Some remarks on permanent dominant conjecture

TL;DR

This paper establishes a new identity linking determinants and generalized matrix functions, and offers a criterion for positive semi-definite matrices that supports the permanent dominant conjecture, generating numerous classes of such matrices.

Contribution

It introduces a novel identity and a new criterion for positive semi-definite matrices related to the permanent dominant conjecture, expanding the set of known applicable matrices.

Findings

Derived an identity between determinant and generalized matrix function.

Provided a criterion affirming the permanent dominant conjecture.

Generated infinitely many classes of positive semi-definite matrices supporting the conjecture.

Abstract

In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite classes of positive semi-definite serving the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

Equations

$$d_{\chi }^G(A)=\sum _{g\in G}\chi (g)\prod _{i=1}^n{A}_{ig(i)}.$$

$${\bar{d}}_{\chi }^G(A)\ge \det (A),$$

$$\mathrm{per}(A)\ge h(A),$$

$$\mathrm{per}(A)\ge \mathrm{per}(B)\mathrm{per}(C),$$

$${\bar{d}}_{\chi }^G(A)\le \mathrm{per}(A),$$

$${\bar{d}}_{\chi }^G(A)=\frac{1}{\chi (e)}\sum _{g\in G}\chi (g)l_g(A)\le \sum _{g\in G}\frac{|\chi (g)|}{\chi (e)}|l_g(A)|\le \sum _{g\in G}|l_g(A)|\le \sum _{\sigma \in S_n}|l_{\sigma }(A)|.$$

$$\sum _{\sigma \in A_n}{l}_{\sigma }(A)-\sum _{\sigma \in O_n}{l}_{\sigma }(A)\le \sum _{\sigma \in A_n}{l}_{\sigma }(A)+\sum _{\sigma \in O_n}{l}_{\sigma }(A),$$

$$d_1^{A_n}(A)\le \mathrm{per}(A).$$

$$l_{\sigma }(A)=\prod _{i=1}^n{A}_{i\sigma (i)}=\prod _{i=1}^n{a}_i{\bar{a}}_{\sigma (i)}=(\prod _{i=1}^n{a}_i)(\prod _{i=1}^n{\bar{a}}_{\sigma (i)})=(\prod _{i=1}^n{a}_i)\overline{(\prod _{i=1}^n{a}_i)}=|\prod _{i=1}^n{a}_i{|}^2\ge 0.$$

$$l_{\sigma }(A)=\prod _{i=1}^n(a_i{\bar{a}}_{\sigma (i)}+{\delta }_{i\sigma (i)}{d}_i)=\sum _{I\subseteq [n]}(\prod _{i\in I}{a}_i{\bar{a}}_{\sigma (i)})(\prod _{i\notin I}{\delta }_{i\sigma (i)}{d}_i).$$

$$l_{\sigma }(A)=\sum _{I\in N^c}(\prod _{i\in I}{a}_i{\bar{a}}_{\sigma (i)})(\prod _{i\notin I}{d}_i)=\sum _{I\in N^c}(\prod _{i\in I}{a}_i)(\overline{\prod _{i\in I}{a}_i})(\prod _{i\notin I}{d}_i)=\sum _{I\in N^c}|\prod _{i\in I}{a}_i){|}^2(\prod _{i\notin I}{d}_i)\ge 0.$$

$$d_{\chi }^G((AB)[\alpha |\beta ])=\frac{\chi (e)}{|G|}\sum _{\gamma \in \Omega }{d}_{\chi }^G(A[\alpha |\gamma ])d_{\chi }^G(B[\gamma |\beta ]),$$

$$d_{\chi }^G(L^{\ast }[\gamma |(n)])$$

$$d_{\chi }^G(A)=\frac{\chi (e)}{|G|}\sum _{\gamma \in \Omega }{d}_{\chi }^G(L[(n)|\gamma ])d_{\chi }^G(L^{\ast }[\gamma |(n)])=\frac{\chi (e)}{|G|}\sum _{\gamma \in \Omega }|d_{\chi }^G(L[(n)|\gamma ]){|}^2;$$

$${\bar{d}}_{\chi }^G(A)=\frac{1}{|G|}\sum _{\gamma \in \Omega }|d_{\chi }^G(L[(n)|\gamma ]){|}^2.$$

$$d_{\chi }^G(L^{g\cdot \gamma })=d_{\chi }^G(L^{\gamma }{P}_g)$$

$$\sum _{\omega \in [\gamma ]}|d_{\chi }^G(L^{\omega }){|}^2=\sum _{g\in S_{\gamma }}|d_{\chi }^G(L^{\gamma }{P}_g){|}^2.$$

