Permanental inequalities for totally positive matrices

Given an $n\times n$ matrix $A=(a_{i,j})$ and subsets $I,J\subseteq[n]:=\{1,\dotsc,n\}$, let $A_{I,J}=(a_{i,j})_{i\in I,j\in J}$ denote the $(I,J)$-submatrix of $A$. For $|I|=|J|$, call $\det(A_{I,J})$ the $(I,J)$-minor of $A$. A real $n\times n$ matrix $A$ is called totally positive (totally nonnegative) if every minor of $A$ is positive (nonnegative). Let $\mathcal{M}^{\textsc{TP}}_{n}\subset\mathcal{M}^{\textsc{TNN}}_{n}$ denote these sets of matrices.

These and the set $\mathcal{M}^{\textsc{HPS}}_{n}$ of $n\times n$ Hermitian positive semidefinite matrices arise in many areas of mathematics, and for more than a century mathematicians have been studying inequalities satisfied by their matrix entries. (See, e.g., [8].) Many such inequalities involve minors and permanents. For instance inequalities of Fischer [9], Fan [4], and Lieb [15] state that for all matrices $A\in\mathcal{M}^{\textsc{TNN}}_{n}\cup\mathcal{M}^{\textsc{HPS}}_{n}$, and for all $I\subseteq[n]$ and $I^{c}:=[n]\smallsetminus I$, we have

(1)

$$\begin{gathered}\det(A)\leq\det(A_{I,I})\det(A_{I^{c},I^{c}}),\\ \mathrm{per}(A)\geq\mathrm{per}(A_{I,I})\;\mathrm{per}(A_{I^{c},I^{c}}).\end{gathered}$$

Koteljanskii’s inequality [13], [14] states that for $A\in\mathcal{M}^{\textsc{TNN}}_{n}\cup\mathcal{M}^{\textsc{HPS}}_{n}$ and for all $I,J\subseteq[n]$ we have

(2)

$$\det(A_{I\cup J,I\cup J})\det(A_{I\cap J,I\cap J})\leq\det(A_{I,I})\det(A_{J,J}).$$

Many open questions about inequalities exist and seem difficult. For instance, it is known which $8$-tuples $(I,J,K,L,I^{\prime},J^{\prime},K^{\prime},L^{\prime})$ of subsets satisfy

(3)

$$\det(A_{I,I^{\prime}})\det(A_{J,J^{\prime}})\leq\det(A_{K,K^{\prime}})\det(A_{L,L^{\prime}})$$

for all $A\in\mathcal{M}^{\textsc{TNN}}_{n}$ [7], [16], but few permanental analogs of such inequalities are known. While some of these $8$-tuples also satisfiy

(4)

$$\mathrm{per}(A_{I,I^{\prime}})\;\mathrm{per}(A_{J,J^{\prime}})\geq\mathrm{per}(A_{K,K^{\prime}})\;\mathrm{per}(A_{L,L^{\prime}}),$$

this second inequality is not true in general. For example, the natural permanental analog

(5)

$$\mathrm{per}(A_{I\cup J,I\cup J})\;\mathrm{per}(A_{I\cap J,I\cap J})\geq\mathrm{per}(A_{I,I})\;\mathrm{per}(A_{J,J}),$$

of (2) holds neither for all $A\in\mathcal{M}^{\textsc{HPS}}_{n}$ nor for all $A\in\mathcal{M}^{\textsc{TNN}}_{n}$. (See [17, §6] for a counterexample with $n=3$.)

