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\title{Schur polynomials, entrywise positivity preservers, and weak
majorization}
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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Apoorva
Khare\thanks{\href{mailto:khare@iisc.ac.in}{khare@iisc.ac.in}. A.K.~is
partially supported by a Young Investigator Award from the Infosys
Foundation, and a Ramanujan Fellowship and MATRICS Grant from SERB,
India.}\addressmark{1} \and Terence
Tao\thanks{\href{mailto:tao@math.ucla.edu}{tao@math.ucla.edu}. T.T.~was
supported by NSF grant DMS-1266164 and by a Simons Investigator
Award.}\addressmark{2}}
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\address{\addressmark{1}Indian Institute of Science; Analysis and
Probability Research Group; Bangalore, India \\
\addressmark{2}University of California at Los Angeles, USA}
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\received{\today}
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%\revised{}
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\abstract{We prove a monotonicity phenomenon for ratios of Schur
polynomials. In this we are motivated by -- and apply our result to --
understanding polynomials and power series that preserve positive
semidefiniteness (psd) when applied entrywise to psd matrices. We then
extend these results to classify polynomial preservers of total
positivity. As a further application, we extend a conjecture of Cuttler,
Greene, and Skandera (2011) to obtain a novel characterization of weak
majorization using Schur polynomials. Our proofs proceed through
a Schur positivity result of Lam, Postnikov, and Pylyavskyy (2007), and
computing the leading terms of Schur polynomials.}
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%\resume{\lipsum[2]}
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\keywords{Schur polynomial, Cauchy--Binet formula, Schur positivity,
positive semidefinite matrix, totally positive matrix, weak majorization}
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\begin{document}
\maketitle
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%{{{1 Section 0 - History
\noindent \textit{Notation:}
Given integers $1 \leqslant k \leqslant N$ and a domain $I \subset
\mathbb{R}$, let $\bp_N^k(I)$ denote the positive semidefinite Hermitian
$N \times N$ matrices $A = (a_{jk})$, with all entries $a_{jk}$ in $I$
and rank at most $k$; and let $\bp_N(I) := \bp_N^N(I)$. We will mostly
be concerned with the case $I \subset [0,\infty)$.
A function $f : I \to \R$ acts \textit{entrywise} on matrices in
$\bp_N(I)$ via: $f[A] := (f(a_{jk}))_{j,k=1}^N$. We say $f$ is
\textit{entrywise positivity preserving}\footnote{We work with the
notions of positivity (i.e., positive semidefiniteness), total
positivity, and Schur positivity in this note.} if $f[A]$ is positive
semidefinite whenever $A$ is. The present work is motivated by the
classical question of classifying the entrywise positivity preserving
functions; this question has been studied for over a century.
By the Schur product theorem \cite{Schur1911}, and the fact that
$\bp_N(\R)$ is a convex closed cone, it follows that if $f : [0,\rho) \to
\R$ is of the form $f(x) = \sum_{k \geqslant 0}c_k x^k$, and $f$ is
\textit{absolutely monotonic} -- i.e., $c_k \geqslant 0\ \forall k$ --
then $f$ is entrywise positivity preserving on $\bp_N$ for all $N$. The
converse was proved for continuous functions by Schur's student,
Schoenberg:
\begin{theorem}[Schoenberg, \cite{Schoenberg42}]\label{Tschoenberg}
Suppose $f : (-1,1) \to \R$ is continuous and $f[-]$ preserves positivity
on $\bp_N((-1,1)$. Then $f$ is absolutely monotonic.
\end{theorem}
Schoenberg's theorem has subsequently been generalized to other settings,
including $(-\rho,\rho), [0,\rho),$ and $(0,\rho)$ for $0 < \rho
\leqslant \infty$, and the robustness of the absolute monotonicity
condition is valid in each of these settings. We mention the noteworthy
sequel by Rudin \cite{Rudin59}, who strengthened the result by showing
that (i)~the continuity hypothesis can be removed, and (ii)~one only
needs to preserve positivity on Toeplitz matrices of all dimensions.
Rudin was motivated by his work with Kahane (and Helson and Katznelson)
on preservers of Fourier--Stieltjes sequences for positive measures on
the torus.
\begin{remark}
In a parallel vein to Rudin's work, the recent work \cite{BGKP-hankel}
studied preservers of moment sequences for positive measures on the line.
As it shows, Theorem \ref{Tschoenberg} holds upon (i)~preserving
positivity on Hankel matrices of all dimensions, and (ii)~without the
continuity hypothesis.
\end{remark}
A natural and challenging mathematical refinement of the above problem is
to classify entrywise positivity preservers in \textit{fixed} dimension.
This problem has also attained modern relevance owing to its connections
to high-dimensional covariance estimation; for more details see the
discussion and references in \cite{BGKP-fixeddim}. While it has been the
subject of significant research, and (relatively straightforward)
characterizations are known for $N=2$, the problem remains open for all
$N \geqslant 3$.
%}}}
%{{{1 Section 1 - Polynomials preserving positive semidefiniteness
\section{Polynomials preserving positive semidefiniteness}
In this note we focus on the case of polynomials and power series $f(x) =
\sum_{k \geqslant 0} c_k x^k$ that preserve positivity on $\bp_N(I)$ for
fixed $N$. As we reveal, the study of positivity and its preservation by
such functions has remarkable connections to type $A$ representation
theory: Schur polynomials, Schur positivity, Gelfand--Tsetlin patterns,
and the Harish-Chandra--Itzykson--Zuber formula. For full proofs of the
results below, we refer the reader to the paper \cite{KT}, of which this
note is an extended abstract.
As mentioned above, a full classification of the entrywise endomorphisms
of $\bp_N$ for fixed $N$ remains elusive to date. Essentially the only
known necessary condition in fixed dimension is due to Horn, who in his
thesis \cite{horn} ascribes the result to his advisor, Loewner. The
result states that if $I = (0,\infty)$ and $f$ is entrywise positivity
preserving on $\bp_N(I)$ for $N \geqslant 3$, then $f \in C^{N-3}(I)$ and
$f$ has non-negative derivatives on $I$ of orders $0, \dots, N-3$. In
a similar vein, for polynomials and power series, a straightforward
Taylor series argument shows similar conclusions under weaker hypotheses
on the test sets:
\begin{lemma}[Horn-type necessary conditions]\label{horn-type}
Let $N \geqslant 2$ and $0 < \rho \leqslant \infty$, and $f(x) = \sum_{k
\geqslant 0} c_k x^k$ is a convergent power series on $(0,\rho)$.
\begin{itemize}
\item[(i)] (See \cite[Lemma 2.4]{BGKP-fixeddim}.)