$${\bar{d}}_{\chi }^G(A)=\sum _{\gamma \in {\bar{\Delta }}_G}\frac{1}{|G|}\sum _{g\in S_{\gamma }}|d_{\chi }^G(L^{\gamma }{P}_g){|}^2.$$

$$\frac{1}{|G|}\sum _{g\in S_{{\gamma }^{\circ }}}|d_{\chi }^G(L^{{\gamma }^{\circ }}{P}_g){|}^2$$

$$1=(\chi ,\chi {)}_G:=\frac{1}{|G|}\sum _{g\in G}\overline{\chi (g)}\chi (g)=\frac{1}{|G|}\sum _{g\in G}|\chi (g){|}^2.$$

$$\frac{1}{|G|}\sum _{g\in S_{{\gamma }^{\circ }}}|d_{\chi }^G(L^{{\gamma }^{\circ }}{P}_g){|}^2=\frac{1}{|G|}\sum _{g\in G}|d_{\chi }^G(L{P}_g){|}^2=\det (A).$$

$$\frac{1}{|G|}\sum _{g\in S_{\sigma \cdot {\gamma }^{\circ }}}|d_{\chi }^G(L^{\sigma \cdot {\gamma }^{\circ }}{P}_g){|}^2$$

$$\frac{1}{|G|}\sum _{g\in S_{{\gamma }^k}}|d_1^G(L^{{\gamma }^k}{P}_g){|}^2$$

$${\bar{\Delta }}_G^{\ast }={\bar{\Delta }}_G-\{\sigma \cdot {\gamma }^{\circ }|\sigma \in S_G\}-\{{\gamma }^k|k=1,2,\dots ,n\},$$

$${\bar{d}}_{\chi }^G(A)=\det (A)+{\delta }_{\chi ,1}|G|\prod _{i=1}^n|L_{i1}{|}^2+\sum _{\gamma \in {\bar{\Delta }}_G^{\ast }}\frac{1}{|G|}\sum _{g\in S_{\gamma }}|d_{\chi }^G(L^{\gamma }{P}_g){|}^2,$$

$${\bar{d}}_{\chi }^G(A)\ge \det (A),$$

$$d_{\chi }^G(L^{\gamma }{P}_g)=\chi (g^{-1})\sum _{\theta \in G}\chi (\theta )\prod _{i=1}^n{L}_{i\theta (i)}^{\gamma }=\chi (g^{-1})d_{\chi }^G(L^{\gamma }).$$

$$\sum _{g\in S_{\gamma }}|d_{\chi }^G(L^{\gamma }{P}_g){|}^2=\sum _{g\in S_{\gamma }}|\chi (g^{-1})d_{\chi }^G(L^{\gamma }){|}^2=\frac{|G|}{|G_{\gamma }|}|d_{\chi }^G(L^{\gamma }){|}^2.$$

$${\bar{d}}_{\chi }^G(A)=\det (A)+{\delta }_{\chi ,1}|G|\prod _{i=1}^n|L_{i1}{|}^2+\sum _{\gamma \in {\bar{\Delta }}_G^{\ast }}\frac{1}{|G_{\gamma }|}|d_{\chi }^G(L^{\gamma }){|}^2.$$

$$\mathrm{per}(A)=\det (A)+n!\prod _{i=1}^n|L_{i1}{|}^2+(n-1)!\prod _{i=1}^{n-1}|L_{i1}{|}^2|L_{nn}{|}^2+\sum _{\gamma \in {\bar{\Delta }}_{S_n}^{\ast \ast }}\frac{1}{|(S_n{)}_{\gamma }|}|\mathrm{per}(L^{\gamma }){|}^2,$$

Full text

MSC(2010): 15A15, 15B48.

Keywords: Permanent dominant conjecture, Generalized Cauchy-Binet Theorem.

E-mail addresses: kijtir@nu.ac.th (Kijti Rodtes).

