Tropicalizing Principal Minors of Positive Definite Matrices

Abeer Al Ahmadieh School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA. [email protected] , Felipe Rincón School of Mathematical Sciences, Queen Mary University of London, London, UK. [email protected] , Cynthia Vinzant School of Mathematics, University of Washington, Seattle, WA, USA. [email protected] and Josephine Yu School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA. [email protected]

Abstract.

We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of $M$ -concave functions on the discrete $n$ -dimensional cube. We show it coincides with the intersection of the affine tropical flag variety with the submodular cone. In particular, any cell in the regular subdivision of the cube induced by a point in this tropicalization can be subdivided into base polytopes of realizable matroids. We also use this tropicalization as a guide to discover new algebraic inequalities among the principal minors of positive definite matrices of a fixed size.

1. Introduction

For an $n\times n$ matrix $A$ and subsets $S,T\subset[n]=\{1,2,\dots,n\}$ of the same size, we will denote by $A(S,T)$ the determinant of the submatrix of $A$ with rows indexed by $S$ and columns indexed by $T$ . For a square matrix $A$ , we use $A_{S}$ to denote the principal minor $A(S,S)$ . A real symmetric or Hermitian matrix is called positive definite if all of its principal minors are positive.

The tropicalization of a subset $\mathcal{S}\subset\mathbb{R}_{+}^{N}$ is defined as the logarithmic limit set as in [Ale13]:

\operatorname{trop}(\mathcal{S})=\lim_{t\rightarrow\infty}\{(\log_{t}(x_{1}),\dots,\log_{t}(x_{N})):x\in\mathcal{S}\}.

It is a way to track the exponential behavior of a set as coordinates go to $0$ or $\infty$ . When $\mathcal{S}$ is semialgebraic, the tropicalization is a rational polyhedral fan that coincides with the closure of the image of $\mathcal{S}_{\mathbb{R}\{\!\{t\}\!\}}$ under coordinate-wise valuation, where $\mathcal{S}_{\mathbb{R}\{\!\{t\}\!\}}$ denotes the extension of $\mathcal{S}$ to the field of real Puiseux series. See Section 2.1 for further discussion.

The principal minors of an $n\times n$ positive definite matrix $A$ can be encoded by the degree- $n$ homogeneous polynomial in $n+1$ variables

f_{A}=\det({\rm diag}(x_{1},\ldots,x_{n})+yA)=\sum_{S\subseteq[n]}A_{S}\cdot{\bf x}^{[n]\backslash S}y^{|S|}\in\mathbb{R}[x_{1},\dots,x_{n},y],

where ${\bf x}^{T}:=\prod_{i\in T}x_{i}$ . This polynomial is stable [BB06], which implies that the tropicalization of the set of principal minors of positive definite matrices is a subset of the set of $M^{\natural}$ -concave functions on $\{0,1\}^{n}$ [Brä10]. We see in Corollary 4.3 that for $n\geq 6$ not every $M^{\natural}$ -concave function arises this way from positive definite matrices. Our main result is a description of this tropicalization in terms of the affine tropical flag variety.

Theorem 4.1.

The tropicalization of the set of principal minors of $n\times n$ positive definite real symmetric (resp. Hermitian) matrices equals the intersection of the affine tropical flag variety over $\mathbb{R}$ (resp. $\mathbb{C}$ ) with the cone of submodular functions on the hypercube $\{0,1\}^{n}$ .

The study of the tropical flag variety was initiated by Haque in $2012$ [Haq12]. Recently, Brandt, Eur, and Zhang described it in terms of flag matroid polytope subdivisions [BEZ21]. The totally non-negative part of the tropical flag variety was studied in [Bor22].

An $M^{\natural}$ -concave function induces a regular subdivision of the unit cube $\{0,1\}^{n}$ that coarsens a subdivision in which every cell is a dehomogenized matroid polytope, i.e., a 0/1 polytope with edges of the form $e_{i}-e_{j}$ or $e_{i}$ contained in a layer $\{x\in[0,1]^{n}:k\leq\sum_{i}x_{i}\leq k+1\}$ for some $k$ . When the $M^{\natural}$ -concave function arises as the tropicalization of the principal minors of a positive definite matrix, we show that these matroid polytopes correspond to realizable matroids.

Theorem 4.5.

The regular subdivision of $[0,1]^{n}$ induced by the tropicalization of the principal minors of a real symmetric (or Hermitian) positive definite matrix is a coarsening of a subdivision of $[0,1]^{n}$ into dehomogenized matroid polytopes of matroids realizable over $\mathbb{R}$ (or $\mathbb{C}$ , respectively).

Understanding determinantal inequalities for positive definite matrices is a classical topic of study, see for example [JB93, CFB95], and it continues to be a topic of interest [HJ08, Cho16, JZC19, LS20, DWH22, BS23]. In [HJ08], for instance, Hall and Johnson use cone theoretic techniques to characterize the semigroup of ratios of products of principal minors over all positive definite matrices.

Any tropical polynomial inequality valid on the tropicalization of a semialgebraic set $\mathcal{S}$ can be lifted to a polynomial inequality valid on the set $\mathcal{S}$ itself [JSY22]. We use our description of the tropicalization of the set $\mathcal{S}$ of principal minors of positive definite matrices as a guide to understanding the possible polynomial inequalities valid on $\mathcal{S}$ ; in particular, we study lifts of the $M^{\natural}$ -concavity constraints to polynomial inequalities on $\mathcal{S}$ .

Example 1.1.

Let $\mathrm{PD}_{3}$ be the set of $3\times 3$ real symmetric positive definite matrices, and let $\mathcal{S}_{3}\subset\mathbb{R}_{+}^{2^{[3]}}$ be the image of $\mathrm{PD}_{3}$ under the map sending a matrix $A$ to its vector of principal minors $(A_{S})_{S\subset[3]}$ . Consider the set $\operatorname{cone}(\mathcal{S}_{3})=\{\lambda\,x\mid\lambda\in\mathbb{R}_{+}\text{ and }x\in\mathcal{S}_{3}\}$ , where the coordinate indexed by the subset $\emptyset$ is not required to be equal to $1$ . Theorem 4.1 says that $\operatorname{trop}(\operatorname{cone}(\mathcal{S}_{3}))$ consists of all vectors $w\in\mathbb{R}^{2^{[3]}}$ that satisfy the submodular inequalities and the tropical incidence relation

(1)

\max(w_{1}+w_{23},w_{2}+w_{13},w_{3}+w_{12})\text{ is attained at least twice}.

Consider now the projection $\mathcal{P}$ of the semialgebraic set $\operatorname{cone}(\mathcal{S}_{3})$ onto the six coordinates indexed by the subsets $1,2,3,12,13,23$ . The set $\mathcal{P}$ is a full-dimensional semialgebraic set of $\mathbb{R}_{+}^{6}$ ; indeed, the six $1\times 1$ and $2\times 2$ principal minors of a positive definite matrix are algebraically independent. However, its tropicalization $\operatorname{trop}(\mathcal{P})$ satisfies the tropical equation (1), and thus it is only a $5$ -dimensional polyhedral subset of $\mathbb{R}^{6}$ . This tropical equation cannot be lifted to an algebraic equation satisfied by $\mathcal{P}$ , but, as we explore in Section 5, it is instead a consequence of $\mathcal{P}$ satisfying the following three algebraic inequalities:

A_{1}A_{23}+A_{2}A_{13}\geq\frac{1}{2}A_{3}A_{12},\quad A_{1}A_{23}+A_{3}A_{12}\geq\frac{1}{2}A_{2}A_{13},\quad A_{2}A_{13}+A_{3}A_{12}\geq\frac{1}{2}A_{1}A_{23}.

Other inequalities of this form with different coefficients, which also hold for all Lorentzian polynomials, are presented in Theorem 5.1. $\diamond$

Organization

The paper is organized as follows. We introduce relevant background and definitions in Section 2, including various characterizations of $M^{\natural}$ -concave functions and their relation to the affine flag Dressian. In Section 3, we discuss a connection between the tropicalization of the flag variety and a slice of the $(n,2n)$ Grassmannian over arbitrary valuated fields. In Section 4, we specialize these results to real closed and algebraically closed fields of characteristic zero and show that the resulting sets coincide with the tropicalization of the positive definite cone under the principal minor map. In Section 5, we explore consequences for polynomial inequalities on the principal minors of positive definite matrices. Finally, in Section 6, we prove a technical lemma used in Section 3.

Acknowledgements

We are grateful to Jonathan Boretsky, H. Tracy Hall, Yassine El Maazouz, and Bernd Sturmfels for helpful discussions and insights. CV is partially supported by the NSF DMS grant #2153746. JY is partially supported by the NSF DMS grants #1855726 and #2348701.

2. Definitions and Background

2.1. Tropicalization

A (nonarchimedean) valuation on a field $\mathcal{K}$ is a map $\operatorname{val}:\mathcal{K}^{*}\rightarrow\mathbb{R}$ satisfying

\operatorname{val}(ab)=\operatorname{val}(a)+\operatorname{val}(b),\text{ and }\operatorname{val}(a+b)\geq\min(\operatorname{val}(a),\operatorname{val}(b)).

Here we denote $\mathcal{K}^{*}=\mathcal{K}\setminus\{0\}$ . The image $\Gamma=\operatorname{val}(\mathcal{K}^{*})$ of $\operatorname{val}$ is an additive subgroup of $\mathbb{R}$ called the value group. The valuation is non-trivial if $\Gamma\neq\{0\}$ . In order to use the max convention, we will take $\nu:\mathcal{K}^{*}\to\mathbb{R}$ to be $\nu(a)=-\operatorname{val}(a)$ so that

	$\displaystyle\nu(ab)$	$\displaystyle=\nu(a)+\nu(b)$
	$\displaystyle\nu(a+b)$	$\displaystyle\leq\max(\nu(a),\nu(b)).$

We use $\mathbbm{k}$ to denote the residue field of $\mathcal{K}$ , which is the quotient of the valuation ring $\mathcal{O}=\{a\in\mathcal{K}:\operatorname{val}(a)\geq 0\}$ by its unique maximal ideal $\mathfrak{m}=\{a\in\mathcal{K}:\operatorname{val}(a)>0\}$ .

Throughout the paper, we use $\mathcal{K}$ to denote a field with a nontrivial valuation with an infinite residue field $\mathbbm{k}$ . The valuation has a splitting or a cross-section if there is a group homomorphism $\phi$ from the value group $\Gamma$ to the multiplicative group $\mathcal{K}^{*}$ such that $\operatorname{val}\circ\,\phi$ is the identity on $\Gamma$ . If $\mathcal{K}$ is real closed or algebraically closed, then there is a splitting by [AGS20, Lemma 2.4] and [MS21, Lemma 2.1.15]. The splitting $\phi$ allows us to talk about the leading coefficient of an element $a\in\mathcal{K}^{*}$ as the element $\overline{t^{-\operatorname{val}(a)}a}$ in the residue field, where $t=\phi(1)$ . Typical examples include the fields of rational functions, Laurent series, or Puiseux series over a field $\mathbbm{k}$ . The Puiseux series field $\mathbbm{k}\{\!\{t\}\!\}$ is real closed if $\mathbbm{k}$ is real closed and is algebraically closed if $\mathbbm{k}$ is algebraically closed of characteristic $0$ .