If $f$ is entrywise positivity preserving on $\bp^1_N((0,\rho))$, and
$c_{n_0} < 0$ for some $n_0$, then $c_n > 0$ for at least $N$
values of $n < n_0$. (Thus the first $N$ non-zero Maclaurin coefficients
of $f$, if they exist, must be positive.)
\item[(ii)]
Suppose $\rho = \infty$, and $f$ is convergent on $(0,\infty)$ and
entrywise positivity preserving on $\bp^1_N((0,\infty))$. If $c_{n_0} <
0$ for some $n_0$, then $c_n > 0$ for at least $N$ values of $n < n_0$
and at least $N$ values of $n > n_0$. (Hence if $f$ is a polynomial, then
the first $N$ non-zero coefficients and the last $N$ non-zero
coefficients of $f$, if they exist, must be positive.)
\end{itemize}
\end{lemma}
The question now arises, if any other coefficient of a preserver $f$ can
be negative. (Certainly if no coefficient is negative then $f[-] : \bp_n
\to \bp_n\ \forall n \geqslant 1$ by the Schur product theorem, as
explained above.) Until very recently, not a single example was known of
a power series which preserved $\bp_N((0,\rho))$ entrywise for any $N
\geqslant 3$. The only such `atomic' examples were discovered in recent
work \cite{BGKP-fixeddim,BGKP-FPSAC}, for polynomials
\[
f(x) = x^r (c_0 + c_1 x + \cdots + c_{N-1} x^{N-1} + c_M x^M)
\]
with real coefficients (where $M \geqslant N \geqslant 3,\ r \in
\Z^{\geqslant 0}$), which preserve positivity on $\bp_N((0,\rho))$ for
bounded domains, i.e., $\rho < \infty$.
However, the methods used in \cite{BGKP-fixeddim, BGKP-FPSAC} provably do
not extend to any other set of (non-consecutive) powers; nor to the case
of unbounded domains. Consequently, there were no other examples known to
date, nor whether such examples can even exist. In particular, apart from
Lemma \ref{horn-type}(i), no other constraint on the coefficients was
known.
In this note, we not only address this gap and resolve it completely, but
more strongly, settle the question: \textit{What are all possible sign
patterns of power series that entrywise preserve positivity on
$\bp_N((0,\rho))$, for finite $\rho$?}\footnote{We also answer the
question for $\rho = \infty$ in the full paper \cite{KT}; here we
restrict ourselves to $\rho < \infty$.} Explicitly, we show:
\begin{theorem}\label{main-1}
Fix integers $N > 0$ and $0 \leqslant n_0 < n_1 < \cdots < n_{N-1}$, as
well as a sign $\epsilon_M \in \{-1,0,+1\}$ for each $M > n_{N-1}$. Given
positive reals $0 < \rho < \infty$ and $c_{n_0},\dots,c_{n_{N-1}}$, there
exists a convergent power series on $(0,\rho)$
that is an entrywise positivity preserver on $\bp_N((0,\rho))$:
\[
f(x) = c_{n_0} x^{n_0} + c_{n_1} x^{n_1} + \dots + c_{n_{N-1}}
x^{n_{N-1}} + \sum_{M > n_{N-1}} c_M x^M,
\]
such that $c_M$ has the sign of $\epsilon_M$ for every $M > n_{N-1}$.
\end{theorem}
Theorem \ref{main-1} shows that the necessary Horn-type condition in
Lemma \ref{horn-type}(i) is sharp.\footnote{Similarly for the unbounded
domain case $\rho = \infty$, we show in \cite{KT} that the necessary
condition in Lemma \ref{horn-type}(ii) is also sharp, i.e.~the only
restriction in the possible sign patterns.} In particular, it
demonstrates the existence of polynomials and power series that preserve
positivity on $\bp_N((0,\rho))$ but not on $\bp_{N+1}((0,\rho))$.
To prove Theorem \ref{main-1}, we first present a `fewnomial' version
that is equivalent. Namely, suppose we can show the result for exactly
one $c_M < 0$ and all other $c_M = 0$ for $M > n_{N-1}$, i.e.,
\begin{align*}
\textit{There exists } c_M &\ \textit{ with the same sign as } \epsilon_M
\textit{ such that }\\
f_M(x) &\ := \sum_{j=0}^{N-1} c_{n_j} x^{n_j} + c_M x^M \textit{
entrywise preserves positivity on } \bp_N((0,\rho)).
\end{align*}
Then by suitably choosing each $c_M$ and rescaling $f_M$ by
$2^{n_{N-1}-M}$, and adding together the rescaled functions (together
with the Schur product theorem), one obtains the desired power series in
Theorem \ref{main-1}.
Thus, it suffices to prove the above polynomial version. The following
result achieves this goal; and moreover, obtains the exact threshold
bound for $c_M < 0$.
\begin{theorem}\label{Treal-rank1}
Fix integers $N > 0$ and $0 \leqslant n_0 < n_1 < \cdots < n_{N-1} < M$,
as well as a real scalar $0 < \rho < \infty$.
Given real scalars $c_{n_0}, \dots, c_{n_{N-1}}, c_M$, define
$\displaystyle f(x) := \sum_{j=0}^{N-1} c_{n_j} x^{n_j} + c_M x^M$.
Then the following are equivalent:
\begin{enumerate}
\item The function $f$ entrywise preserves positivity on rank-one
matrices in $\bp_N((0,\rho))$.