Let us put aside $\mathcal{M}^{\textsc{HPS}}_{n}$ and consider conjectured inequalities of the form

(6)

$$\mathrm{product}_{1}\leq\mathrm{product}_{2}$$

involving minors and permanents of matrices in $\mathcal{M}^{\textsc{TNN}}_{n}$ and $\mathcal{M}^{\textsc{TP}}_{n}$. One strategy for studying (6) is to view the difference $\mathrm{product}_{2}-\mathrm{product}_{1}$ as a polynomial

(7)

$$f(x):=f(x_{1,1},x_{1,2},\dotsc,x_{n,n})\in\mathbb{Z}[x]:=\mathbb{Z}[x_{1,1},x_{1,2},\dotsc,x_{n,n}]$$

in matrix entries. Then the validity of the inequality (6) is equivalent to the statement that for all $A=(a_{i,j})\in\mathcal{M}^{\textsc{TNN}}_{n}$, we have

(8)

$$f(A):=f(a_{1,1},a_{1,2},\dotsc,a_{n,n})\geq 0.$$

We call a polynomial (7) with this property a totally nonnegative polynomial. Since $\mathcal{M}^{\textsc{TP}}_{n}$ is dense in $\mathcal{M}^{\textsc{TNN}}_{n}$, the inequality (8) holds for all $A\in\mathcal{M}^{\textsc{TP}}_{n}$ if and only if it holds for all $A\in\mathcal{M}^{\textsc{TNN}}_{n}$.

A second strategy for studying (variations of) a potential inequality (6) is to ask for which positive constants $k_{1}$, $k_{2}$ the modified inequalities

(9)

$$k_{1}\cdot\mathrm{product}_{1}\leq\mathrm{product}_{2}\leq k_{2}\cdot\mathrm{product}_{1}$$

hold for all $A\in\mathcal{M}^{\textsc{TNN}}_{n}$. Bounds of $k_{1}=1$ or $k_{2}=1$ imply the inequality (6) or its reverse to hold; other bounds give information not apparent in the proof or disproof of (6). Equivalently, we may view the ratio of $\mathrm{product}_{2}$ to $\mathrm{product}_{1}$ as a rational function

(10)

$$R(x):=R(x_{1,1},x_{1,2},\dotsc,x_{n,n})\in\mathbb{Q}(x):=\mathbb{Q}(x_{1,1},x_{1,2},\dotsc,x_{n,n})$$

in matrix entries, and we may ask for upper and lower bounds as $x$ varies over $\mathcal{M}^{\textsc{TP}}_{n}$. While a ratio (10) is not defined everywhere on $\mathcal{M}^{\textsc{TNN}}_{n}$, the density of $\mathcal{M}^{\textsc{TP}}_{n}$ in $\mathcal{M}^{\textsc{TNN}}_{n}$ allows us to restrict our attention to $\mathcal{M}^{\textsc{TP}}_{n}$: we have

(11)

$$k_{1}\leq R(x)\leq k_{2}$$

for all $x\in\mathcal{M}^{\textsc{TP}}_{n}$ if and only if the same inequalities hold for all $x\in\mathcal{M}^{\textsc{TNN}}_{n}$ such that $R(x)$ is defined. Clearly the lower bound $k_{1}$ is interesting only when positive, since products of minors and permanents of totally nonnegative matrices are trivially bounded below by $0$.

A characterization of all ratios of the form

(12)

$$\frac{\det(x_{I,I^{\prime}})\det(x_{J,J^{\prime}})}{\det(x_{K,K^{\prime}})\det(x_{L,L^{\prime}})},\qquad I,I^{\prime},\dotsc,L,L^{\prime}\subset[n],$$

which are bounded above and/or nontrivially bounded below on $\mathcal{M}^{\textsc{TP}}_{n}$ follows from work in [7] and [16]. Each ratio (12) is bounded above and/or below by $1$, and for each $n$, factors as a product of elements of a finite set of indecomposable ratios. This result was extended in [11] to include ratios of products of arbitrarily many minors

(13)

$$\frac{\det(x_{I_{1},I^{\prime}_{1}})\cdots\det(x_{I_{p},I^{\prime}_{p}})}{\det(x_{J_{1},J^{\prime}_{1}})\cdots\det(x_{J_{p},J^{\prime}_{p}})}.$$

Again, each of these factors as a product of elements belonging to a finite set of indecomposable ratios. For $n=3$, each ratio (13) is bounded above and/or below by $1$; for $n\geq 4$, such bounds are conjectured [3].