Given a subset $\mathcal{S}\subset(\mathcal{K}^{*})^{n}$ , we define

\nu(\mathcal{S})=\{(\nu(x_{1}),\ldots,\nu(x_{n})):x\in\mathcal{S}\}.

We will denote by $\mathcal{R}$ a real closed field with a nontrivial valuation. Any real closed field has a unique total ordering $\geq$ compatible with the field operations, where $a\geq b$ if $a-b$ is a complete square. We will always assume that the valuation is compatible with the order, namely, if $a,b\in\mathcal{R}$ with $0<a<b$ , then $\nu(a)\leq\nu(b)$ . An example is the field of real Puiseux series $\mathbb{R}\{\!\{t\}\!\}$ , where a series is positive if its leading coefficient is positive.

Finally, we use $\mathcal{C}$ to denote the degree-two field extension $\mathcal{C}=\mathcal{R}(\mathbbm{i})$ where $\mathbbm{i}^{2}=-1$ . Since $\mathcal{R}$ is real-closed, $\mathcal{C}$ is algebraically closed. This comes with complex conjugation $\overline{a+\mathbbm{i}b}=a-\mathbbm{i}b$ where $a,b\in\mathcal{R}$ . The valuation on $\mathcal{R}$ naturally extends to $\mathcal{C}$ by $\nu(a+\mathbbm{i}b)=\max\{\nu(a),\nu(b)\}$ .

For a semialgebraic subset $\mathcal{S}\subset\mathbb{R}_{+}^{n}$ , let $\mathcal{S}_{\mathcal{R}}$ be the semialgebraic subset of $\mathcal{R}^{n}$ defined by the same semialgebraic expression (a first-order formula in the language of ordered rings) defining $\mathcal{S}$ . By [Ale13, JSY22], the image of $S_{\mathcal{R}}$ under coordinate-wise valuation coincides with the tropicalization of $\mathcal{S}$ (up to closure):

\operatorname{trop}(\mathcal{S})=\overline{\nu(\mathcal{S}_{\mathcal{R}})}

Here closure is taken in the Euclidean topology on $\mathbb{R}^{n}$ . In particular, this set does not depend on the choice of the extension $\mathcal{R}$ or the choice of semialgebraic expression defining $\mathcal{S}$ . Moreover we have $\operatorname{trop}(\mathcal{S})\cap\Gamma^{n}=\nu(\mathcal{S}_{\mathcal{R}})$ [Ale13, Theorem 4.2], and $\operatorname{trop}(\mathcal{S})$ is a union of polyhedra [AGS20, Theorem 3.1].

If $X\subset(\mathbb{C}^{*})^{n}$ is an algebraic variety, then its image under coordinate-wise absolute-value is a semialgebraic subset $|X|$ of $\mathbb{R}_{+}^{n}$ . The tropicalization of $X$ can be defined as the tropicalization of $|X|$ . We can similarly define $X_{\mathcal{C}}$ to be the algebraic variety of $\mathcal{C}^{n}$ defined by the vanishing of all polynomials that vanish on $X$ . A part of the fundamental theorem of tropical geometry states that the logarithmic limit set of $|X|$ coincides with the closure of the image of $X_{\mathcal{C}}$ under coordinate-wise valuation. That is, $\operatorname{trop}(X)=\overline{\nu(X)}$ . A proof can be found, for example, using [Ber71, Theorem 2] and [Gub13, Proposition 3.8].

A symmetric matrix over a real field $\mathcal{R}$ , or a Hermitian matrix over $\mathcal{C}$ , is called positive definite or PD if all of its principal minors are positive. Thus the principal minors give a map from the set of $n\times n$ matrices to $\mathcal{R}^{2^{[n]}}$ , sending $A\mapsto(A_{S})_{S\subset[n]}$ . The image of the sets of symmetric and Hermitian matrices have been studied in [HS07, Oed11, LS09, AAV24a, AAV24b] and it has many applications in probability and statistics, see for instance [EM22]. Let $\operatorname{PM}^{+}_{n}(\mathcal{R})$ and $\operatorname{PM}^{+}_{n}(\mathcal{C})\subset\mathcal{R}^{2^{[n]}}$ denote the image of the symmetric and Hermitian PD cone respectively under this principal minor map. Because the image of the principal minors of positive definite matrices is a semialgebraic set, the theorems above imply that

\operatorname{trop}(\operatorname{PM}^{+}_{n}(\mathbb{R}))=\overline{\nu(\operatorname{PM}^{+}_{n}(\mathcal{R}))}\text{ and }\operatorname{trop}(\operatorname{PM}^{+}_{n}(\mathbb{C}))=\overline{\nu(\operatorname{PM}^{+}_{n}(\mathcal{C}))}.

2.2. Discrete Concavity

For a set $S\subset[n]$ and an element $i\in[n]$ , let us use the shorthand $Si$ for $S\cup\{i\}$ and $S\backslash i$ for the set difference $S\backslash\{i\}$ . A function $F:2^{[n]}\rightarrow\mathbb{R}$ is submodular if for all $S,T\subseteq[n]$ ,

F(S\cap T)+F(S\cup T)\leq F(S)+F(T).

Submodularity is implied by the following “local” submodularity condition that for all $S\subset[n]$ and distinct $i,j\in[n]\setminus S$ ,

F(S)+F(Sij)\leq F(Si)+F(Sj).

A function $F:2^{[n]}\rightarrow\mathbb{R}$ induces two regular subdivisions of the unit cube $[0,1]^{n}$ by lifting each of the vertices of the cube by height $F$ in a new dimension, and then projecting back to $\mathbb{R}^{n}$ either the upper or the lower convex hull of the lift. The alcoved triangulation of a cube is the triangulation of the unit cube $[0,1]^{n}$ into $n!$ simplices where each simplex is the convex hull of an edge path from $(0,0,\dots,0)$ to $(1,1,\dots,1)$ whose sum of coordinates is monotonously increasing along the path. Equivalently, it is given by dicing the cube with the type- $A$ braid arrangement hyperplanes $x_{i}=x_{j}$ . This triangulation is “dual” to the permutohedron. Lovász showed that a function is submodular if and only if the regular subdivision of the unit cube induced by the lower convex hull is a coarsening of the alcoved triangulation [Lov83]. In particular, the edges in the lower convex hull correspond to pairs of subsets $S,T\subset[n]$ with $S\subset T$ . Thus the edges with directions $e_{i}-e_{j}$ cannot appear in the lower hull and can only appear in the upper hull.

The notions of $M$ -convex and $M^{\natural}$ -convex functions were introduced by Murota and collaborators [Mur98]. $M$ -concave functions are also called valuated polymatroids. A function $F:2^{[n]}\rightarrow\mathbb{R}$ is $M^{\natural}$ -concave if for all $S,T\subseteq[n]$ and all $i\in S\backslash T$ , either

•

$F(S)+F(T)\leq F(S\backslash i)+F(Ti)$ , or
•

there exists $j\in T\backslash S$ such that $F(S)+F(T)\leq F(Sj\backslash i)+F(Ti\backslash j)$ .

We will see some more characterizations of $M^{\natural}$ -concave functions below in Proposition 2.2.

Our interest in $M^{\natural}$ -concave functions comes from the following fact.

Proposition 2.1.

The tropicalization of the image of the positive definite cone under the principal minor map is a subset of the set of $M^{\natural}$ -concave functions on $\{0,1\}^{n}$ .

Proof.

If $A\in\mathbb{R}\{\!\{t\}\!\}^{n\times n}$ is a positive definite matrix then, using [BB06], the polynomial $f=\det({\rm diag}(x_{1},\ldots,x_{n})+A)=\sum_{S\subseteq[n]}A_{S}{\bf x}^{[n]\backslash S}$ is stable and has positive coefficients, so $(\nu(A_{S}))_{S\subseteq[n]}$ is an $M^{\natural}$ -concave function by [Brä10]. ∎

2.3. $M^{\natural}$ -concavity and valuated matroids

In this subsection we compile various characterizations of $M^{\natural}$ -concave functions in terms of valuated matroids and polytope subdivisions.

For integers $1\leq k\leq n-1$ , the (affine) Dressian $\operatorname{Dr}(k,n)$ is the polyhedral subset of $\mathbb{R}^{\binom{n}{k}}$ consisting of all functions $p:\binom{[n]}{k}\to\mathbb{R}$ such that for any subsets $S\in\binom{[n]}{k-1}$ and $T\in\binom{[n]}{k+1}$ ,

\max_{i\in T\setminus S}\,(p(Si)+p(T\backslash i))\quad\text{is attained at least twice}.

Points in the Dressian $\operatorname{Dr}(k,n)$ are, up to scaling, in one-to-one correspondence with rank- $k$ valuated matroids on the ground set $[n]$ or equivalently, $k$ -dimensional tropical linear spaces in $\mathbb{R}^{n}$ .

The affine flag Dressian $\operatorname{FlDr}(n)$ is the polyhedral subset of $\mathbb{R}^{2^{[n]}}$ consisting of all functions $p:2^{[n]}\to\mathbb{R}$ such that for any subsets $S,T\subset[n]$ with $|S|\leq|T|-2$ ,

\max_{i\in T\setminus S}(p(Si)+p(T\backslash i))\quad\text{is attained at least twice}.

When $|S|=|T|-2$ the conditions above are simply the Plücker relations on the points $(p(S))_{S\in\binom{[n]}{k}}$ with $k=|S|+1$ , while when $|S|<|T|-2$ they are incidence relations among the corresponding tropical linear spaces. Flag Dressians have been studied in [BEZ21].

For a function $F:2^{[n]}\to\mathbb{R}$ , its multisymmetric lift is the function $\hat{F}:\binom{[2n]}{n}\to\mathbb{R}$ given by $\hat{F}(T):=F(T\cap[n])$ . The homogenization of the $k$ to $k+1$ layer is the function $\tilde{F}_{k}:\binom{n+1}{k+1}\to\mathbb{R}$ given by

\tilde{F}_{k}(S):=\begin{cases}F(S)&\text{if }n+1\notin S\\ F(S\backslash\!(n+1))&\text{if }n+1\in S.\end{cases}

Proposition 2.2.

For a function $F:2^{[n]}\rightarrow\mathbb{R}$ , the following are equivalent.

(1)

The function $F$ is $M^{\natural}$ -concave.
(2)

The multisymmetric lift $\hat{F}$ belongs to the Dressian $\operatorname{Dr}(n,2n)$ .
(3)

The function $F$ is submodular and, for every $1\leq k\leq n-2$ , the homogenized layer $\tilde{F}_{k}$ belongs to the Dressian $\operatorname{Dr}(k+1,n+1)$ .
(4)

The function $F$ is submodular and belongs to the affine flag dressian $\operatorname{FlDr}(n)$ .
(5)

The function $F$ induces a regular subdivision of the unit cube $[0,1]^{n}$ (via upper hull) in which every edge has direction $e_{i}-e_{j}$ or $e_{i}$ .