\item Either all $c_{n_j}, c_M \geqslant 0$; or
$c_{n_j} > 0\ \forall j$, and $c_M \geqslant -\mathcal{C}^{-1}$, where
\begin{equation}\label{Esharp}
\mathcal{C} = \sum_{j=0}^{N-1} \frac{V(\bn_j)^2}{V(\bn)^2}
\frac{\rho^{M - n_j}}{c_{n_j}}.
\end{equation}
\noindent Here $\bn := (n_0, \dots, n_{N-1})^T$, $\bn_j :=
(n_0, \dots, n_{j-1}, n_{j+1}, \dots, n_{N-1}, M)^T$, and given a vector
${\bf t} = (t_0, \dots, t_{k-1})^T$, its `Vandermonde determinant' is
$V({\bf t}) := \prod_{1 \leqslant i < j \leqslant k} (t_j - t_i)$.
\item The function $f$ entrywise preserves positivity on
$\bp_N([0,\rho])$.
\end{enumerate}
\end{theorem}
Theorem \ref{Treal-rank1} provides a quantitative form of Schoenberg's
theorem in fixed dimension. As mentioned above, it moreover provides the
first examples (for non-consecutive powers $n_j$) of polynomials
preserving positivity on $\bp_N$ but not on $\bp_{N+1}$. Also note
that the threshold $\mathcal{C}$ in \eqref{Esharp} is attained on the
boundary of the cone, on rank-one matrices.
We conclude this section by recording the following strengthening of
Theorem \ref{main-1}. First note that Theorem \ref{Treal-rank1} can be
reformulated as a linear matrix inequality:
\begin{equation}\label{am}
A^{\circ M} \preceq {\mathcal C} \sum_{j=0}^{N-1} c_{n_j} A^{\circ n_j},
\qquad \forall A \in \bp_N([0,\rho]),
\end{equation}
where $\preceq$ denotes the positive semidefinite ordering: $A \preceq B
\Leftrightarrow B-A \in \bp_N(\R)$. The sharp bound $\mathcal{C}$ is also
tight enough, to enable generalizing \eqref{am} to arbitrary power
series:
\begin{cor}[Analytic functions]\label{Tanalytic}
Notation as in Theorem \ref{Treal-rank1}.
Given a power series $g(x) = \sum_{M > n_{N-1}} g_M x^M$ that is
convergent at $\rho$, there exists a finite threshold $\mathcal{K}$
such that the function
$\displaystyle x \mapsto \mathcal{K} \sum_{j=0}^{N-1} c_{n_j} x^{n_j} -
g(x)$
is entrywise positivity preserving on $\bp_N([0,\rho])$, i.e.,
\begin{equation}\label{amp}
g[A] \preceq {\mathcal K} \sum_{j=0}^{N-1} c_{n_j}
A^{\circ n_j}, \qquad \forall A \in \bp_N([0,\rho]).
\end{equation}
\end{cor}
\noindent The proof can be found in \cite[Section 3]{KT}.
%}}}
%{{{1 Section 2 - Schur positivity and ratios of Schur polynomials
\section{Schur positivity and ratios of Schur polynomials}
This section is devoted to proving Theorem \ref{Treal-rank1}. The key
ingredient in the proof, missing from previous work in the literature, is
the use of Schur polynomials; in some sense, the classification of sign
patterns starts with the Schur product theorem, and comes back full
circle to Schur, via Schur polynomials and Schur positivity.
We now set notation. Fix an integer $N>0$, and define $\bn_{\min}$ to be
the vector $(0,1,\dots,N-1)^T$. Given a vector $\bn = (n_0, \dots,
n_{N-1})^T$ of strictly increasing non-negative integers, define the
Schur polynomial $s_\bn$ in the vector $\bu = (u_1, \dots, u_N)^T$ by the
formula
\begin{equation}\label{sdef}
s_\bn(\bu) := \sum_T \prod_{j=0}^{N-1} u_{j+1}^{a_j},
\end{equation}
where $T$ runs over all column-strict Young tableaux of shape $(n_{N-1} -
(N-1), \dots, n_0 - 0)$ and with cell entries $1,\dots,N$, and $a_j$ is
the number of occurrences of $j$ in the tableau $T$. Schur polynomials
are homogeneous symmetric polynomials of total degree $\sum_j n_j -
\binom{N}{2}$, and are characters of irreducible representations of
$\mathfrak{sl}_N(\mathbb{C})$. We now mention two further properties of
Schur polynomials that will be of use below: (i) their relation to
generalized Vandermonde determinants, and (ii) the `Weyl Dimension
Formula':
\begin{equation}\label{Ewdf}
\det (\bu^{\circ n_0} | \cdots | \bu^{\circ n_{N-1}}) = s_\bn(\bu)
V(\bu), \qquad s_\bn((1,\dots,1)^T) = \frac{V(\bn)}{V(\bn_{\min})}.
\end{equation}
As we explain below, a key step in the proof of Theorem \ref{Treal-rank1}
is the following monotonicity phenomenon for Schur polynomials, which is
interesting for additional reasons explained in the final section.
\begin{proposition}\label{Pschur-ratio}
Fix integers $0 \leqslant n_0 < \cdots < n_{N-1}$
and $0 \leqslant m_0 < \cdots < m_{N-1}$, such that $n_j \leqslant m_j\
\forall j$. Define the function $f : (0,\infty)^N \to \R$ via:
$f(\bu) := \displaystyle \frac{s_\bm(\bu)}{s_\bn(\bu)}$.
Then $f$ is non-decreasing in each coordinate.
\end{proposition}
We show that this result is in fact the analytical shadow of a deeper,
algebraic Schur positivity phenomenon. To show the result, by
the quotient rule and symmetry it suffices to show that the polynomial
$P_{\bm,\bn}(\bu) := s_\bn \cdot \partial_{u_1}(s_\bm) - s_\bm \cdot
\partial_{u_1}(s_\bn)$
sends $(0,\infty)^N$ to $[0,\infty)$. This is assured if $P_{\bm,\bn}$ is
a $\Z^{\geqslant 0}$-linear combination of monomials,
i.e.~monomial-positive. Even stronger: note that expanding $s_\bn$ as a
polynomial in $u_1$, the coefficient of $u_1^k$ is a skew-Schur
polynomial $s_{\bn/ (k)}((u_2,\dots,u_N)^T)$; and similarly for
$s_\bm(\bu)$. (See \cite[Chapter I.5]{Macdonald} for details and further
properties.) Now we claim:
\begin{proposition}
Writing
$P_{\bm,\bn}(\bu) := s_\bn \cdot \partial_{u_1}(s_\bm) - s_\bm \cdot
\partial_{u_1}(s_\bn)$
as a polynomial in $u_1$, the coefficient of every power of $u_1$ is {\em
Schur positive}, i.e., a non-negative integer linear combination of Schur
polynomials in $u_2, \dots, u_N$.