While the permanental version (5) of Koteljanskii’s inequality is false, we will show in Section 3 that the corresponding ratio is bounded above and nontrivially below. Specifically,

(14)

$$\frac{1}{|I\cup J|!~{}|I\cap J|!}\leq\frac{\mathrm{per}(x_{I,I})\;\mathrm{per}(x_{J,J})}{\mathrm{per}(x_{I\cup J,I\cup J})\;\mathrm{per}(x_{I\cap J,I\cap J})}\leq|I|!~{}|J|!$$

for all $I,J\subseteq[n]$ and $x\in\mathcal{M}^{\textsc{TP}}_{n}$. The failure of (5), combined with (14), exposes a difference between ratios of minors and of permanents: unlike the bounded ratios in (12), not all bounded ratios of permanents are bounded by $1$. Thus it is natural to ask which ratios

(15)

$$R(x)=\frac{\text{per}(x_{I_{1},I^{\prime}_{1}})\text{per}(x_{I_{2},I^{\prime}_{2}})\cdots\text{per}(x_{I_{r},I^{\prime}_{r}})}{\text{per}(x_{J_{1},J^{\prime}_{1}})\text{per}(x_{J_{2},J^{\prime}_{2}})\cdots\text{per}(x_{J_{q},J^{\prime}_{q}})}$$

are bounded above and/or nontrivially below as real-valued functions on $\mathcal{M}^{\textsc{TP}}_{n}$, and to state bounds.

In Section 2 we describe a multigrading of the coordinate ring $\mathbb{Z}[x]$ of $n\times n$ matrices. Extending work in [6], we define a partial order on the monomials in $\mathbb{Z}[x]$ which characterizes the differences $\smash{\prod x_{i,j}^{c_{i,j}}-\prod x_{i,j}^{d_{i,j}}}$ which are totally nonnegative polynomials. This leads to our main results in Section 3 which characterize ratios (15) which are bounded above and nontrivially below as real-valued functions on $\mathcal{M}^{\textsc{TP}}_{n}$. We provide some such bounds, which are not necessarily tight. We finish in Section 4 with some open questions.

We will find it convenient to view degree-$r$ monomials in $\mathbb{Z}[x]$ in terms of permutations in the symmetric group $\mathfrak{S}_{r}$ and multisets of $[n]$. In particular, given permutations $v,w\in\mathfrak{S}_{r}$ define the monomial

$$x^{v,w}:=x_{v_{1},w_{1}}\cdots x_{v_{r},w_{r}}.$$

Define an $r$-element multiset of $[n]$ to be a nondecreasing $r$-tuple of elements of $[n]$. In exponential notation, we write $i^{k}$ to represent $k$ consecutive occurrences of $i$ in such an $r$-tuple, e.g.,

(16)

$$(1,1,2,3)=1^{2}2^{1}3^{1},\qquad(1,2,2,2)=1^{1}2^{3}.$$

Two $r$-element multisets

(17)

$$M=(m_{1},\dotsc,m_{r})=1^{\alpha_{1}}\cdots n^{\alpha_{n}},\qquad O=(o_{1},\dotsc,o_{r})=1^{\beta_{1}}\cdots n^{\beta_{n}},$$

determine a generalized submatrix $x_{M,O}$ of $x$ by $(x_{M,O})_{i,j}:=x_{m_{i},o_{j}}$. For example, when $n=3$, we have the $4\times 4$ generalized submatrix and monomial

(18)

$$x_{1123,1222}=\begin{bmatrix}x_{1,1}&x_{1,2}&x_{1,2}&x_{1,2}\\ x_{1,1}&x_{1,2}&x_{1,2}&x_{1,2}\\ x_{2,1}&x_{2,2}&x_{2,2}&x_{2,2}\\ x_{3,1}&x_{3,2}&x_{3,2}&x_{3,2}\end{bmatrix}\negthickspace,\qquad(x_{1123,1222})^{1234,4312}=x_{1,2}x_{1,2}x_{2,1}x_{3,2}.$$