Proof.

The equivalence $(1)\iff(2)$ follows from [GRSU24, Proposition 1.4].

The equivalence $(1)\iff(3)$ follows from [RvGP02, Theorem 10], which says that $F$ is $M^{\natural}$ -concave if and only if $F$ is submodular and for all $S\subset[n]$ and $i,j,k,l\in[n]\setminus S$ , each of the maxima

	$\displaystyle\max\{F(Sij)+F(Sk),F(Sik)+F(Sj),F(Sjk)+F(Si)\}$
	$\displaystyle\max\{F(Sij)+F(Sk\ell),F(Sik)+F(Sj\ell),F(Sjk)+F(Si\ell)\}$

is attained at least twice.

The equivalence $(3)\iff(4)$ follows from [BEZ21, Theorem 5.1.2].

The equivalence $(1)\iff(5)$ holds more generally for functions on subsets of $\mathbb{Z}^{n}$ . By [Mur98, Theorem 6.30], a function is $M$ -concave on a finite subset of $\mathbb{Z}^{n}$ with a fixed coordinate sum if and only if every cell in the regular subdivision induced by the function via the upper hull is an $M$ -convex set. By [Mur98, Theorem 4.15] and [MUWY18, Lemma 2.3], $M$ -convex sets are integer points in generalized permutohedra, which are polytopes with edges in directions $e_{i}-e_{j}$ . Projecting out one of the coordinates gives the desired result for $M^{\natural}$ -concave functions. ∎

Example 2.3.

Consider the matrix

A=\begin{pmatrix}1&1&1\\ 1&1+t^{4}&1+t^{3}\\ 1&1+t^{3}&1+t^{2}+t^{4}\end{pmatrix}=B^{T}B\ \ \text{ where }\ \ B=\begin{pmatrix}1&1&1\\ 0&t^{2}&t\\ 0&0&t^{2}\end{pmatrix}.

The matrix $A$ is positive definite since $A=B^{T}B$ with $\det(B)\neq 0$ . For $w_{S}=-\operatorname{val}(A_{S})$ , we have $w_{\emptyset}=w_{1}=w_{2}=w_{3}=0$ , $w_{12}=-4$ , $w_{13}=w_{23}=-2$ , and $w_{123}=-8$ . The function $S\mapsto w_{S}$ is $M^{\natural}$ -concave and induces the regular subdivision of the cube shown in Figure 1. $\diamond$

Figure 1. The regular subdivision induced by the matrix in Example 2.3.

We call a function $F:2^{[n]}\to\mathbb{R}$ strictly submodular if

F(S)+F(Sij)<F(Si)+F(Sj)

for all $S\subset[n]$ and distinct $i,j\in[n]\backslash S$ .

Lemma 2.4.

Let $F:2^{[n]}\to\mathbb{R}$ be $M^{\natural}$ -concave. The function $F$ is strictly submodular if and only if each cell of the regular subdivision of $[0,1]^{n}$ induced by $F$ via upper hull is contained in $\{x\in[0,1]^{n}:k\leq\sum_{i=1}^{n}x_{i}\leq k+1\}$ for some $k=0,\ldots,n-1$ .

Proof.

( $\Leftarrow$ ) Suppose that every cell of the subdivision induced by $F$ has the desired form. Consider $S\subset[n]$ with $|S|=k\leq n-2$ and $i,j\in[n]\backslash S$ . By assumption, no cell of the subdivision can contain both points $e_{S}$ and $e_{Sij}$ . Therefore, on the square with vertices $e_{S},e_{Si},e_{Sj},e_{Sij}$ , $F$ must induce the subdivision with an edge between $e_{Si}$ and $e_{Sj}$ . It follows that $F(Si)+F(Sj)>F(S)+F(Sij)$ .

( $\Rightarrow$ ) Suppose that $F$ is strictly submodular and suppose, for the sake of contradiction, that a cell $C$ of the subdivision induced by $F$ is not contained in a slice of the cube with the desired form. Let $S\subset[n]$ be of minimum size $|S|=k$ with $e_{S}\in C$ . Let $|x|$ denote $\sum_{i}x_{i}$ . By assumption, $\max\{|x|:x\in C\}\geq k+2$ . Since $e_{S}$ is a vertex of $C$ , there is some edge $[e_{S},e_{S}+v]$ of $C$ for which $|e_{S}+v|>|e_{S}|$ . By $M^{\natural}$ -concavity, this edge direction must be parallel to $e_{i}$ for some $i\in[n]\backslash S$ . Then $e_{Si}$ is also a vertex of $C$ . Since this does not achieve the maximum of $|x|$ over $C$ , by the same reasoning, there exists some $j\in[n]\backslash(Si)$ for which $e_{Sij}\in C$ . By $M^{\natural}$ -concavity, $[e_{S},e_{Sij}]$ cannot be an edge of $C$ , implying that $e_{Sj}\in C$ as well. The only way for these four points to belong to the same cell $C$ is for $F(Si)+F(Sj)=F(S)+F(Sij)$ , contradicting the strict submodularity of $F$ . ∎

Remark 2.5.

The assumption of $M^{\natural}$ -concavity is necessary in Lemma 2.4. For example, consider the function $F:2^{[4]}\to\mathbb{R}$ defined by $F(\emptyset)=F([4])=-6$ , $F(i)=-3$ for $i\neq 1$ and $F(1)=0$ , $F(S)=-1$ for $|S|=2$ , and $F(S)=-3$ for $|S|=3$ with $S\neq 234$ and $F(234)=0$ . One can check that $F$ is strictly submodular but that the edge $[e_{1},e_{234}]$ appears in the induced subdivision via the upper hull.

3. Tropical Grassmannians and Tropical Flag Varieties

In this section we will show a new relationship between tropical Grassmannians and tropical flag varieties, analogous to the equivalence $(2)\iff(4)$ in Proposition 2.2.

The (complete) flag variety is a subvariety of a product of projective spaces $\prod_{k=1}^{n}\mathbb{P}^{\binom{n}{k}-1}$ . An element $(p_{k})_{k\in[n]}$ belongs to the flag variety if there is a complete flag of linear spaces $\{0\}=L_{0}\subset L_{1}\subset L_{2}\subset\ldots\subset L_{n}=\mathcal{K}^{n}$ so that $p_{k}$ is the vector of Plücker coordinates of $L_{k}$ . The (complete) affine flag variety is the variety ${\rm Fl}_{\mathcal{K}}(n)\subset\mathcal{K}^{2^{n}}=\prod_{k=0}^{n}K^{\binom{n}{k}}$ consisting of vectors $(q_{S})_{S\subseteq[n]}$ which, when considered as a sequence $(q_{\varnothing}),(q_{S}:|S|=1),(q_{S}:|S|=2),\dots,(q_{S}:|S|=n-1),(q_{[n]})$ , belong to the flag variety. Each component $(q_{S}:|S|=k)$ can be scaled independently by nonzero elements of $\mathcal{K}$ . In particular we allow $(q_{\varnothing})$ and $(q_{[n]})$ to take arbitrary values in $\mathcal{K}^{*}$ .

Consider the Plücker embedding of the partial flag variety

\{(L_{k},L_{k+1})\in\operatorname{Gr}_{\mathcal{K}}(k,n)\times\operatorname{Gr}_{\mathcal{K}}(k+1,n):L_{k}\subset L_{k+1}\}.

Its image under the homogenizing map taking $q_{S}$ to $q_{S(n+1)}$ if $|S|=k$ and to $q_{S}$ if $|S|=k+1$ coincides with the Plücker embedding of $\operatorname{Gr}_{\mathcal{K}}(k+1,n+1)$ . An analogous homogenization appeared in Proposition 2.2(3). Concretely, if $L_{k}$ and $L_{k+1}$ are the span of the top $k$ and $k+1$ rows of a $(k+1)\times n$ matrix $M$ , then the corresponding subspace in $\operatorname{Gr}_{\mathcal{K}}(k+1,n+1)$ is obtained as the rowspan of the $(k+1)\times n$ matrix obtained by appending the column $e_{k+1}$ to $M$ .

Let $\nu({\rm Fl}_{\mathcal{K}}(n))\subset\Gamma^{2^{[n]}}\subset\mathbb{R}^{2^{[n]}}$ be the $\nu$ values of the points in the affine flag variety ${\rm Fl}_{\mathcal{K}}(n)$ , none of whose Plücker coordinates are zero. This set has lineality space containing the tropical scaling of each factor; that is, for any $w:2^{[n]}\to\mathbb{R}$ in $\nu({\rm Fl}_{\mathcal{K}}(n))$ and any vector of scalars $\lambda=(\lambda_{0},\ldots,\lambda_{n})\in\Gamma^{n+1}$ , the function $\lambda\cdot w:2^{[n]}\to\mathbb{R}$ given by

(2)

(\lambda\cdot w)(S)=\lambda_{|S|}+w(S)

also belongs to $\nu({\rm Fl}_{\mathcal{K}}(n))$ . The following lemma says that the tropical flag variety $\nu({\rm Fl}_{\mathcal{K}}(n))$ can be recovered from its intersection with the submodular cone using lineality.

Lemma 3.1.

For any $w:2^{[n]}\to\mathbb{R}$ , there are tropical scalars $\lambda=(\lambda_{0},\ldots,\lambda_{n})\in\Gamma^{n+1}$ so that $\lambda\cdot w$ is strictly submodular. Moreover, if $w$ is already submodular and $\Gamma$ is dense in $\mathbb{R}$ , then $\lambda$ can be chosen to be arbitrarily small.

Proof.

Let $\lambda_{0}=\lambda_{1}=0$ and for each $k=2,\ldots,n$ , inductively choose $\lambda_{k}$ so that

\lambda_{k}<\min_{\stackrel{{\scriptstyle S\in{\binom{[n]}{k}}}}{{i,j\in S}}}\ w(S\backslash i)+w(S\backslash j)+2\lambda_{k-1}-w(S\backslash\{i,j\})-w(S)-\lambda_{k-2},

which implies that $\lambda\cdot w$ is strictly submodular.

If $w$ is already submodular, then $w(S\backslash i)+w(S\backslash j)-w(S\backslash\{i,j\})-w(S)\geq 0$ . We can choose $\lambda_{0}=\lambda_{1}=0$ and $\lambda_{2}=-\varepsilon$ where $\varepsilon>0$ is arbitrarily small. For $k\geq 3$ we can choose $\lambda_{k}$ to satisfy $3\lambda_{k-1}<\lambda_{k}<\min\{0,2\lambda_{k-1}-\lambda_{k-2}\}$ . Inductively, this gives $|\lambda_{k}|<3^{k-1}\varepsilon$ , thus $\lambda$ can be arbitrarily small. ∎

Consider the linear subspace $\mathcal{L}_{n}$ of $\mathbb{R}^{\binom{[2n]}{n}}$ defined as

(3)

\mathcal{L}_{n}=\left\{p\in\mathbb{R}^{\binom{[2n]}{n}}:p(C)=p(D)\text{ whenever }C\cap[n]=D\cap[n]\right\}.