\end{proposition}
Clearly, this result implies Proposition \ref{Pschur-ratio} by the above
discussion.
\begin{proof}[Sketch of proof]
Write $\bu' := (u_2, \dots, u_N)^T \in \R^{N-1}$ and the vector
$(0,\dots,0,1)^T \in \Z^N$ as ${\bf e}_N$. Then $s_\bn(\bu) = \sum_{j
\geqslant 0} s_{(\bn - \bn_{\min}) / j {\bf e}_N}(\bu')$, where the
right-hand summand is understood to vanish whenever $n_{N-1} - (N-1) <
j$. Similarly one writes out $s_\bm(\bu)$ in terms of skew-Schur
polynomials. Now a symmetrization procedure shows that
\begin{align*}
P_{\bm,\bn}(\bu) =
\sum_{k > j \geqslant 0} \left( s_{(\bn - \bn_{\min}) / j {\bf
e}_N}(\bu') s_{(\bm - \bn_{\min}) / k {\bf e}_N}(\bu') - s_{(\bn -
\bn_{\min})/ k {\bf e}_N}(\bu') s_{(\bm - \bn_{\min})/j {\bf
e}_N}(\bu') \right)\\
\times (k-j) u_1^{j+k-1}.
\end{align*}
Thus it suffices to show that each summand is Schur positive when $k>j$
and $\bm - \bn_{\min}$ dominates $\bn - \bn_{\min}$ coordinatewise, where
$\bn_{\min} = (0,\dots,N-1)$ as above. But this is a special case of a
Schur positivity result by Lam, Postnikov, and Pylyavskyy \cite[Theorem
4]{LPP}. Notice that if $n_{N-1} - (N-1) < k$ then the `negative
coefficient' summand above vanishes, and then the Schur positivity of
$s_{(\bn - \bn_{\min}) / j {\bf e}_N}(\bu') s_{(\bm - \bn_{\min}) / k
{\bf e}_N}(\bu')$ already follows from the Littlewood--Richardson rule
\cite[Chapter I, Equations (5.2), (5.3)]{Macdonald}, since it implies
that skew-Schur polynomials are Schur positive, whence so are their
products.
\end{proof}
With Proposition \ref{Pschur-ratio} in hand, we now complete the proof of
Theorem \ref{Treal-rank1}.
\begin{defn}\label{Dwedge}
For $S \subset \R$ a subset, $S^N_<$ comprises all vectors in $S^N$ with
pairwise distinct coordinates that are sorted in increasing order.
\end{defn}
\begin{proof}[Sketch of proof of Theorem \ref{Treal-rank1}]
Clearly $(3) \implies (1)$.
Next, suppose $(1)$ holds. If $c_{n_j}, c_M$ are not all non-negative,
then $c_{n_j} > 0\ \forall j$ by Lemma \ref{horn-type}(i). We claim:
\begin{lemma}
Suppose $\F$ is a field, and $h(x) = \sum_{n \in S} c_n x^n \in \F[x]$ is
a polynomial, with $S \subset \Z^{\geqslant 0}$ having at least $n$
elements. Then for any $\bu, \bv \in (\F^N)^T$, we have:
\begin{equation}\label{Ecauchybinet}
\det h[\bu \bv^T] = \sum_{\bn \in S^N_<} s_\bn(\bu) s_\bn(\bv) V(\bu)
V(\bv) \prod_{n \in \bn} c_n.
\end{equation}
\end{lemma}
\begin{proof}
Write $S = \{ n_1, \dots, n_K \}$ where the $n_j$ are in increasing
order. Then,
\[
h[\bu \bv^T] = \sum_{j=1}^K c_{n_j} \bu^{\circ \bn_j} (\bv^{\circ
\bn_j})^T = (\bu^{\circ n_1} | \dots | \bu^{\circ n_K})
{\rm diag}(c_{n_1}, \dots, c_{n_K}) (\bv^{\circ n_1} | \dots | \bv^{\circ
n_K}).
\]
Now \eqref{Ecauchybinet} follows from the Cauchy--Binet formula.
\end{proof}
The next step is to reformulate hypothesis $(1)$, by assuming $c_M < 0$.
Set $t := |c_M|^{-1}$ and define
\begin{equation}
p_t(x) := t \, h(x) - x^M, \qquad \text{where} \qquad h(x) :=
\sum_{j=0}^{N-1} c_{n_j} x^{n_j}.
\end{equation}
Then hypothesis $(1)$ in the theorem is equivalent to assuming that
$p_t[\bu \bu^T]$ is positive semidefinite for $\bu \in
(0,\sqrt{\rho})^N$, and hypothesis $(2)$ seeks to find the smallest
threshold $t$ that works for all such vectors $\bu$.
Our approach is to (a) first produce the optimal threshold for a single
vector $\bu$ (i.e., matrix $\bu \bu^T$), and then to (b) maximize over a
suitable set of vectors $\bu$ to obtain the constant in \eqref{Esharp}.
The first of these steps will follow from the following basic result.
\begin{lemma}\label{Lupdate}
Fix a vector $\bw \in \R^N$ and a positive definite (real symmetric)
matrix $H$. Define $P_t := t H - \bw \bw^T$ for $t \in \R$. Then the
following are equivalent:
\begin{enumerate}
\item $P_t$ is positive semidefinite.
\item $\det P_t \geqslant 0$.
\item $\displaystyle t \geqslant \bw^T H^{-1} \bw =
1 - \frac{\det (H - \bw \bw^T)}{\det H}$.
