ along disjoint intervals of $\RR$. This statement is what we call the \defn{divisor form} of the secant conjecture, since it arises from intersecting Schubert varieties of codimension one, i.e., \emph{Schubert divisors}; the general form of the secant conjecture involves intersecting Schubert varieties of arbitrary codimension. Note that this case of the secant conjecture is a generalization of the Shapiro--Shapiro conjecture, since an osculating plane to $\gamma$ is a special case of a generalized secant.

The secant conjecture appeared in \cite{ruffo_sivan_soprunova_sottile06} (cf.\ \cite[Section~13.4]{sottile11}), and it was extensively tested experimentally in a project led by Sottile \cite{garcia-puente_hein_hillar_martin_del_campo_ruffo_sottile_teitler12}, as described in \cite{hillar_garcia-puente_martin_del_campo_ruffo_teitler_johnson_sottile10}. It has also been proved in special cases: Eremenko, Gabrielov, Shapiro, and Vainshtein \cite[Section~3]{eremenko_gabrielov_shapiro_vainshtein06} established the case $m-d\le 2$; and Mukhin, Tarasov, and Varchenko \cite{mukhin_tarasov_varchenko09c} (cf.\ \cite[Section~3.1]{garcia-puente_hein_hillar_martin_del_campo_ruffo_sottile_teitler12}) verified the case of the divisor form when there exists $r > 0$ such that every $W_i$ is a (non-generalized) secant where $w_1, \dots, w_{m-d} \in I_i$ are an arithmetic progression of step size $r$.

We show that the divisor form of the secant conjecture is true in general:
\begin{theorem}[Secant conjecture, divisor form]
\label{thm:secant}
Let $I_1, \dots, I_n  \subseteq \RR$ be pairwise disjoint real intervals.
For $i =1, \dots, n$, let $W_i \in \Gr(m-d,m)$ be a generalized secant
to $\gamma$ along $I_i$.
Then there are exactly $\numsyt{\nu}$ distinct solutions to the Schubert
problem \eqref{eq:schubertproblem}, and all solutions are real.
\end{theorem}

This verifies the secant conjecture in the first non-trivial case of interest for a Schubert problem on an arbitrary Grassmannian. We do not yet know how to address the general form of the secant conjecture with our methods.

%We mention that when $\nu = \rectangle$ (and $n = d(m-d)$), \cref{thm:secant} remains true 
%if $I_1, \dots, I_n$ are cyclic intervals of
%$\RP^1$, i.e., one of the intervals is allowed to wrap
%around infinity.  The secant conjecture is sometimes phrased in this way.
%This follows from \cref{thm:secant} as stated, using the $\PGL_2$-action on $\Gr(d,m)$.

\subsection{The disconjugacy conjecture}

Suppose that $V$ is a $d$-dimensional vector space of real analytic functions, 
defined on an interval $I\subseteq\RR$.  Disconjugacy is concerned with 
the question of how many zeros a function in $V$ can have.
By linear algebra, there always exists a nonzero function 
$f \in V$ such that $f$ has at least $d-1$ zeros on $I$.  We say that $V$ is \defn{disconjugate}
on $I$ if every nonzero function in $V$ has at most $d-1$ 
zeros on $I$ (counted with multiplicities). Disconjugacy has long been studied because it is related to explicit solutions for linear differential equations; see \cite{coppel71}, as well as \cite[Section~4.1]{karp} and the references therein.

It is not always straightforward to decide if $V$ is disconjugate on $I$.
However, a necessary condition is that $\Wr(V)$ has no zeros on $I$.
This is because $\Wr(V)$ has a zero at $w$ if and only if
there exists a nonzero $f \in V$ such that $f$ has a zero at $w$ of 
multiplicity at least $d$.  In general, the converse is false; for example, $V = \langle \cos u, \sin u \rangle$ is not disconjugate on $I = \RR$, and $\Wr(V) = 1$. Eremenko \cite{eremenko15,eremenko19} conjectured that the converse statement is actually correct
under very special circumstances.  This is known as the 
\defn{disconjugacy conjecture}, which we state now as a theorem:

\begin{theorem}[Disconjugacy conjecture]
\label{thm:disconj}
Let $V \subseteq \RR[u]$ be a finite-dimensional vector space of polynomials such that
$\Wr(V)$ has only real zeros.  Then $V$ is disconjugate on
every interval which avoids the zeros of $\Wr(V)$.
\end{theorem}

The disconjugacy conjecture was previously verified in the case that $\dim(V) \le 2$ \cite{eremenko_gabrielov_shapiro_vainshtein06} (cf.\ \cite[p.~341]{eremenko15}).

\subsection{Positivity conjectures}

A $d$-plane $V \in \Gr(d,m)$ is called \defn{totally nonnegative} if all of its Pl\"ucker coordinates are real and nonnegative (up to rescaling). Similarly, $V$ is called \defn{totally positive in $\scellnu$} if $V\in\scellnu$ and all of its \Plucker coordinates which are not trivially zero on $\scellnu$ are positive, i.e.,
\begin{equation}
\label{eq:positivenupluckers}
\Delta^\lambda > 0 \;\text{ for all } \lambda\subseteq\nu \qquad \text{and} \qquad \Delta^\lambda = 0 \;\text{ for all } \lambda\not\subseteq\nu\,.
\end{equation}
For example, each element $V\in\Gr(2,4)$ from \cref{ex:main} is totally nonnegative if and only if $z_1, z_2 \ge 0$, and is totally positive in its Schubert cell if and only if $z_1, z_2 > 0$.