The ring $\mathbb{Z}[x]$ has a natural multigrading

(19)

$$\mathbb{Z}[x]=\bigoplus_{r\geq 0}\bigoplus_{M,O}\mathcal{A}_{M,O},$$

where the second direct sum is over pairs $(M,O)$ of $r$-element multisets of $[n]$, and

(20)

$$\mathcal{A}_{M,O}:=\mathrm{span}_{\mathbb{Z}}\{(x_{M,O})^{e,w}\,|\,w\in\mathfrak{S}_{r}\}.$$

More precisely, for $M$, $O$ as in (17), a basis for $\mathcal{A}_{M,O}$ is given by all monomials

(21)

$$\prod_{i,j\in[n]}\negthickspace x_{i,j}^{c_{i,j}}$$

with $C=(c_{i,j})\in\mathrm{Mat}_{n\times n}(\mathbb{N})$ satisfying

(22)

$$c_{i,1}+\cdots+c_{i,n}=\alpha_{i},\quad c_{1,j}+\cdots+c_{n,j}=\beta_{j}\quad\text{for }i,j=1,\dotsc,n.$$

One may express a monomial (21) in the form $(x_{M,O})^{e,w}$ by the following algorithm.

For example, it is easy to check that for $n=3$ and multisets $(1123,1222)=(1^{2}2^{1}3^{1},1^{1}2^{3}3^{0})$ of $\{1,2,3\}$, the graded component $\mathcal{A}_{1123,1222}$ of $\mathbb{Z}[x_{1,1},x_{1,2},\dotsc,x_{3,3}]$ is spanned by monomials (21), where $C=(c_{i,j})$ is one of the matrices

(23)

$$\begin{bmatrix}1&1&0\\ 0&1&0\\ 0&1&0\end{bmatrix}\negthickspace,\quad\begin{bmatrix}0&2&0\\ 1&0&0\\ 0&1&0\end{bmatrix}\negthickspace,\quad\begin{bmatrix}0&2&0\\ 0&1&0\\ 1&0&0\end{bmatrix}$$

having row sums $(2,1,1)$ and column sums $(1,3,0)$. These are

(24)

$$x_{1,1}x_{1,2}x_{2,2}x_{3,2},\quad x_{\smash{1,2}}^{2}x_{2,1}x_{3,2},\quad\smash{x_{1,2}^{2}}x_{2,2}x_{3,1},$$

with column index sequences equal to the rearrangements $1222$, $2212$, $2221$ of $1222$. Algorithm 2.1 then produces permutations $1234$, $2314$, $2341$ in $\mathfrak{S}_{4}$, and we may express the monomials (24) as

(25)

$$(x_{1123,1222})^{1234,1234},\quad(x_{1123,1222})^{1234,2314},\quad(x_{1123,1222})^{1234,2341}.$$

For $r$-element multisets $M$, $O$ of $[n]$, the monomials in $\mathcal{A}_{M,O}$ are closely related to parabolic subgroups of $\mathfrak{S}_{r}$ with standard generators $s_{1},\dotsc,s_{r-1}$, and double cosets of the form $W_{\iota(M)}wW_{\iota(O)}$ where $w$ belongs to $\mathfrak{S}_{r}$, $W_{J}$ is the subgroup of $\mathfrak{S}_{r}$ generated by $J$, and

(26)

$$\begin{gathered}\iota(M):=\{s_{1},\dotsc,s_{r-1}\}\smallsetminus\{s_{\alpha_{1}},s_{\alpha_{1}+\alpha_{2}},\dotsc,s_{r-\alpha_{n}}\}=\{s_{j}\,|\,m_{j}=m_{j+1}\},\\ \iota(O):=\{s_{1},\dotsc,s_{r-1}\}\smallsetminus\{s_{\beta_{1}},s_{\beta_{1}+\beta_{2}},\dotsc,s_{r-\beta_{n}}\}=\{s_{j}\,|\,o_{j}=o_{j+1}\}.\end{gathered}$$