\end{enumerate}
\end{lemma}
%\begin{proof}
%Clearly $(1) \implies (2)$. To show $(2) \implies (3)$, expand the
%expression $\det \begin{pmatrix} 1 & -\bw^T \\ \bw & tH \end{pmatrix}$ in
%two ways using Schur complements, to get:
%\[
%(1 - t^{-1} \bw^T H^{-1} \bw) \det tH = \det (tH - \bw \bw^T).
%\]
%Specializing to $t=1$ proves the equality in $(3)$; the inequality is
%also immediate from the above equality. Finally, suppose $(3)$ holds, and
%set $\bv := H^{-1/2} \bw$. Then,
%\[
%H^{-1/2} P_t H^{-1/2} = t \, {\rm Id} - \bv \bv^T,
%\]
%so $P_t$ is positive semidefinite if and only if $t \geqslant \bv^T \bv =
%\bw^T H^{-1} \bw$.
%\end{proof}
Returning to the proof of $(1) \implies (2)$ in the theorem, define the
vectors
\begin{equation}\label{Evector}
\bu(\epsilon) := (1, \epsilon, \dots, \epsilon^{N-1})^T, \quad
\bu_\epsilon := \sqrt{\rho \epsilon} \, \bu(\epsilon), \qquad \epsilon
\in (0,1).
\end{equation}
Now apply Lemma \ref{Lupdate} with $H := \sum_{j=0}^{N-1} c_{n_j}
(\bu_\epsilon \bu_\epsilon^T)^{\circ n_j}$ and $\bw :=
\bu_\epsilon^{\circ M}$, noting via Vandermonde determinants that $H$ is
nonsingular for all $\epsilon \in (0,1)$. Expanding the formula in Lemma
\ref{Lupdate}(3) and using \eqref{Ewdf}, it follows that $t$ must exceed
the quantity
\[
1 - \prod_{j=0}^{N-1} c_{n_j}^{-1}
V(\bu_\epsilon)^{-2} s_\bn(\bu_\epsilon)^{-2} \left( \prod_{j=0}^{N-1}
c_{n_j} V(\bu_\epsilon)^2 s_\bn(\bu_\epsilon)^2 - \sum_{j=0}^{N-1}
\prod_{k \neq j} c_{n_k} V(\bu_\epsilon)^2 s_{\bn_j}(\bu_\epsilon)^2
\right)
\]
for every $\epsilon \in (0,1)$. This expression simplifies to yield:
\[
t \geqslant \sup_{\epsilon \in (0,1)} \sum_{j=0}^{N-1}
\frac{s_{\bn_j}(\bu_\epsilon)^2}{c_{n_j} s_\bn(\bu_\epsilon)^2}.
\]
Applying Proposition \ref{Pschur-ratio}, the supremum is attained as
$\epsilon \to 1^-$, and by \eqref{Ewdf} yields precisely the constant
$\mathcal{C}$ in \eqref{Esharp}. This proves $(2)$.
Conversely, assuming $(2)$, the proof of $(1)$ proceeds similarly using
Lemma \ref{Lupdate}. By continuity and symmetry, it suffices to prove
$p_t[\bu \bu^T] \in \bp_N$ for all $\bu \in (0,\sqrt{\rho})^N$ with
strictly increasing coordinates, if $t \geqslant \mathcal{C}$ as in
\eqref{Esharp}. But again by Proposition \ref{Pschur-ratio}, for such $t$
it follows that
\[
t \geqslant \sup_{\bu \in (0,\sqrt{\rho})^N_<} \sum_{j=0}^{N-1}
\frac{s_{\bn_j}(\bu)^2}{c_{n_j} s_\bn(\bu)^2}.
\]
Repeating (in reverse order) the arguments following \eqref{Evector},
the assertion $(1)$ follows.
Finally, the proof of $(1) \implies (3)$ uses the following `extension
principle':
\begin{theorem}
Fix $0 < \rho \leqslant \infty$ and a continuously differentiable
function $h : (0,\rho) \to \R$. If $h$ and $h'$ are entrywise positivity
preserving on $\bp_N^1((0,\rho))$ and $\bp_{N-1}((0,\rho))$ respectively,
then $h$ does the same on all of $\bp_N((0,\rho))$.
\end{theorem}
We refer the reader to \cite[Section 3]{KT} for more on this result
and how it completes the proof of Theorem \ref{Treal-rank1}.
\end{proof}
We conclude this section by remarking that the recent works
\cite{BGKP-fixeddim,BGKP-FPSAC} prove Theorem \ref{Treal-rank1} in the
special case when $\bn = \bn_{\min}$, i.e., $n_j = j\ \forall 0 \leqslant
j < N$. In that case, the hypotheses in the theorem are in fact
equivalent to:
\begin{enumerate}
\item[$(3')$] \textit{The entrywise map $f$ preserves positivity on
$\bp_N(D(0,\rho))$, where $D(0,\rho)$ is the complex disc centered at the
origin and of radius $\rho \in (0,\infty)$; and $\bp_N$ here denotes
complex Hermitian positive semidefinite matrices.}
\end{enumerate}
However, as we discuss in \cite[Section 7]{KT}, such a general statement
necessarily does not hold if $\bn$ is not an integer translate of
$\bn_{\min}$. In fact for \textit{every} $\bn \neq n_0 + \bn_{\min}$,
there are infinitely many $M > n_{N-1}$ for which $f[-]$ fails to
preserve positivity on $\bp_N(D(0,\rho))$. As a simple example, even with
negative real entries one cannot have a `structured' classification of
sign patterns as in Lemma \ref{horn-type}(i). Consider the polynomials
\[
p_{k,t}(x) := t(1 + x^2 + \cdots + x^{2k}) - x^{2k+1}, \qquad
k \geqslant 0, \ t > 0,
\]
acting on $\bp_2((-\rho,\rho))$. Setting $\bu := (1,-1)^T$ and $A =
(\rho/2) \bu \bu^T \in \bp_2((-\rho,\rho))$, one computes: $\bu^T
p_{k/2}[A] \bu = -4(\rho/2)^{2k+1} < 0$. Consequently, $p_{k,t}$ is not
entrywise positivity preserving for any $t>0$ and integer $k \geqslant
1$.