The linear subspace $\mathcal{L}_{n}$ is isomorphic to $\mathbb{R}^{2^{[n]}}$ via the map

\phi:\mathcal{L}_{n}\xrightarrow[]{\ \ \cong\ \ }\mathbb{R}^{2^{[n]}}

sending any $p\in\mathcal{L}_{n}$ to $\phi(p)\in\mathbb{R}^{2^{[n]}}$ given by $\phi(p)(S)=p(S\cup T)$ with $T$ any subset of $\{n+1,\dots,2n\}$ of size $n-|S|$ .

The equivalence $(2)\iff(4)$ in Proposition 2.2 can be rephrased as saying

{\rm Dr}(n,2n)\cap\mathcal{L}_{n}\ \ =\ \ {\rm FlDr}(n)\cap{\rm SUBMOD}_{n},

where ${\rm SUBMOD}_{n}$ denotes the cone of submodular functions on $\{0,1\}^{n}$ and we identify points in $\mathcal{L}_{n}$ with points in $\mathbb{R}^{2^{[n]}}$ as above. Our main result in this section, stated below, says that the realizable parts of these sets coincide. In the next section we show that these sets coincide with the tropicalization of the set of principal minors of positive definite matrices when $\mathcal{K}$ is equal to either $\mathcal{R}$ or $\mathcal{C}$ . However, the result holds for more general fields $\mathcal{K}$ and may be of independent interest.

Theorem 3.2.

Let $\mathcal{K}$ be a field with a nonarchimedean valuation with a splitting and an infinite residue field. For any positive integer $n$ we have

\nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}\ \ =\ \ \nu({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}.

If $\mathcal{K}$ is algebraically closed or real closed, we thus have

(4)

\operatorname{trop}({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}\ \ =\ \ \operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}.

Before providing the proof of our main result, we remark on the dimension of these polyhedral complexes.

Proposition 3.3.

If $\mathcal{K}$ is algebraically closed or real closed, the polyhedral complex in Equation (4) in Theorem 3.2 has dimension $\binom{n+1}{2}+1$ .

Proof.

The affine tropical flag variety $\operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))\subset\prod_{k=0}^{n}\mathbb{R}^{\binom{n}{k}}$ is invariant under the tropical scaling of each factor as in Equation (2). Therefore, by Lemma 3.1, any polyhedral cone in the affine tropical flag variety has the same dimension as its intersection with the submodular cone. Thus it suffices to show that $\operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))$ has dimension $\binom{n+1}{2}+1$ .

The flag variety has dimension $\binom{n}{2}$ , as a dense subset of it can be parametrized by upper triangular matrices with ones on the diagonal. Thus the affine flag variety ${\rm Fl}_{\mathcal{K}}(n)$ has dimension $\binom{n+1}{2}+1=\binom{n}{2}+n+1$ , accounting for the $n+1$ extra dimensions from scaling. If $\mathcal{K}$ is algebraically closed, this implies that the tropicalization $\operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))$ is a pure polyhedral complex of the same dimension $\binom{n+1}{2}+1$ by the Structure Theorem of Tropical Geometry.

If $\mathcal{K}$ is not algebraically closed, it is a priori possible that $\operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))$ is a (not necessarily pure) polyhedral complex of smaller dimension. However, when $\mathcal{K}$ is real closed, it was shown in [Bor22, Proposition $4.10^{\rm trop}$ ] that the tropicalization of the positive part ${\rm Fl}_{\mathcal{K}}^{>0}(n)$ of the flag variety, which is contained in $\operatorname{trop}({\rm Fl}_{\mathcal{K}}(n))$ , has dimension $\binom{n+1}{2}+1$ , by showing that the Marsh-Rietsch (bijective) parameterization $\mathcal{K}_{>0}^{\binom{n+1}{2}+1}\to{\rm Fl}_{\mathcal{K}}^{>0}(n)$ tropicalizes to a (bijective) parametrization $\mathbb{R}_{>0}^{\binom{n+1}{2}+1}\to\operatorname{trop}({\rm Fl}_{\mathcal{K}}^{>0}(n))$ . In [Bor22], the tropical flag variety is considered projectively, so the dimensions used there do not account for the $n+1$ extra dimensions from scaling. The map easily extends to the affine setting. ∎

Proof of Theorem 3.2( $\subseteq$ ).

Since both $\nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}$ and $\nu({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}$ are closed under global tropical scaling (i.e., adding a constant function), we may assume all the functions $w\in 2^{[n]}$ under consideration have $w(\varnothing)=0$ .

Let $M\in\mathcal{K}^{n\times 2n}$ be such that the valuation of its $n\times n$ minors belongs to $\mathcal{L}_{n}$ . The submatrix $M([n],\{n+1,\dots,2n\})$ has valuation $0$ by our assumption, so we can assume that $M$ has the form $\begin{pmatrix}B&I\end{pmatrix}$ for some $n\times n$ matrix ${B}$ and the $n\times n$ identity matrix $I$ . Since the valuation of the $n\times n$ minors of $M$ belongs to $\mathcal{L}_{n}$ , the valuation of the minor ${B}(T,S)$ is determined by $S$ independently of $T$ .

Consider the flag $L_{1}\subset L_{2}\subset\dots\subset L_{n}=\mathcal{K}^{n}$ where $L_{k}$ is the span of the first $k$ rows of ${B}$ . Its Plücker coordinates are given by the minors $({B}(\{1,\ldots,k\},S):S\in\binom{[n]}{k})$ , which are equal to the corresponding $n\times n$ minors of $M$ . This shows that the valuation of the $n\times n$ minors if $M$ belongs to $\nu({\rm Fl}_{\mathcal{K}}(n))$ . Moreover, since $\nu({\rm Gr}_{\mathcal{K}}(n,2n))$ is contained in the Dressian ${\rm Dr}(n,2n)$ , submodularity follows from the $(5)\implies(4)$ part of Proposition 2.2. ∎

For the other inclusion, we need the following technical lemmas. Recall that $\nu$ is the negative of the valuation.

Lemma 3.4.

Let $B\in\mathcal{K}^{n\times n}$ be an upper triangular matrix such that the function $2^{[n]}\rightarrow\mathbb{R}$ given by $S\mapsto\nu(B(\{1,\ldots,|S|\},S))$ is submodular. Then

(5)

\nu(B(\{1,\ldots,|S|\},S))\geq\nu(B(T,S))

for all $S,T\subset[n]$ with $|S|=|T|$ .

We defer the proof of Lemma 3.4 to Section 6. We will say that $B$ is top heavy if it is upper triangular and satisfies the condition (5).

Lemma 3.5.

Let $\mathcal{K}$ be a valued field with a splitting and an infinite residue field. Let $B,C\in\mathcal{K}^{n\times n}$ , $\tilde{B}=CB$ and suppose $C$ is generic with entry-wise valuation zero. Then the valuation of a minor of $\tilde{B}$ depends only on the choice of the columns, not on the choice of the rows. More precisely, for any subsets $S,T\subseteq[n]$ of size $|S|=|T|=k$ ,

\nu(\tilde{B}(T,S))=\max_{T^{\prime}\in\binom{[n]}{k}}\nu(B(T^{\prime},S)).

In particular, the negative valuation of the $n\times n$ minors of the matrix $\begin{pmatrix}\tilde{B}&I_{n}\end{pmatrix}$ belongs to $\nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}$ .

If, in addition, $B$ is a top heavy matrix as defined by (5) and $C$ is generic lower triangular with valuation zero for the nonzero entries, then

\nu(\tilde{B}(T,S))=\nu(B([k],S))

for all subsets $S,T\subseteq[n]$ of the same size $k$ .

Here “generic” means that the leading coefficients of the entries of $C$ lie in a nonempty Zariski open set over the residue field. We are defining the leading coefficient using the splitting. If $\mathcal{K}$ is a Puiseux series field over a ground field $\mathbbm{k}$ , then we can take $C$ to be a generic matrix over $\mathbbm{k}$ .

Proof.

Consider two $k\times k$ submatrices of $\tilde{B}$ on the same set of columns. We can transform one to the other by swapping two rows at a time, so let us assume that these two submatrices differ only at one row. Each row of $\tilde{B}$ is a generic linear combination of rows of $B$ . By linearity of determinant, each of the two minors is a generic linear combination of the same $n$ minors where the row being swapped is replaced by each of the original rows. By genericity there is no cancellation of leading terms so the linear combinations must have the same valuation, as desired.

Now suppose $B$ is top heavy and $C$ is lower triangular. Then each row of $\tilde{B}$ is a generic linear combination of a top-justified subset of rows of $B$ . When we repeatedly expand a $k\times k$ minor of $\tilde{B}$ using the linearity of determinants, we see that its valuation agrees with the valuation of the top justified minor by the top-heaviness of $B$ . ∎

Proof of Theorem 3.2( $\supseteq$ ).

Suppose that the function $w:2^{[n]}\to\mathbb{R}$ belongs to ${\nu}({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}$ . We will show that it belongs to $\nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}$ . Since both of these sets are closed under tropical scaling (adding a constant function to $w$ ), we may assume that $w(\varnothing)=0$ . Then

w(S)=\nu(B([k],S))-\lambda_{k},

where $B$ is an upper triangular $n\times n$ matrix over $\mathcal{K}$ and $\lambda_{1},\ldots\lambda_{n}$ are in the value group $\Gamma$ . The rows $v_{1},\ldots,v_{n}$ of $B$ represent the flag

{\rm span}\{v_{1}\}\subset{\rm span}\{v_{1},v_{2}\}\subset\dots\subset{\rm span}\{v_{1},\ldots,v_{n}\}=\mathcal{K}^{n}

in the flag variety and $\lambda_{1},\ldots\lambda_{n}$ results from scaling of the Plücker coordinates of each linear space in the flag. Since $w(S)$ is finite for every $S$ , the minors $B([k],[k])$ are nonsingular. After rescaling the rows $v_{1}\rightarrow t^{\lambda_{1}}v_{1}$ , $v_{2}\rightarrow t^{\lambda_{2}-\lambda_{1}}v_{2}$ , $v_{3}\rightarrow t^{\lambda_{3}-\lambda_{2}}v_{3},\dots,v_{n}\rightarrow t^{\lambda_{n}-\lambda_{n-1}}v_{n}$ we may assume that

w(S)=\nu(B([k],S)).

By Lemma 3.4, $B$ is top heavy. Let $\tilde{B}=LB$ where $L$ is a generic lower triangular matrix as in Lemma 3.5, so for any $S\subset[n]$ with $|S|=k$

w(S)=\nu(B([k],S))=\nu(\tilde{B}(T,S))

for all $T\in{\binom{[n]}{k}}$ by Lemma 3.5. Since $w(\varnothing)=0$ , the vector $w$ coincides with the $\nu$ values of the $n\times n$ minors of the matrix $(\tilde{B}~{}I)$ under the identification of $\mathbb{R}^{2^{[n]}}$ with the linear space $\mathcal{L}_{n}$ as in (3), and so $w$ belongs to $\nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}$ . ∎

Remark 3.6.