It is easy to see that the map $M\mapsto\iota(M)$ is bijective: one recovers $M=1^{\alpha_{1}}\cdots n^{\alpha_{n}}$ from the generators not in $\iota(M)$ as in (26). It is known that each double coset has unique minimal and maximal elements with respect to the Bruhat order on $\mathfrak{S}_{r}$, defined by declaring $v\leq w$ if each reduced expression $s_{i_{1}}\cdots s_{i_{\ell}}$ for $w$ contains a subword which is a reduced expression for $v$. (See, e.g., [2], [5].) Let $W_{\iota(M)}\backslash W/W_{\iota(O)}$ denote the set of all double cosets of $W=\mathfrak{S}_{r}$ determined by $r$-element multisets $M$, $O$.

For any subsets $I$, $J$ of generators of $\mathfrak{S}_{r}$, the Bruhat order on $\mathfrak{S}_{r}$ induces a poset structure on $W_{I}\backslash W/W_{J}$ as follows. We declare $W_{I}vW_{J}\leq W_{I}wW_{J}$ if elements of the cosets satisfy any of the three (equivalent) inequalities in the Bruhat order on $\mathfrak{S}_{r}$ [5, Lem. 2.2].

1. (i)

   The minimal element of $W_{I}vW_{J}$ is less than or equal to the minimal element of $W_{I}wW_{J}$.

2. (ii)

   The maximal element of $W_{I}vW_{J}$ is less than or equal to the maximal element of $W_{I}wW_{J}$.

3. (iii)

   At least one element of $W_{I}vW_{J}$ is less than or equal to at least one element of $W_{I}wW_{J}$.

We call this poset the Bruhat order on $W_{I}\backslash W/W_{J}$. A fourth equivalent inequality can be stated in terms of matrices of exponents defined by monomials in $\mathcal{A}_{M,O}$. (See, e.g., [12].) Given a matrix $C=(c_{i,j})\in\mathrm{Mat}_{n\times n}(\mathbb{N})$, define the matrix $C^{*}=(c^{*}_{i,j})\in\mathrm{Mat}_{n\times n}(\mathbb{N})$ by

(27)

$$c^{*}_{i,j}=\text{sum of entries of $C_{[i],[j]}$}.$$

The Bruhat order on $W_{\iota(M)}\backslash W/W_{\iota(O)}$ is closely related to certain totally nonnegative polynomials in $\mathcal{A}_{M,O}$. Indeed, when $M=O=1^{n}$, totally nonnegative polynomials of the form $x^{e,v}-x^{e,w}$ are characterized by the Bruhat order on $\mathfrak{S}_{n}$ [6].

We will now extend this result to all monomials in $\mathbb{Z}[x]$. Let us define a partial order $\leq_{T}$ on all monomials in $\mathbb{Z}[x]$ by declaring $(x_{M,O})^{e,v}\leq_{T}(x_{P,Q})^{e,w}$ if $(x_{P,Q})^{e,w}-(x_{M,O})^{e,v}$ is a totally nonnegative polynomial. We call this the total nonnegativity order on monomials in $\mathbb{Z}[x]$. It is not hard to show that the total nonnegativity order is a disjoint union of its restrictions to the multigraded components (19) of $\mathbb{Z}[x]$.

For example, let us revisit the monomials (24) – (25) in the graded component $\mathcal{A}_{1123,1222}$ of $\mathbb{Z}[x_{1,1},x_{1,2},\dotsc,x_{3,3}]$. It is easy to see that $1234<2314<2341$ in the Bruhat order on $\mathfrak{S}_{4}$ and that the application of (27) to the corresponding matrices in (23) yields the componentwise comparisons

(30)

$$\begin{bmatrix}1&2&2\\ 1&3&3\\ 1&4&4\end{bmatrix}\geq\begin{bmatrix}0&2&2\\ 1&3&3\\ 1&4&4\end{bmatrix}\geq\begin{bmatrix}0&2&2\\ 0&3&3\\ 1&4&4\end{bmatrix}\negthickspace.$$