Thus, the general problem is provably harder than the one studied in
\cite{BGKP-fixeddim,BGKP-FPSAC}, and new methods were required to solve
it.
\begin{remark}
If one is merely interested in classifying the sign patterns of
positivity preserving power series on $\bp_N((0,\rho))$, then the bound
in \eqref{Esharp} is stronger than what is required. As explained in
\cite[Section 3]{KT}, a more `qualitative' approach suffices to show
Theorem \ref{main-1}. The key result required is a (novel) `first-order
approximation' of every Schur polynomial -- see Proposition
\ref{Pleading}, which is used below to prove an extension of the
Cuttler--Greene--Skandera conjecture \cite{CGS}.
\end{remark}
\begin{remark}
In fact Theorem \ref{Treal-rank1} holds for all \textbf{real} powers that
lie in $\Z^{\geqslant 0} \cup [N-2,\infty)$ (for the reasons behind this
set of powers, see \cite{FitzHorn}). One proof involves first extending
Proposition \ref{Pschur-ratio} to real powers, i.e., using generalized
Vandermonde determinants. A second, `qualitative' proof involves
obtaining `first-order approximations' for such determinants
(generalizing the ones mentioned in the previous remark) using
Gelfand--Tsetlin polytopes and the Harish-Chandra--Itzykson--Zuber
integral. For details, see \cite[Sections 5, 8]{KT}.
\end{remark}
%}}}
%{{{1 Section 3 - Application 1: Polynomials preserving total positivity
\section{Application 1: Polynomials preserving total positivity}
A rectangular real matrix is \textit{totally positive} \cite{karlin} if
every minor is non-negative. Such matrices appear in representation
theory, discrete mathematics, stochastic processes, and other areas; for
instance, generalized Vandermonde matrices are (strictly) totally
positive.
It was recently shown in \cite{BGKP-hankel} that, in the spirit of
Schoenberg and Rudin's theorems, an entrywise map $f : [0,\infty) \to \R$
preserves total positivity on \textit{Hankel} matrices of all sizes, if
and only if $f|_{(0,\infty)}$ is absolutely monotonic and $0 \leqslant
f(0) \leqslant \lim_{\epsilon \to 0^+} f(\epsilon)$. Thus, totally
positive Hankel matrices serve as a `well-behaved' test set. (In fact,
the Schur product theorem also holds for this class of matrices.) In
contrast, if we work with the larger set of all symmetric (equivalently,
positive semidefinite) totally positive matrices, then preservers $f$ of
total positivity on this set are necessarily constant or linear; and this
holds even if we restrict to just the $5 \times 5$ symmetric matrices in
this set and $f$ analytic.
In light of these remarks, we work only with the Hankel totally positive
matrices with entries in an interval $I = (0,\rho)$ or $[0,\rho]$.
Denoting such sets by $\HTN_N(I)$, we have:
\begin{theorem}\label{Thtn1}
Notation as in Theorem \ref{Treal-rank1}. The following are equivalent.
\begin{enumerate}
\item The entrywise map $f[-]$ preserves total positivity on rank-one
matrices in $\HTN_N((0,\rho))$.
\item The entrywise map $f[-]$ preserves positivity on rank-one matrices
in $\HTN_N((0,\rho))$.
\item Either all $c_{n_j}, c' \geqslant 0$; or
$c_{n_j} > 0\ \forall j$, and $c' \geqslant -\mathcal{C}^{-1}$, where
$\displaystyle \mathcal{C} = \sum_{j=0}^{N-1} \frac{V(\bn_j)^2}{V(\bn)^2}
\frac{\rho^{M - n_j}}{c_{n_j}}$.
\item The entrywise map $f[-]$ preserves total positivity on
$\HTN_N([0,\rho])$.
\end{enumerate}
\end{theorem}
Thus, preserving total positivity on rank-one Hankel matrices in
$\bp_N((0,\rho))$ is the same as preserving positivity on this test set,
and also equivalent to the hypotheses in Theorem \ref{Treal-rank1}.
To prove the result, we make use of the following connection between
positive and totally positive Hankel matrices.
\begin{lemma}[{see \cite[Corollary 3.5]{FJS}}]\label{Lhtn}
Let $A_{N \times N}$ be a Hankel matrix. Then $A$ is totally positive
if and only if $A$ and its truncation $A^{(1)}$ have non-negative
principal minors. Here, $A^{(1)}$ denotes the submatrix of $A$ with the
first column and last row removed.
\end{lemma}
\begin{proof}[Sketch of proof of Theorem \ref{Thtn1}]
Clearly $(4) \implies (1) \implies (2)$. Now observe from the
computations following Lemma \ref{Lupdate} and equation \eqref{Evector}
that one only needs to work with the matrices $\bu_\epsilon
\bu_\epsilon^T$, and these are all rank-one matrices in
$\HTN_N((0,\rho))$ by Lemma \ref{Lhtn}. Similarly, Lemma
\ref{horn-type}(i) can be shown working only with $\bu_\epsilon
\bu_\epsilon^T$. Thus, $(2) \implies (3)$.
Finally, if $(3)$ holds, the map $f[-]$ preserves positivity on
$\bp_N([0,\rho))$ by Theorem \ref{Treal-rank1}. Now we are done by the
following `extension principle', which can be shown using Lemma
\ref{Lhtn}:
\textit{If $0 < \rho \leqslant \infty$ and $f : [0,\rho) \to \R$
entrywise preserves positivity on $\bp_N([0,\rho))$, then $f[-]$
preserves total positivity on $\HTN_N([0,\rho)) \cap \bp_N([0,\rho))$.}
\end{proof}
\begin{remark}
The extension principle in the proof just above, allows one to also
classify the sign patterns of all power series that preserve total
positivity in fixed dimension, for both bounded and unbounded domains.
These results follow from their counterparts for entrywise positivity
preservers; see \cite[Section 9]{KT} for details.