El Maazouz [EM22] defines the entropy map associated to a lattice $\Lambda=B\mathcal{O}^{n}\subset\mathcal{K}^{n}$ to be the function on $\{0,1\}^{n}$ defined by $I\mapsto\min_{|J|=|I|}\operatorname{val}(\det(B(I,J)))$ , which coincides with the formula in Lemma 3.5 above up to sign and transpose. Interestingly, in [EM22], this function is primarily considered over fields $\mathcal{K}$ with finite residue field, for which Lemma 3.5 does not apply. For fixed $n$ , one should be able to amend the arguments above to apply when the residue field $\mathbbm{k}$ is finite but sufficiently large.

Example 3.7 (Example 2.3 continued).

Consider the matrix $B$ from Example 2.3. One can check that the function $S\mapsto\nu(B([|S|],S)$ on $\{0,1\}^{3}$ is submodular. Since $B$ is also upper triangular, it follows from Lemma 3.4 that $B$ is top heavy. Consider a “generic” lower triangular matrix

C=\begin{pmatrix}1&0&0\\ 1&1&0\\ 2&1&1\end{pmatrix}\ \ \text{ with }\ \ \tilde{B}=CB=\begin{pmatrix}1&1&1\\ 1&1+t^{2}&1+t\\ 2&2+t^{2}&2+t+t^{2}\end{pmatrix}.

Then by the Cauchy-Binet identity,

	$\displaystyle\tilde{B}(23,23)$	$\displaystyle=C(23,12)B(12,23)+C(23,13)B(13,23)+C(23,23)B(23,23)$
		$\displaystyle=(-1)(t-t^{2})+(1)(t^{2})+(1)(t^{4})=-t+2t^{2}+t^{4}.$

In particular, $\nu(\tilde{B}(23,23))=-1$ is equal to the maximum $\nu$ value of a minor $B(ij,23)$ , which is achieved by $ij=12$ , since $B$ is top-heavy. Similarly, one can check that $\nu(\tilde{B}(ij,23))=-1$ for every pair $ij$ . $\diamond$

4. Tropicalizing principal minors of positive definite matrices

In this section we show that the tropicalization of the image of the positive definite cone under the principal minor map coincides with the subsets of tropical Grassmannians and tropical flag varieties studied in the previous section. As before we will use $\mathcal{R}$ and $\mathcal{C}$ to denote a real closed field and an algebraically closed field respectively, with nontrivial nonarchimedean valuation.

Theorem 4.1.

For any positive integer $n$ and a field $\mathcal{K}=\mathcal{R}$ or $\mathcal{C}$ , we have

\nu({\rm cone}(\operatorname{PM}^{+}_{n}(\mathcal{K}))\ \ =\ \ \nu({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}\ \ =\ \ \nu({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n},

where ${\rm cone}(\operatorname{PM}^{+}_{n}(\mathcal{K}))=\{\lambda a:\lambda\in\mathcal{K},a\in\operatorname{PM}^{+}_{n}(\mathcal{K})\}$ . Taking closures gives

\operatorname{trop}({\rm cone}(\operatorname{PM}^{+}_{n}(\mathcal{K}))\ \ =\ \ \operatorname{trop}({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}\ \ =\ \ {\rm trop}({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}.

Proof.

Let $\mathcal{K}=\mathcal{R}$ or $\mathcal{C}$ , and let $\mathbbm{k}$ denote its residue field. The equality of the last two sets follows from Theorem 3.2. We will now show that the first two sets are equal.

( $\subseteq$ ) Let $A$ be a positive definite matrix over $\mathcal{K}$ . Then $A=B^{*}B$ for some $B\in\mathcal{K}^{n\times n}$ . By Cauchy-Binet,

A(S,S)=\sum_{T\in\binom{[n]}{k}}B^{*}(S,T)B(T,S)=\sum_{T\in\binom{[n]}{k}}\overline{B(T,S)}B(T,S).

Since all of the terms in the first sum are nonnegative, the tropicalization ( $\nu$ values) of the sum is the tropical sum (maximum) of the tropicalization of the summands. Moreover, for any $a\in\mathcal{K}$ , $\nu(a)=\nu(\overline{a})$ , so $\nu(\overline{a}a)=2\nu(a)$ . All together this gives

(6)

\nu(A(S,S))=\max_{T\in\binom{[n]}{k}}2\nu(B(T,S)).

Let $U\in\mathbbm{k}^{n\times n}$ be generic and let $\tilde{B}=UB$ . Then it follows from (6) and Lemma 3.5 that

\operatorname{val}(A(S,S))=2\operatorname{val}(\tilde{B}(T,S))

for any $T\subset[n]$ with $|T|=|S|$ . The values $\nu(\tilde{B}(T,S))$ are tropicalizations of $n\times n$ minors of the $n\times 2n$ matrix $(\tilde{B}~{}I)$ , and they lie in the linear subspace $\mathcal{L}_{n}$ . Thus

\frac{1}{2}\operatorname{val}(A(S,S))\in\operatorname{trop}({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}.

However $\operatorname{trop}({\rm Gr}_{\mathcal{K}}(n,2n))$ and $\mathcal{L}_{n}$ are invariant under scaling, so we also have

\operatorname{val}(A(S,S))\in\operatorname{trop}({\rm Gr}_{\mathcal{K}}(n,2n))\cap\mathcal{L}_{n}.

( $\supseteq$ ) Suppose $M\in\mathcal{K}^{n\times 2n}$ is such that the $\nu$ values of the $n\times n$ minors of $M$ belong to $\mathcal{L}_{n}$ . We may assume that the valuation of the minor $M([n],\{n+1,\ldots,2n\})$ is zero, and that $M$ has the form $\begin{pmatrix}B&I\end{pmatrix}$ for some $n\times n$ matrix $B$ . Then the valuation of the minors $B(T,S)$ are independent of $T$ . Consider the positive definite matrix $A=B^{T}B$ . Then for any $S,T\subset[n]$ with $|S|=|T|=k$ , by (6) we have

\nu(B(T,S))=\frac{1}{2}\max_{T^{\prime}\in\binom{[n]}{k}}2\nu(B(T^{\prime},S))=\frac{1}{2}\nu(A(S,S)),

completing the proof. ∎

Corollary 4.2.

For $\mathcal{K}=\mathcal{R}$ or $\mathcal{C}$ , the set $\operatorname{trop}(\operatorname{PM}_{n}^{+}(\mathcal{K}))$ has dimension $\binom{n+1}{2}$ .

Proof.

The set $\operatorname{cone}(\operatorname{PM}_{n}^{+}(\mathcal{K}))$ is obtained from $\operatorname{PM}_{n}^{+}(\mathcal{K})$ by global scaling. Its dimension and the dimension of its tropicalization is therefore one more than the respective dimensions for $\operatorname{PM}_{n}^{+}(\mathcal{K})$ . The result then follows from Proposition 3.3. ∎

Corollary 4.3.

The set of tropicalized principal minors of $n\times n$ positive definite matrices is contained in the set of $M^{\natural}$ -concave functions on $2^{[n]}$ with value zero on $\varnothing$ . This containment is an equality for $n\leq 5$ , but it is strict for $n\geq 6$ .

Proof.

By Theorem 4.1, the set of tropicalized principal minors of $n\times n$ positive definite matrices coincides with the tropical flag variety inside the submodular cone. The $M^{\natural}$ -concave functions coincide with the flag Dressian by Proposition 2.2. By Section 5 of [BEZ21], the tropical flag variety is contained in the flag Dressian; this containment is an equality for $n\leq 5$ and is strict for $n\geq 6$ . This is still the case after intersecting with the submodular cone by Lemma 3.1. ∎

If $w\in\mathbb{R}^{2^{[n]}}$ , for each $k\in[n]$ let $w_{k}$ denote the restriction of $w$ to subsets of size $k$ . As discussed in Section 3, the fact that $\nu(\operatorname{PM}^{+}_{n}(\mathcal{K}))\subset\nu({\rm Fl}_{\mathcal{K}}(n))$ implies the following corollary.

Corollary 4.4.

For $w\in\nu(\operatorname{PM}^{+}_{n}(\mathcal{K}))$ , the homogenization of the restriction $(w_{k},w_{k+1})$ belongs to $\nu(\operatorname{Gr}_{\mathcal{K}}(k+1,n+1))$ for any $0\leq k\leq n-1$ .

Recall from Lemma 2.4 that if a function $w:2^{[n]}\rightarrow\mathbb{R}$ is $M^{\natural}$ -concave and strictly submodular, then each cell in the regular subdivision of $\{0,1\}^{n}$ induced by $w$ via upper hull is contained in a layer $\{x\in[0,1]^{n}:k\leq\sum_{i}x_{i}\leq k+1\}$ for some $k$ . Even when $w\in\nu(\operatorname{PM}^{+}_{n}(\mathcal{K}))$ is not strictly submodular, we can perturb $w$ and get a subdivision where every cell is contained in one of these layers.

Theorem 4.5.

The regular subdivision of $[0,1]^{n}$ induced by the tropicalization of the principal minors of any positive definite matrix is a coarsening of a subdivision of $[0,1]^{n}$ into dehomogenized realizable matroid polytopes. Specifically, for points in $\nu(\operatorname{PM}^{+}_{n}(\mathcal{R}))$ or $\nu(\operatorname{PM}^{+}_{n}(\mathcal{C}))$ , the matroids are realizable over $\mathbb{R}$ or $\mathbb{C}$ , respectively.

Proof.

Let $w\in\nu(\operatorname{PM}^{+}_{n}(\mathcal{K}))$ for $\mathcal{K}=\mathcal{R}$ or $\mathcal{C}$ . By Theorem 4.1, $w$ belongs to $\nu({\rm Fl}_{\mathcal{K}}(n))\cap{\rm SUBMOD}_{n}$ . By Lemma 3.1, for arbitrarily small $\lambda\in\Gamma^{n+1}$ , the point $\lambda\cdot w$ is strictly submodular. It also belongs to $\nu({\rm Fl}_{\mathcal{K}}(n))$ and, in particular, is $M^{\natural}$ -concave. By Lemma 2.4, each cell of the subdivision induced by $\lambda\cdot w$ is a subset of $\{x\in[0,1]^{n}:k\leq\sum_{i}x_{i}\leq k+1\}$ for some $k$ . Because $\lambda$ was taken arbitarily small, the subdivision induced by $\lambda\cdot w$ is a refinement of that induced by $w$ . We conclude by checking that each cell induced by $\lambda\cdot w$ is a dehomogenized realizable matroid polytope.