The totally nonnegative part of $\Gr(d,m)$ is a totally nonnegative partial flag variety in the sense of Lusztig \cite{lusztig94,lusztig98} (see \cite[Section~1]{bloch_karp23} for further discussion), and was studied combinatorially by Postnikov \cite{postnikov06}. Total positivity in Schubert cells was considered by Berenstein and Zelevinsky \cite{berenstein_zelevinsky97}. These and similar totally positive spaces have been extensively studied in the past few decades, with connections to representation theory \cite{lusztig94}, combinatorics \cite{postnikov06}, cluster algebras \cite{fomin_williams_zelevinsky}, soliton solutions to the KP equation \cite{kodama_williams14}, scattering amplitudes \cite{arkani-hamed_bourjaily_cachazo_goncharov_postnikov_trnka16}, Schubert calculus \cite{knutson14}, topology \cite{galashin_karp_lam22}, and many other topics.

Mukhin--Tarasov and Karp conjectured that the reality statements from \cref{sec:shapiroshapiro,sec:secantconjecture} have totally positive analogues. We verify this in slightly greater generality:
\begin{theorem}[Positive Shapiro--Shapiro conjecture]
\label{thm:positive}
Let $z_1, \dots, z_n$ and $W_1, \dots, W_n$ be as in \cref{thm:ssc}.
\begin{enumerate}[label=(\roman*)]
\item\label{positive1} If $z_1, \dots, z_n \in [0,\infty)$, then all solutions to the Schubert
problem \eqref{eq:schubertproblem} are real and totally nonnegative.

\item\label{positive2} If $z_1, \dots, z_n \in (0,\infty)$, then
all solutions to the Schubert problem \eqref{eq:schubertproblem} 
are real and totally positive in $\scellnu$.
\end{enumerate}
\end{theorem}

\begin{theorem}[Positive secant conjecture, divisor form]
\label{thm:positivesecant}
Let $I_1, \dots, I_n$ and $W_1, \dots, W_n$ be as in \cref{thm:secant}.
\begin{enumerate}[label=(\roman*)]
\item\label{positivesecant1} If $I_1, \dots, I_n  \subseteq [0,\infty)$, 
then there are exactly $\numsyt{\nu}$ distinct solutions to the Schubert
problem \eqref{eq:schubertproblem}, and all solutions are real and totally nonnegative.

\item\label{positivesecant2} If $I_1, \dots, I_n \subseteq (0,\infty)$, 
then there are exactly $\numsyt{\nu}$ distinct solutions to the Schubert
problem \eqref{eq:schubertproblem}, and all solutions are real and totally positive in $\scellnu$.
\end{enumerate}

\end{theorem}

In the special case $\nu = \rectangle$, \cref{thm:positive}\ref{positive1} was conjectured by Evgeny Mukhin and Vitaly Tarasov in 2017, and \cref{thm:positive,thm:positivesecant} were conjectured independently in \cite{karp}.

\subsection{Relationships between conjectures}

We now explain how the conjectures stated in this section are related to each other, and why they follow from our main result \cref{thm:main}. We have already noted that the divisor form of the secant conjecture (\cref{thm:secant}) implies the Shapiro--Shapiro conjecture (\cref{thm:ssc}). Eremenko showed that the disconjugacy conjecture (\cref{thm:disconj}) implies the divisor form of the secant conjecture; in fact, his motivation was to generalize the argument used to prove the $m-d \le 2$ case of the secant conjecture \cite[Section~3]{eremenko_gabrielov_shapiro_vainshtein06}. Moreover, it was shown in \cite{karp} using topological arguments that the four statements in \cref{thm:positive,thm:positivesecant} in the case $\nu=\rectangle$ are all pairwise equivalent, and that they are moreover equivalent to the disconjugacy conjecture. We can similarly show that \cref{thm:positive} implies \cref{thm:positivesecant}. Therefore to prove all of these statements, it suffices to establish \cref{thm:positive}. This is a direct consequence of \cref{thm:main}; we briefly sketch the argument (see \cite[Section~1]{karp_purbhoo} for the details).

If $W \in \Gr(m-d,m)$ osculates $\gamma$ at $w\in\CC$, then $V \cap W \neq \{0\}$ if and only if $-w$ is a zero of $\Wr(V)$. Hence in the setting of \cref{thm:positive}, the Schubert problem~\eqref{eq:schubertproblem} is equivalent to $V\in\scellnu$ and $\Wr(V) = g$, where $g(u) = (u+z_1) \dotsm (u+z_n)$. By \cref{thm:main}\ref{main_eigenspace}, we can write any such solution $V$ as $V_E$ for some eigenspace $E \subseteq \spechtnu$ of $\bethenu$. This means that the \Plucker coordinates $[\Delta^\lambda : \lambda \subseteq \rectangle]$ of $V$ are the eigenvalues of the operators $\beta^\lambda$ on $E$. If $z_1, \dots, z_n \in [0,\infty)$, then one can show that each $\beta^\lambda$ is positive semidefinite. Therefore the eigenvalues of $\beta^\lambda$ are real and nonnegative, so $V$ is totally nonnegative. This proves part \ref{positive1} of \cref{thm:positive}. Similarly, if $z_1, \dots, z_n\in (0,\infty)$, then each $\beta^\lambda$ with $\lambda\subseteq\nu$ is positive definite, and hence has positive eigenvalues. This implies that $V$ is totally positive in $\scellnu$, proving part \ref{positive2}.

\acknowledgements{We thank Frank Sottile for helpful discussions.  Calculations for
this project were carried out using Sage~\cite{sagemath}; 
we thank Mike Zabrocki for assistance with some parts of the code.}

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