\end{remark}
%}}}
%{{{1 Section 4 - Application 2: The CGS conjecture, and weak majorization via Schur polynomials
\section{Application 2: The Cuttler-Greene-Skandera conjecture, and
weak majorization via Schur polynomials}
Given a real vector $\bu = (u_1, \dots, u_N)^T$, denote its decreasing
rearrangement by $u_{[1]} \geqslant \cdots \geqslant u_{[N]}$. We say
$\bu$ \textit{weakly majorizes} $\bv$ for vectors $\bu, \bv \in \R^N$ --
and write $\bu \succ_w \bv$ -- if
\begin{equation}\label{Eweak}
\sum_{j=1}^k u_{[j]} \geqslant \sum_{j=1}^k v_{[j]}, \ \forall 0 < k < N,
\qquad \sum_{j=1}^N u_{[j]} \geqslant \sum_{j=1}^N v_{[j]}.
\end{equation}
If moreover the final inequality is an equality, we say $\bu$
\textit{majorizes} $\bv$.
We begin by recalling a conjecture by Cuttler, Greene, and Skandera
\cite[Conjecture 7.4]{CGS}, which says that given $\bm, \bn \in
(\Z^{\geqslant 0})^N_<$ (see Definition \ref{Dwedge}),
\begin{equation}\label{Ecgs}
\frac{s_\bm(\bu)}{s_\bn(\bu)} \geqslant
\frac{s_\bm((1,\dots,1)^T)}{s_\bn((1,\dots,1)^T)},
\qquad \forall \bu \in (0,\infty)^N,
\end{equation}
if $\bm$ majorizes $\bn$. The conjecture was very recently proved in
\cite{Sra} and also in \cite{Blossom2}.
In a parallel direction, observe that if $\bm$ dominates $\bn$
coordinatewise, and $\bm \neq \bn$, then \eqref{Ecgs} cannot hold at
points $\epsilon (1,\dots,1)^T$ for $\epsilon \in (0,1)$, by homogeneity
considerations. However, \eqref{Ecgs} holds on $[1,\infty)^N$, as an
immediate corollary of Proposition \ref{Pschur-ratio}:
\begin{equation}
m_j \geqslant n_j\ \forall j \quad \implies \quad
\frac{s_\bm(\bu)}{s_\bn(\bu)} \geqslant
\frac{s_\bm((1,\dots,1)^T)}{s_\bn((1,\dots,1)^T)} =
\frac{V(\bm)}{V(\bn)}, \qquad \forall \bu \in [1,\infty)^N.
\end{equation}
A common unification of both of these settings is thus a natural question
-- restricting to $\bu \in [1,\infty)^N$ as above. The aforementioned
works \cite{Blossom2, CGS, Sra} all assume $\sum_j m_j = \sum_j n_j$;
replacing this by an inequality allows us to achieve the desired common
generalization, and to show the converse. In fact, this
\textit{characterizes} weak majorization, for real tuples:
\begin{theorem}\label{Tcgs}
Given vectors $\bm, \bn \in (\R^{\geqslant 0})^N_<$ , we have
\begin{equation}\label{Ecgs-revised}
\frac{\det (\bu^{\circ m_0} | \dots | \bu^{\circ m_{N-1}})}{V(\bm)}
\geqslant \frac{\det (\bu^{\circ n_0} | \dots | \bu^{\circ
n_{N-1}})}{V(\bn)}, \qquad \forall \bu \in [1,\infty)^N
\end{equation}
if and only if $\bm$ weakly majorizes $\bn$.
\end{theorem}
While the integer tuple case was our main motivation, it is more
convenient to work with the more general real tuples in (one half of) the
proof. The (other half of the) proof proceeds through a `first-order
approximation' of generalized Vandermonde determinants; here we write
down its special case for Schur polynomials.
\begin{proposition}\label{Pleading}
Fix integers $N > 0$ and $0 \leqslant n_0 < \cdots < n_{N-1}$, and
scalars $0 \leqslant u_1 \leqslant \cdots \leqslant u_N$. With $\bn, \bu$
as in \eqref{sdef}, we have the following two sharp inequalities:
\begin{equation}\label{Eleading}
1 \times \bu^{\bn - \bn_{\min}} \leqslant s_\bn(\bu) \leqslant
\frac{V(\bn)}{V(\bn_{\min})}
\times \bu^{\bn - \bn_{\min}}.
\end{equation}
\end{proposition}
\begin{proof}
By \eqref{Ewdf}, $s_\bn(\bu)$ is the sum of
$\frac{V(\bn)}{V(\bn_{\min})}$ monomials, one of which equals $\bu^{\bn -
\bn_{\min}}$, and this dominates all other monomials. The sharpness can
be shown by using the principal specialization of the Weyl Character
Formula \cite[Chapter I.3]{Macdonald}.
\end{proof}
\begin{proof}[Sketch of proof of Theorem \ref{Tcgs}]
Suppose \eqref{Ecgs-revised} holds. Define the
partial sums of $\bm, \bn$:
\[
\widetilde{n}_j := n_{N-j} + \cdots + n_{N-1}, \qquad
\widetilde{m}_j := m_{N-j} + \cdots + m_{N-1}, \qquad
0 \leqslant j \leqslant N-1.
\]
\noindent Now fix $j$ and set $\bu = \bu(t) :=
(1,\dots,N-j,(N-j+1)t,\dots,Nt)$ for $t \in [1,\infty)$. Using
\eqref{Ecgs-revised} and Proposition \ref{Pleading}, we compute for all
$t \geqslant 1$:
\[
t^{\widetilde{n}_j} \prod_{k=1}^N k^{n_{k-1}} = \bu^\bn
\leqslant \bu^{\bn_{\min}} s_\bn(\bu)
\leqslant \bu^{\bn_{\min}} \frac{V(\bn)}{V(\bm)} s_\bm(\bu)
\leqslant \frac{V(\bn) \bu^\bm}{V(\bn_{\min})} = t^{\widetilde{m}_j}
\frac{V(\bn)}{V(\bn_{\min})} \prod_{k=1}^N k^{m_{k-1}}.