Let $w^{\prime}=(w^{\prime}_{k},w^{\prime}_{k+1})$ denote the restriction of $\lambda\cdot w$ to subsets of size $k$ and $k+1$ . Since $\lambda\cdot w\in\nu({\rm Fl}_{\mathcal{K}}(n))$ , the homogenization of $w^{\prime}$ is an element of $\nu(\operatorname{Gr}_{\mathcal{K}}(k+1,n+1))$ as described in Section 3. In particular, any cell in the subdivision of $\{x\in[0,1]^{n+1}:\sum_{i}x_{i}=k+1\}$ induced by the homogenization of $w^{\prime}$ is the matroid polytope of a matroid realizable over the residue field of $\mathcal{K}$ . For $\mathcal{K}=\mathcal{R}$ or $\mathcal{C}$ , it follows that the matroid is realizable over $\mathbb{R}$ or $\mathbb{C}$ , respectively. Dehomogenizing, i.e., dropping the coordinate $x_{n+1}$ , gives the result. ∎

5. Inequalities from $M^{\natural}$ -concavity

By the Lifting Lemma [JSY22], for any semi-algebraic set $S$ , any tropical polynomial inequality valid on $\operatorname{trop}(S)$ can be lifted to a usual polynomial inequality valid on $S$ . In this section we use the theory of Lorentzian polynomials [BH20, AGV18] to derive new inequalities on principal minors on positive definite matrices that lift the $M^{\natural}$ -concavity inequalities satisfied by the tropicalization, based on the fact that the polynomial $f=\det({\rm diag}(x_{1},\ldots,x_{n})+A)$ is Lorentzian for any positive definite matrix $A$ . Furthermore, we provide examples to show that these inequalities are tight.

For $D=\{d_{1},\ldots,d_{k}\}\subset[n]$ , let $\partial^{D}f$ denote the partial derivatives of $f$ with respect to the variables $x_{d_{1}},\ldots,x_{d_{k}}$ . For $S\subset[n]$ , let $A_{S}$ denote the principal minor indexed on the rows and columns by $S$ . To simplify notation, we write $Sij=S\cup\{i,j\}$ . We first derive inequalities on the coefficients of arbitrary Lorentzian polynomials.

Theorem 5.1.

Let $f=\sum_{S\subseteq[n]}c_{S}\,{\bf x}^{[n]\backslash S}y^{|S|}$ be a Lorentzian polynomial in the variables $x_{1},\dots,x_{n},y$ . If $n\geq 4$ , then for any $S\subseteq[n]\setminus\{1,2,3,4\}$ and any $r\in\mathbb{R}$ , the coefficients of $f$ satisfy

(r+1)\,c_{S14}c_{S23}+r(r+1)\,c_{S13}c_{S24}\geq r\,c_{S12}c_{S34}

and all analogous inequalities obtained by permutations of indices. Similarly, if $n\geq 3$ , then for any $S\subseteq[n]\setminus\{1,2,3\}$ and any $r\in\mathbb{R}$ , the coefficients satisfy

(r+1)\,c_{S1}c_{S23}+r(r+1)\,c_{S2}c_{S13}\geq r\,c_{S3}c_{S12}

and all analogous inequalities obtained by permutations of indices.

Moreover, both of these inequalities are tight for the subset of Lorentzian polynomials whose coefficients $c_{S}$ are the principal minors $A_{S}$ of a positive semidefinite matrix $A$ .

We will make use of the following lemma.

Lemma 5.2.

If $q=c_{34}x_{1}x_{2}+c_{24}x_{1}x_{3}+c_{23}x_{1}x_{4}+c_{14}x_{2}x_{3}+c_{13}x_{2}x_{4}+c_{12}x_{3}x_{4}$ is Lorentzian, then for any $r\in\mathbb{R}$

(r+1)\,c_{14}c_{23}+r(r+1)\,c_{13}c_{24}\geq r\,c_{12}c_{34}.

Proof.

By the definition of Lorentzian polynomials, the coefficients $c_{ij}$ are all nonnegative and the symmetric matrix

Q=\nabla^{2}q=\begin{pmatrix}0&c_{34}&c_{24}&c_{23}\\ c_{34}&0&c_{14}&c_{13}\\ c_{24}&c_{14}&0&c_{12}\\ c_{23}&c_{13}&c_{12}&0\end{pmatrix}

representing the quadratic form $q$ has at most one positive eigenvalue. It follows that $\det(Q)\leq 0$ .

The determinant of $Q$ is equal to the discriminant of the quadratic polynomial

(7)

c_{13}c_{24}\,r^{2}+(c_{13}c_{24}+c_{14}c_{23}-c_{12}c_{34})\,r+c_{14}c_{23}=(r+1)\,c_{14}c_{23}+r(r+1)\,c_{13}c_{24}-r\,c_{12}c_{34}

with respect to $r$ .

Since the extremal coefficients $c_{13}c_{24}$ and $c_{14}c_{23}$ and nonnegative and the discriminant is $\det(Q)\leq 0$ , it follows that the quadratic polynomial (7) is nonnegative for all $r\in\mathbb{R}$ . ∎

Proof of Theorem 5.1.

We first show that the first inequality implies the second. If $f$ is Lorentzian, then so is its polarization

{\rm Pol}(f)=\sum_{S\subseteq[n]}\frac{c_{S}}{\binom{n}{|S|}}{\bf x}^{[n]\backslash S}e_{|S|}(y_{1},\ldots,y_{n}),

where $e_{j}(y_{1},\ldots,y_{n})$ is the $j$ th elementary symmetric polynomial in $y_{1},\ldots,y_{n}$ ; see [BH20, Prop. 3.1]. For any set $S\subseteq[n]\setminus\{1,2,3\}$ with $|S|=k-1\leq n-3$ and $i,j\in\{1,2,3\}$ , the coefficient of ${\bf x}^{[n]\backslash Sij}{\bf y}^{[k-1]}$ in ${\rm Pol}(f)$ is $\binom{n}{k-1}^{-1}c_{Sij}$ and the coefficient of ${\bf x}^{[n]\backslash Si}{\bf y}^{[k]}$ in ${\rm Pol}(f)$ is $\binom{n}{k}^{-1}c_{Si}$ . Assuming the first inequality holds for all Lorentzian polynomials and applying it to ${\rm Pol}(f)$ with the variables $(x_{1},x_{2},x_{3},y_{k})$ in place of $(x_{1},x_{2},x_{3},x_{4})$ and $\{x_{i}:i\in S\}\cup\{y_{1},\ldots,y_{k-1}\}$ in place of $\{x_{i}:i\in S\}$ gives the desired inequality.

Now we show the first inequality. For general $D\subseteq[n]$ , $\partial^{D}f=\sum_{S}c_{S}{\bf x}^{[n]\backslash(S\cup D)y^{|S|}}$ where the sum is taken over $S\subseteq[n]$ for which $S\cap D=\emptyset$ . Now we fix $S\subseteq[n]\backslash\{1,2,3,4\}$ and let $D=[n]\backslash(S\cup\{1,2,3,4\})$ . Consider the polynomial $g$ obtained by specializing the derivative $\partial^{|S|+2}_{y}\partial^{D}f$ to $y=0$ and $x_{i}=0$ for $i\in S$ :

g=\left(\partial^{|S|+2}_{y}\partial^{D}f\right)|_{\{y=0,\ x_{i}=0\ \forall i\in S\}}.

Taking derivatives and specializing variables to zero preserves Lorentzian-ity by [BH20, Thm. 2.30], so $g$ is Lorentzian. One can check that $g$ is given by

g=(|S|+2)!(c_{34S}x_{1}x_{2}+c_{24S}x_{1}x_{3}+c_{23S}x_{1}x_{4}+c_{14S}x_{2}x_{3}+c_{13S}x_{2}x_{4}+c_{12S}x_{3}x_{4}).

The desired inequality then holds by Lemma 5.2.

Finally, in Examples 5.4 and 5.5 we will see that these inequalities are tight for the minors of positive semidefinite matrices when $n=4$ and $3$ , respectively, and $S=\emptyset$ . Appending identity matrices of arbitrary size shows that these inequalities are tight for general $n$ and $S$ . ∎

Corollary 5.3.

Let $A$ be an $n\times n$ positive semidefinite matrix. If $n\geq 4$ , then for any $S\subseteq[n]\backslash\{1,2,3,4\}$ and any $r\in\mathbb{R}$ ,

(r+1)\,A_{S14}A_{S23}+r(r+1)\,A_{S13}A_{S24}\geq r\,A_{S12}A_{S34}

Similarly, if $n\geq 3$ , then for any $S\subseteq[n]\backslash\{1,2,3\}$ and any $r\in\mathbb{R}$ ,

(r+1)\,A_{S1}A_{S23}+r(r+1)\,A_{S2}A_{S13}\geq r\,A_{S3}A_{S12}.

Moreover these inequalities are tight for any $r\in\mathbb{R}$ .

Proof.

The polynomial $f=\det({\rm diag}(x_{1},\ldots,x_{n})+yA)=\sum_{S\subseteq[n]}A_{S}{\bf x}^{[n]\backslash S}y^{|S|}$ is stable [BB06] and hence Lorentzian [BH20]. The result then follows from Theorem 5.1. ∎

Example 5.4.

For $r\in\mathbb{R}$ , consider the matrix

A=\left(\begin{array}[]{cccc}1&1&1&r\\ 1&2&-r+2&2r-1\\ 1&-r+2&r^{2}-2r+2&-r^{2}+3r-1\\ r&2r-1&-r^{2}+3r-1&2r^{2}-2r+1\end{array}\right).

One can check that $A$ is a positive semidefinite matrix of rank two. For example, all of the principal minors are sums of squares in $r$ . Moreover, the $2\times 2$ minors of $A$ satisfy

(r+1)A_{14}A_{23}+r(r+1)A_{13}A_{24}=rA_{12}A_{34},

showing the inequalities are tight. $\diamond$

Example 5.5.

Interestingly, there is not a parametrized family of $3\times 3$ positive semidefinite matrices for which the second inequality of Theorem 5.1 holds with equality. The image of the set of $3\times 3$ positive semidefinite matrices under the map $A\mapsto(A_{1}A_{23},A_{2}A_{13},A_{3}A_{12})$ is not closed. We see that the equality is only attained in the limit.

Fix $r\in\mathbb{R}$ and consider the following parameterized collection of $3\times 3$ matrices:

A=\begin{pmatrix}1+\varepsilon&\sqrt{1-\lambda(r+1)^{2}}&\sqrt{1-\lambda}\\ \sqrt{1-\lambda(r+1)^{2}}&1+\varepsilon&\sqrt{1-\lambda r^{2}}\\ \sqrt{1-\lambda}&\sqrt{1-\lambda r^{2}}&1+\varepsilon\end{pmatrix}.

For $0<\lambda<\min\{1/(r+1)^{2},1/r^{2},1\}$ and $\varepsilon>0$ , this matrix has real entries and positive $1\times 1$ and $2\times 2$ principal minors. Moreover, for fixed $\varepsilon>0$ , the limit of $\det(A)$ as $\lambda\to 0$ is $\varepsilon^{2}(3+\varepsilon)$ . Thus for sufficiently small $\lambda>0$ , the matrix $A$ is positive definite. One can compute that

(r+1)\,A_{1}A_{23}+r(r+1)\,A_{2}A_{13}-r\,A_{3}A_{12}=(1+r+r^{2})\varepsilon(1+\varepsilon)(2+\varepsilon).