Thus we have $(x_{1123,1222})^{1234,1234}\geq_{T}(x_{1123,1222})^{1234,2314}\geq_{T}(x_{1123,1222})^{1234,2341}$, i.e.,

$$x_{1,1}x_{1,2}x_{2,2}x_{3,2}\ \geq_{T}\ x_{\smash{1,2}}^{2}x_{2,1}x_{3,2}\ \geq_{T}\ \smash{x_{1,2}^{2}}x_{2,2}x_{3,1}.$$

Furthermore, the chain $1234<2134<2314<2341$ in the Bruhat order on $\mathfrak{S}_{4}$ with

(31)

$$2134=(1,2)1234,\quad 2314=(2,3)2134,\quad 2341=(3,4)2314$$

allows us to write $x_{1,1}x_{1,2}x_{2,2}x_{3,2}-x_{\smash{1,2}}^{2}x_{2,1}x_{3,2}$ as

$$\begin{gathered}\Big{(}\negthinspace(x_{1123,1222})^{1234,1234}-(x_{1123,1222})^{1234,2134}\Big{)}\,+\,\Big{(}\negthinspace(x_{1123,1222})^{1234,2134}-(x_{1123,1222})^{1234,2314}\Big{)}\\ =\det\negthinspace\begin{bmatrix}x_{1,1}&x_{1,2}\\ x_{1,1}&x_{2,2}\end{bmatrix}\negthinspace x_{2,2}x_{3,2}\,+\,x_{1,2}\det\negthinspace\begin{bmatrix}x_{1,1}&x_{1,2}\\ x_{2,1}&x_{2,2}\end{bmatrix}\negthinspace x_{3,2},\end{gathered}$$

and to write $x_{\smash{1,2}}^{2}x_{2,1}x_{3,2}-\smash{x_{1,2}^{2}}x_{2,2}x_{3,1}$ as

$$\Big{(}\negthinspace(x_{1123,1222})^{1234,2314}-(x_{1123,1222})^{1234,2341}\Big{)}=x_{1,2}^{2}\det\negthinspace\begin{bmatrix}x_{2,1}&x_{2,2}\\ x_{3,1}&x_{3,2}\end{bmatrix}\negthinspace.$$

Let $\mathcal{M}^{\textsc{TP}}_{n}$ be the set of totally positive $n\times n$ matrices. To characterize ratios of products of permanents which are bounded above and/or nontrivially bounded below on the set $\mathcal{M}^{\textsc{TP}}_{n}$, we first consider necessary conditions on the multisets of rows and columns appearing in such ratios. Let

(32)

$$R(x)=\frac{\mathrm{per}(x_{I_{1},I^{\prime}_{1}})\mathrm{per}(x_{I_{2},I^{\prime}_{2}})\cdots\mathrm{per}(x_{I_{r},I^{\prime}_{r}})}{\mathrm{per}(x_{J_{1},J^{\prime}_{1}})\mathrm{per}(x_{J_{2},J^{\prime}_{2}})\cdots\mathrm{per}(x_{J_{q},J^{\prime}_{q}})},$$

be such a ratio, where

(33)

$$(I_{1},\dotsc,I_{r}),\quad(I^{\prime}_{1},\dotsc,I^{\prime}_{r}),\quad(J_{1},\dotsc,J_{q}),\quad(J^{\prime}_{1},\dotsc,J^{\prime}_{q})$$

are multisets of $[n]$ satisfying $|I_{k}|=|I^{\prime}_{k}|$, $|J_{k}|=|J^{\prime}_{k}|$ for all $k$. In order for $R(x)$ to be bounded above or nontrivially bounded below on $\mathcal{M}^{\textsc{TP}}_{n}$ the multisets (33) must be related in terms of an operation which we call multiset union. Given multisets $M=1^{\alpha_{1}}\cdots n^{\alpha_{n}}$, $O=1^{\beta_{1}}\cdots n^{\beta_{n}}$ of $[n]$, define their multiset union to be

(34)

$$M\Cup O:=1^{\alpha_{1}+\beta_{1}}\cdots n^{\alpha_{n}+\beta_{n}}.$$

For example, $1124\Cup 1233=11122334$.