\]
We infer from taking $t \to \infty$ that the growth rate
$\widetilde{m}_j$ of the right-hand side must dominate that on the left,
which is $\widetilde{n}_j$. Therefore $\bm$ weakly majorizes
$\bn$.\footnote{A similar argument, based on a generalization of
Proposition \ref{Pleading}, works for real powers $\bm, \bn$.}
Conversely, suppose $\bm, \bn$ are non-negative real vectors with $\bm
\succ_w \bn$. Given $\bu \in [1,\infty)^N$, define $F_\bu : [0,\infty)^N
\to \R$ to be the Harish-Chandra--Itzykson--Zuber integral:
\begin{equation}\label{Eweakmaj}
F_\bu(\bm) := \int_{U(N)} \exp \mathrm{tr} \left(
\mathrm{diag}(m_0,\dots,m_{N-1}) U
\mathrm{diag}(\log(u_1),\dots,\log(u_N)) U^* \right) \ dU.
\end{equation}
By continuity, it suffices to show \eqref{Ecgs-revised} for $\bu \in
(1,\infty)^N_<$. If we show $F_\bu(\bm) \geqslant F_\bu(\bn)$, then we
would be done by the Harish-Chandra--Itzykson--Zuber formula. Now this
property of $F_\bu$ follows from \cite[Chapter 3, C.2.d]{MOA}. See
\cite[Section 10]{KT} for details.
\end{proof}
%}}}
%{{{1 Bibliography
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography
\end{document}
\begin{thebibliography}{88}
\bibitem{Blossom2}
R.~Ait-Haddou and M.-L.~Mazure.
\newblock The fundamental blossoming inequality in Chebyshev spaces--I:
Applications to Schur functions.
\newblock \href{http://doi.org/10.1007/s10208-016-9334-8}{\em Found.
Comput. Math.}, in press, 2017.
\bibitem{BGKP-fixeddim}
A.~Belton, D.~Guillot, A.~Khare, and M.~Putinar.
\newblock Matrix positivity preservers in fixed dimension. I.
\newblock \href{http://dx.doi.org/10.1016/j.aim.2016.04.016}{\em Adv.
Math.}, 298:325--368, 2016.
\bibitem{BGKP-FPSAC}
A.~Belton, D.~Guillot, A.~Khare, and M.~Putinar.
\newblock Schur polynomials and positivity preservers.
\newblock FPSAC 2016 Proceedings (DMTCS), Volume BC,
pages 155--166, 2016.
\bibitem{BGKP-hankel}
A.~Belton, D.~Guillot, A.~Khare, and M.~Putinar.
\newblock Moment-sequence transforms.
\newblock {\em Preprint},
\href{http://arxiv.org/abs/1610.05740}{arXiv:1610.05740}, 2016.
\bibitem{CGS}
A.~Cuttler, C.~Greene, and M.~Skandera.
\newblock Inequalities for symmetric means.
\newblock \href{http://dx.doi.org/10.1016/j.ejc.2011.01.020}{\em European
J. Combin.}, 32(6):745--761, 2011.
\bibitem{FJS}
S.~Fallat, C.R.~Johnson, and A.D.~Sokal.
\newblock Total positivity of sums, Hadamard products and Hadamard
powers: Results and counterexamples.
\newblock \href{http://dx.doi.org/10.1016/j.laa.2017.01.013}{\em Linear
Algebra Appl.}, 520:242--259, 2017.
\bibitem{FitzHorn}
C.H.~Fitz{G}erald and R.A.~Horn.
\newblock On fractional {H}adamard powers of positive definite matrices.
\newblock \href{http://dx.doi.org/10.1016/0022-247X(77)90167-6}{\em J.
Math. Anal. Appl.}, 61(3):633--642, 1977.
\bibitem{horn}
R.A.~Horn.
\newblock The theory of infinitely divisible matrices and kernels.
\newblock \href{http://dx.doi.org/10.1090/S0002-9947-1969-0264736-5}{\em
Trans. Amer. Math. Soc.}, 136:269--286, 1969.
\bibitem{karlin}
S.~Karlin.
\newblock {\em Total positivity. Vol. I}.
\newblock Stanford University Press, Stanford, California 1968.
\bibitem{KT}
A.~Khare and T.~Tao.
\newblock On the sign patterns of entrywise positivity preservers in
fixed dimension.
\newblock {\em Preprint},
\href{http://arxiv.org/abs/1708.05197}{arXiv:1708.05197}, 2017.
\bibitem{LPP}
T.~Lam, A.~Postnikov, and P.~Pylyavskyy.
\newblock Schur positivity and Schur log concavity.
\newblock \href{http://www.jstor.org/stable/40068109}{\em Amer. J.
Math.}, 129(6):1611--1622, 2007.
\bibitem{Macdonald}
I.G.~Macdonald.
\newblock {\em Symmetric functions and Hall polynomials}.
\newblock Oxford University Press, 1995.
\newblock With contributions by A. Zelevinsky, Oxford Science Publications.
\bibitem{MOA}
A.W.~Marshall, I.~Olkin, and B.C.~Arnold.
\newblock {\em Inequalities: Theory of majorization and its applications}
(2nd edition).
\newblock Springer Series in Statistics, Springer, 2011.
\bibitem{Rudin59}
W.~Rudin.
\newblock Positive definite sequences and absolutely monotonic functions.
\newblock \href{http://dx.doi.org/10.1215/S0012-7094-59-02659-6}{\em Duke
Math. J}, 26(4):617--622, 1959.
\bibitem{Schoenberg42}
I.J.~Schoenberg.
\newblock Positive definite functions on spheres.
\newblock \href{http://dx.doi.org/10.1215/S0012-7094-42-00908-6}{\em Duke
Math. J.}, 9:96--108, 1942.
\bibitem{Schur1911}
J.~Schur.
\newblock {B}emerkungen zur {T}heorie der beschr{\"a}nkten
{B}ilinearformen mit unendlich vielen {V}er{\"a}nderlichen.
\newblock \href{http://dx.doi.org/10.1515/crll.1911.140.1}{\em J.~reine
angew.~Math.}, 140:1--28, 1911.
\bibitem{Sra}
S.~Sra.
\newblock On inequalities for normalized Schur functions.
\newblock \href{http://doi.org/10.1016/j.ejc.2015.07.005}{\em European J.
Combin.}, 51:492--494, 2016.
\end{thebibliography}
%}}}
\end{document}