In particular, as $\varepsilon\to 0$ , the limit is zero and the desired inequality becomes tight. $\diamond$

Remark 5.6.

One can check that the linear inequalities in Theorem 5.1 cut out a quadratic cone. More precisely, a point $(x,y,z)\in\mathbb{R}^{3}$ satisfies the inequality $(r+1)x+r(r+1)y\geq rz$ for all $r\in\mathbb{R}$ if and only if $2(xy+xz+yz)\geq x^{2}+y^{2}+z^{2}$ and $x+y+z\geq 0$ .

6. Proof of the technical lemma

This section is dedicated to the proof of Lemma 3.4, which was used in the proof of Theorem 3.2. Throughout this section we take $B\in\mathcal{K}^{n\times n}$ to be an upper triangular matrix whose upper justified minors $\det(B(\{1,\ldots,|S|\},S))$ are all nonzero. We consider the function

(8)

w(T,S)=\nu(\det(B(T,S)))

which is defined on pairs $(T,S)$ of subsets of the same size and takes values in $\mathbb{R}\cup\{-\infty\}$ .

Before the proof, we introduce some notation. We define the following partial order on $\binom{[n]}{k}$ . Given $S,T\in\binom{[n]}{k}$ , we say that

S\preceq T\quad\text{ if }\quad\left|\{s\in S:s\leq t_{j}\}\right|\geq j\text{ for all }j=1,\ldots,k

where $t_{1}<t_{2}<\ldots<t_{k}$ are the elements of $T$ . In particular, since $B$ is upper triangular, $\det(B(T,S))=0$ whenever $S\prec T$ . We call $I\subset[n]$ an interval if it is a collection of consecutive numbers. That is, whenever $i,k\in I$ and $i<j<k$ we have $j\in I$ . We use $[a]$ denote the interval $\{1,\ldots,a\}$ .

Proposition 6.1.

The function $w$ in (8) satisfies the following:

(i)

$w(\{1,..,|S|\},S)\neq-\infty$ for all $S$ .
(ii)

$w(T,S)=-\infty$ whenever $S\prec T$ .
(iii)

for any interval $I$ with $\max(I)\leq\min(T\cup S)$ , $w(I\cup T,I\cup S)=\sum_{i\in I}w(i,i)+w(T,S)$ .
(iv)

if $\max(S\cup T)\leq t$ and $\max(S\cup T)\leq s$ , then $w(Tt,Ss)=w(T,S)+w(t,s)$ .
(v)

for any $S$ , the function $T\mapsto w(T,S)$ satisfies the 3-term tropical Plücker relations.

Proof.

Item (i) holds by assumption. Whenever $S\prec T$ , $\det(B(T,S))=0$ and $w(T,S)=-\infty$ , since $B$ is upper triangular, showing (ii). For (iii), the conditions on $I,S,T$ imply that $\det(B(I\cup T,I\cup S))=\left(\prod_{i\in I}b_{ii}\right)\cdot\det(B(T,S))).$ Taking valuations of both sides proves the claim. Under the assumptions of (iv), $B(t,u)=0$ for all $u\in S$ . If $t\leq s$ , then the determinant of $B(Tt,Ss)$ factors as product of $\det(B(T,S))$ and $B(t,s)$ . Otherwise, both $\det(B(Tt,Ss))$ and $B(t,s)$ will be zero. Finally, for any fixed $S$ , the values $w(T,S)$ are the maximal minors of an $n\times|S|$ matrix, which therefore satisfy the three-term Plücker relations. ∎

To prove Lemma 3.4, it therefore suffices to show the following.

Theorem 6.2.

Let $w$ be any function defined on pairs $(T,S)$ of subsets of $[n]$ of the same size and taking values in $\mathbb{R}\cup\{-\infty\}$ . If $w$ satisfies the properties (i)-(v) in Proposition 6.1 and the function $F(S)=w(\{1,..,|S|\},S)$ is submodular, then

w(\{1,\ldots,|S|\},S)\geq w(T,S)

for all $S,T\subseteq[n]$ with $|S|=|T|$ .

For the remainder of the section, we assume $w$ is an arbitrary function satisfying these properties and for which $F(S)=w(\{1,..,|S|\},S)$ is submodular. Our goal is to prove this theorem. We first prove it in the case $|S|=1$ .

Lemma 6.3.

$w(j,k)\geq w(j+1,k)$ for all $1\leq j,k\leq n$ .

Proof.

If $j+1>k$ , then $w(j+1,k)=-\infty$ by property (ii), and the inequality holds trivially. So we can assume that $j+1\leq k$ . Let $I$ denote the interval $\{1,\ldots,j-1\}$ . By submodularity,

	$\displaystyle w(Ij,Ij)+w(Ij,Ik)$	$\displaystyle=F(Ij)+F(Ik)$
		$\displaystyle\geq F(I)+F(I\cup\{j,k\})$
		$\displaystyle=w(I,I)+w(I\cup\{j,j+1\},I\cup\{j,k\}).$

Using property (iv), this gives that

\sum_{i\in I}w(i,i)+w(j,j)+\sum_{i\in I}w(i,i)+w(j,k)\geq\sum_{i\in I}w(i,i)+\sum_{i\in I}w(i,i)+w(j,j)+w(j+1,k).

Canceling out diagonal terms gives $w(j,k)\geq\ w(j+1,k)$ . ∎

For an interval $I$ , we use $1+I$ to denote the shifted interval $\{1+i:i\in I\}$ .

Lemma 6.4.

Let $I$ be an interval and subset $S\subseteq[n]$ of the same size with $I\preceq S$ . Then

w(I,S)-w(1+I,S)\geq w(I\backslash\!\max(I),S\backslash\!\max(S))-w((1+I)\backslash\!\max(1+I),S\backslash\!\max(S))\geq 0.

Proof.

Let $s=\max(S)$ , $i=\min(I)$ , and $j=\max(I)$ . Then $[i-1]\cup I=[j]$ .

First we assume that $i<\min(S)$ , then $[i]\cap S=\phi$ . For the left inequality, we use the submodularity inequality

F([i-1]\cup S)+F([i]\cup S\backslash s)\geq F([i-1]\cup S\backslash s)+F([i]\cup S).

Using property (iv) and canceling our diagonal terms $\sum_{k=1}^{i-1}w(k,k)+\sum_{k=1}^{i}w(k,k)$ , this inequality becomes

w(I,S)+w((I+1)\backslash(j+1)),S\backslash s)\geq w(I\backslash j,S\backslash s)+w(1+I,S).

The claim follows after rearranging terms.

If $\min(S)=i$ , then let $J$ be an interval such that $\min(J)=i$ and $J=I\cap S$ . Let $I^{\prime}=I\backslash J$ and $S^{\prime}=S\backslash J$ . Then, by the previous part, the claim holds for $I^{\prime}$ and $S^{\prime}$ , that is

w(I^{\prime},S^{\prime})-w(1+I^{\prime},S^{\prime})\geq w(I^{\prime}\backslash\!\max(I^{\prime}),S\backslash\!\max(S))-w((1+I^{\prime})\backslash\!\max(1+I^{\prime}),S^{\prime}\backslash\!\max(S^{\prime})).

By (iii), $w(I,S)=\sum_{k=i}^{\max(J)}w(k,k)+w(I^{\prime},S^{\prime})$ and by (iv) we have $w(1+I,S)=\sum_{k=i}^{\max(J)}w(k+1,k)+w(1+I^{\prime},S^{\prime})$ . Thus, adding $\sum_{k=i}^{\max(J)}w(k,k)-w(k+1,k)$ to both sides of the above inequality proves the result.

The right inequality holds by induction on $|S|$ . The base case $|I\backslash\max(I)|=1$ is covered by Lemma 6.3. The inductive step is given by the left inequality. ∎

Lemma 6.5.

Let $I$ be an interval and a subset $S\subseteq[n]$ of the same size. For any $t>\max(I)$ such that $(It\backslash\!\max(I))\preceq S$ ,

w(I,S)\geq w(It\backslash\!\max(I),S)).

Proof.

Let $s=\max(S)$ , $s^{\prime}=\max(S\backslash\!s)$ , $i=\min(I)$ , and $j=\max(I)$ .

( $t=s$ ) By submodularity of $F$ , we have

F([i-1]\cup S)+F([s^{\prime}])\geq F([i-1]\cup S\backslash s)+F([s^{\prime}]s).

Cancelling diagonal terms $\sum_{k=1}^{i-1}w(k,k)+\sum_{k=1}^{s^{\prime}}w(k,k)$ on both sides gives

w(I,S)\geq w(I\backslash j,S\backslash s)+w(s^{\prime}+1,s).

By Lemma 6.3, $w(s^{\prime}+1,s)\geq w(s,s)$ . Then, using property (iv), we find

w(I,S)\geq w(I\backslash j,S\backslash s)+w(s,s)=w((Is\backslash j),S).

( $t<s$ ) We induct on $|S|$ and then $t-\min(I)$ . The case $|S|=1$ is Lemma 6.3. If $t-\min(I)=1$ , then the inequality holds also by Lemma 6.3. We therefore assume $|S|\geq 2$ and $t-\min(I)>1$ .

By property (v), the following maximum is attained at least twice

\max\left\{w(Iab\backslash\{i,j\},S))+w(Ics\backslash\{i,j\},S)\right\},

where the max is taken over all assignments $\{a,b,c\}=\{i,j,t\}$ . We break the argument into two cases, depending on which terms attain this maximum.

(Case 1) If the term with $\{a,b\}=\{i,j\}$ attains the maximum then

w(I,S)+w(Its\backslash\{i,j\},S)\geq w(It\backslash j,S)+w(Is\backslash i,S).

Rearranging terms gives

	$\displaystyle w(I,S))-w(It\backslash j,S)$	$\displaystyle\geq w(Is\backslash i,S)-w(Its\backslash\{i,j\},S)$
		$\displaystyle=w(I\backslash i),S\backslash s)-w(It\backslash\{i,j\},S\backslash s)\geq 0.$

The last inequality follows by induction on $|S|$ .

(Case 2) If the term with $\{a,b\}=\{i,j\}$ does not attain the maximum, then the other two terms are equal. That is,

w(It\backslash j,S)+w(Is\backslash i,S)=w(Is\backslash j,S)+w(It\backslash i,S).

Rearranging terms and assuming that $i<\min(S)$ gives

	$\displaystyle w(It\backslash j,S)-w(It\backslash i,S)$	$\displaystyle=w(Is\backslash j,S)-w(Is\backslash i,S)$
		$\displaystyle=w(I\backslash j,S\backslash s)-w(I\backslash i,S\backslash s)$
		$\displaystyle\leq w(I,S)-w(1+I,S)$
		$\displaystyle\leq w(I,S)-w(((1+I)t\backslash(j+1),S)$
		$\displaystyle=w(I,S)-w(It\backslash i,S).$

The first inequality follows from Lemma 6.4 and the fact that $(1+I)\backslash\!\max(1+I)=I\backslash i$ , and the last inequality follows by induction on $t-\min(I)$ , since $t-\min(1+I)<t-\min(I)$ . We have $1+I\preceq S$ since $i<\min(S)$ . Simplifying the final inequality gives

w(It\backslash j,S)\leq w(I,S),

as desired.

Now if $i=\min(S)$ , then let $J$ be an interval such that $\min(J)=i$ and $J=I\cap S$ . Let $I^{\prime}=I\backslash J$ and $S^{\prime}=S\backslash J$ . Then, by the previous part, the claim holds for $I^{\prime}$ and $S^{\prime}$ , that is

w(I^{\prime}t\backslash j,S^{\prime})\leq w(I^{\prime},S^{\prime}).

By (iii), $w(I,S)=\sum_{k=i}^{\max(J)}w(k,k)+w(I^{\prime},S^{\prime})$ . Thus, adding $\sum_{k=i}^{\max(J)}w(k,k)$ to both sides of the above inequality finishes the proof. ∎

Lemma 6.6.

Let $S,T\subseteq[n]$ with $T\preceq S$ . Let $I$ be the interval with $\min(I)=\min(T)$ and $|I|=|S|=|T|$ . Then $w(I,S)\geq w(T,S)$ .

Proof.

First we assume that $\max(S)\neq\max(T)$ and we proceed by induction on $|T|$ . For $|T|=1$ , $I=T=\{\min(T)\}$ so the statement holds trivially. Therefore we can assume $|S|=|T|\geq 2$ .

Consider the first consecutive run of elements of $T$ . That is, let $J$ be the largest interval with $\min(J)=\min(T)$ and $J\subseteq T$ . We also induct on $|T|-|J|$ . If $|J|=|T|$ , then $I=J=T$ and the inequality holds trivially. The case $|T|-|J|=1$ follows from Lemma 6.5. Therefore we may suppose $|T|-|J|\geq 2$ . Let $j=\max(J)$ . We will show by induction on $|T|$ that

(9)

w(T,S)\leq\max_{t\in T\backslash J}w((T(j+1)\backslash t,S).

Since $(T(j+1)\backslash t)$ has a longer initial consecutive sequence than $T$ , it follows by induction on $|T|-|J|$ that $w((T(j+1)\backslash t,S)\leq w(I,S)$ , implying $w(T,S)\leq w(I,S)$ , as desired.

Let $s=\max(S)$ . Suppose that $U=\{x,y\}$ attains the maximum

\max_{U}w(T(j+1)\backslash U,S\backslash s)

taken over subsets $U\subseteq T\backslash J$ with $|U|=2$ . By property (v), the maximum of

\max_{\{a,b,c\}=\{j+1,x,y\}}w((Tab\backslash\{x,y\}),S)+w((Tcs\backslash\{x,y\}),S)

is attained at least twice. In particular, after possibly relabeling $x,y$ , we have

w(T,S)+w((Ts(j+1)\backslash\{x,y\}),S)\leq w((T(j+1)\backslash y),S)+w(Ts\backslash x,S).

By property (iii), we can cancel the diagonal terms $w(s,s)$ and rearrange to get

w((T(j+1)\backslash\{x,y\}),S\backslash s)-w(T\backslash x,S\backslash s)\leq w((T(j+1)\backslash y),S)-w(T,S).

By induction on $|T|$ , we have that

w(T\backslash x,S\backslash s)\leq\max_{t}w((T(j+1)\backslash\{x,t\}),S\backslash s)

where $t$ runs over $(T\backslash J)\backslash x$ . By assumption on $\{x,y\}$ , this maximum is attained by $t=y$ , giving that

w(T\backslash x,S\backslash s)\leq w(T(j+1)\backslash\{x,y\},S\backslash s).

Together with the inequality from above, we get that

0\leq w(T(j+1)\backslash\{x,y\},S\backslash s)-w(T\backslash x,S\backslash s))\leq w(T(j+1)\backslash y,S))-w(T,S),

showing that

w(T,S)\leq w(T(j+1)\backslash y,S))\leq\max_{t\in T\backslash J}w(T(j+1)\backslash t,S).

If $\max(S)=\max(T)=s$ , let $J$ be the interval such that $\max(J)=\max(S)=\max(T)$ , $J\subset T\cap S$ and it is of maximum length. We induct on the size of $J$ . If $J=1$ , that is if $\max(S\backslash s)\neq\max(T\backslash s)$ , then let $I^{\prime}=I\backslash\!\max(I)$ , $S^{\prime}=S\backslash s$ , and $T^{\prime}=T\backslash s$ . Then by the previous part $w(I^{\prime},S^{\prime})\geq w(T^{\prime},S^{\prime})$ . Using Lemma 6.5, we get

	$\displaystyle w(I,S)$	$\displaystyle\geq w(Is\backslash\!\max(I),S)$
		$\displaystyle=w(I^{\prime}\!s,S)$
		$\displaystyle=w(I^{\prime},S^{\prime})+w(s,s)$
		$\displaystyle\geq w(T^{\prime},S^{\prime})+w(s,s)$
		$\displaystyle=w(T,S).$

For the inductive step, we use similar argument as the one above and this finishes the proof. ∎

Now we are ready to complete the proof of this section.

Proof of Theorem 6.2.

Let $S,T\subseteq[n]$ . By Lemma 6.6, $w(T,S)\leq w(I,S)$ , where $I$ is the interval with $\min(I)=\min(T)$ and $|I|=|T|$ . By Lemma 6.4, $w(1+J,S)\leq w(J,S)$ for every interval $J$ . Inducting then gives $w(k+J,S)\leq w(J,S)$ for arbitrary $k\in\mathbb{Z}_{>0}$ . In particular, for $J=\{1,\ldots,|S|\}$ and $k=\min(T)-1$ , this gives

w(T,S)\leq w(I,S)\leq w(\{1,\ldots,|S|\},S),

as desired. ∎

Remark 6.7.

One might hope for inequalities of the form $w(T,S)\geq w(T^{\prime},S)$ when $T\succeq T^{\prime}$ , but these do not hold in general. For example, consider the matrix

B=\begin{pmatrix}1&1&1&1\\ 0&1&2&3\\ 0&0&1&1+t\\ 0&0&0&1\\ \end{pmatrix}.

This matrix is upper triangular and all of its upper justified minors are nonzero and have valuation $0$ . Therefore the function $F$ is identically $0$ and thus submodular. However $w(\{1,3\},\{3,4\}))=-1$ , whereas $w(\{2,3\},\{3,4\})=0$ .

Example 6.8 ( $n=8,k=4$ ).

To illustrate the strategy of the proof above, consider $S=\{5,6,7,8\}$ , and $T=\{1,4,5,7\}$ . Here $J=\{1\}$ . Inequality (9) states that

w(1457,S)\leq\max\{w(1257,S),w(1247,S),w(1245,S)\}.

Let us show this inequality in the case that $w(127,567)$ achieves the maximum among $\{w(124,567),w(125,567),w(127,567)\}$ . In the proof above, this corresponds to $\{x,y\}=\{4,5\}$ . By the tropical Plücker relations, the maximum

\max_{\{a,b,c\}=\{2,4,5\}}\{w(17ab,S)+w(178c,S)\}\text{ is attained at least twice,}

from which we can conclude that

w(1457,S)+w(1278,S)\leq w(127a,S)+w(178b,S)

for some assignment of $\{a,b\}=\{4,5\}$ . Then $w(xyz8,S)=w(xyz,S^{\prime})+w(8,8)$ for any subset $\{x,y,z\}\subset[7]$ where $S^{\prime}=\{5,6,7\}$ . The inequality above give

	$\displaystyle w(127a,S)-w(1457,S)$	$\displaystyle\geq w(1278,S)-w(178b,S)$
		$\displaystyle=w(127,S^{\prime})-w(17b,S^{\prime}).$

By induction, $w(17b,S^{\prime})$ is bounded above by $\max\{w(127,S^{\prime}),w(12b,S^{\prime})\}$ , which is bounded above by $w(127,S^{\prime})$ , by assumption. Therefore this difference is nonnegative, giving

w(1457,S)\leq w(127a,S)\leq\max_{x,y\in\{4,5,7\}}w(12xy,S).

From this, we can continue by induction. Using (9) again gives

	$\displaystyle w(1257,S)$	$\displaystyle\leq\max\{w(1237,S),w(1235,S)\},$
	$\displaystyle w(1247,S)$	$\displaystyle\leq\max\{w(1237,S),w(1234,S)\},$
	$\displaystyle w(1245,S))$	$\displaystyle\leq\max\{w(1235,S),w(1234,S)\}.$

Finally, by Lemma 6.5, we see that these are all bounded above by $w(1234,S)$ . That is, for $x,y,z\geq 4$

\max_{x,y,z\geq 4}w(1xyz,S)\leq\max_{x,y\geq 4}w(12xy,S)\leq\max_{x\geq 4}w(123y,S)\leq w(1234,S).

$\diamond$

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Tropicalizing Principal Minors of Positive Definite Matrices

Abstract.

1. Introduction

Theorem 4.1.

Theorem 4.5.

Example 1.1.

Organization

Acknowledgements

2. Definitions and Background

2.1. Tropicalization

2.2. Discrete Concavity

Proposition 2.1.

Proof.

2.3. M♮M^{\natural}-concavity and valuated matroids

Proposition 2.2.

Proof.

Example 2.3.

Lemma 2.4.

Proof.

Remark 2.5.

3. Tropical Grassmannians and Tropical Flag Varieties

Lemma 3.1.

Proof.

Theorem 3.2.

Proposition 3.3.

Proof.

Proof of Theorem 3.2(⊆\subseteq).

Lemma 3.4.

Lemma 3.5.

Proof.

Proof of Theorem 3.2(⊇\supseteq).

Remark 3.6.

Example 3.7 (Example 2.3 continued).

4. Tropicalizing principal minors of positive definite matrices

Theorem 4.1.

Proof.

Corollary 4.2.

Proof.

Corollary 4.3.

Proof.

Corollary 4.4.

Theorem 4.5.

Proof.

5. Inequalities from M♮M^{\natural}-concavity

Theorem 5.1.

Lemma 5.2.

Proof.

Proof of Theorem 5.1.

Corollary 5.3.

Proof.

Example 5.4.

Example 5.5.

Remark 5.6.

6. Proof of the technical lemma

Proposition 6.1.

Proof.

Theorem 6.2.

Lemma 6.3.

Proof.

Lemma 6.4.

Proof.

Lemma 6.5.

Proof.

Lemma 6.6.

Proof.

Proof of Theorem 6.2.

Remark 6.7.

Example 6.8 (n=8,k=4n=8,k=4).

References

2.3. $M^{\natural}$ -concavity and valuated matroids

Proof of Theorem 3.2( $\subseteq$ ).

Proof of Theorem 3.2( $\supseteq$ ).

5. Inequalities from $M^{\natural}$ -concavity

Example 6.8 ( $n=8,k=4$ ).