*∗*Research center for Academic Excellence in Mathematics, Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

1. Introduction

Throughout this paper, we denote $M_{n}(\mathbb{C})$ the set of square matrices over the field of complex numbers $\mathbb{C}$. If $A\in M_{n}(\mathbb{C})$ is a positive semi-definite matrix, then it is well known that (see, for example, Chapter 7 in [3]) $h(A)\geq\det(A)\geq 0$ (Hadamard inequality) and $\det(B)\det(C)\geq\det(A)$ (Fischer inequality), where $h(A)=\prod_{i=1}^{n}A_{ii}$ is the product of the main diagonal entries of $A=(A_{ij})$ and $B,C$ are square main diagonal blocks (any size) of $A$. For a finite group $G$ embedded in the full symmetric group $S_{n}$ and character $\chi$ of $G$, Issai Schur ([6], 1918), introduced the notion of generalized matrix function $d^{G}_{\chi}:M_{n}(\mathbb{C})\longrightarrow\mathbb{C}$ as

[TABLE]

If $G=S_{n}$ and $\pi$ is a character of $S_{n}$, then $d^{G}_{\chi}(A)$ is called an immanant. Permanent and determinant are immanants with $\chi=1$ (the principle character) and $\chi=\epsilon$ (alternating character), respectively. Also, $h(A)=d^{\{e\}}_{1}(A)$ and $\det(B)\det(C)=d^{S_{m}\times S_{k}}_{(\epsilon_{1},\epsilon_{2})}(A)$, where $\{e\}$ is the trivial group and $m,k$ are sizes of square diagonal blocks $B,C$ of $A$ and $\epsilon_{1},\epsilon_{2}$ are alternating characters of $S_{m},S_{k}$, respectively. Issai Schur also extended the Hadamard inequality and Fisher inequality as the Schur inequality, [6]:

[TABLE]

for any positive semi-definite matrix $A\in M_{n}(\mathbb{C})$ and character $\chi$ of $G$, where $\bar{d}^{G}_{\chi}(A)$ is the normalized of $d^{G}_{\chi}(A)$, i.e., $\bar{d}^{G}_{\chi}(A):=(1/\chi(e))d^{G}_{\chi}(A)$.

In the opposite direction of the Schur inequality, Marvin Marcus showed in 1963 that, [2],

[TABLE]

for any positive semi-definite matrix $A$. Three years later, Elliott Hershel Lieb obtained that, [1],

[TABLE]

where $B,C$ are square diagonal blocks of the positive semi-definite matrix $A$. He also raised the Permanent Dominant Conjecture as:

[TABLE]

for any positive semi-definite matrix $A\in M_{n}(\mathbb{C})$ and irreducible character $\chi$ of $G\leq S_{n}$.

This conjecture still opens up to this date. There are progresses on this conjecture but most of them concentrate on particular groups and characters; especially on immanants, which is known to satisfy the conjecture when $n\leq 13$, [5]. Also, according to Lieb inequality, the conjecture holds true for any Young subgroup of $S_{n}$ and $\chi=1$, [4]. Further details of the progress to this conjecture along this direction and related conjectures can be found, for example, in [7], [8] and references therein.

On the other hand, it is quite obvious to see that, for any positive semi-definite matrix $A\in M_{n}(\mathbb{C})$, if $A$ is a non-negative real matrix, or $A$ has rank $1$, or $A$ contains a zero row (or column), or $A$ has a zero submatrix of dimension $r\times s$ with $r+s\geq n+1$, then $A$ satisfies the conjecture (see more details in the next section). However, besides these classes of matrices, it seems that there is no new explicit class of positive semi-definite matrices provided to serve the conjecture. In this paper, we use generalized Cauchy-Binet theorem together with character theory to see some relationship (identity) between determinant and generalized matrix functions, in Theorem 3.2. A criterion of positive semi-definite matrices affirming the conjecture is given in Corollary 3.7. As a consequence, infinitely many infinite classes of positive semi-definite serving the conjecture are generated from any positive semi-definite matrix having no zero in the first column, in Theorem 3.8.

2. Some basic remarks

Throughout this section, we let $A=(A_{ij})\in M_{n}(\mathbb{C})$ be a positive semi-definite matrix and let $G\leq S_{n}$ be a subgroup of $S_{n}$. For a character $\chi$ of $G$, it is well known in character theory that $\chi$ is a linear combination of irreducible characters of $G$ with non-negative integer coefficients. We then have that:

Proposition 2.1**.**

If $\bar{d}^{G}_{\chi}(A)\leq\operatorname{per}(A)$ for all irreducible characters $\chi$ of $G$, then $\bar{d}^{G}_{\lambda}(A)\leq\operatorname{per}(A)$ for all characters $\lambda$ of $G$.

For each $g\in G$, denote $l_{g}(A):=\prod_{i=1}^{n}A_{ig(i)}$; so $d^{G}_{\chi}(A)=\sum_{g\in G}\chi(g)l_{g}(A)$. It is also well known in character theory that $|\chi(g)|\leq\chi(e)$ for all $g\in G$, where $e$ is the identity of $G$. Then $|\chi(g)|/\chi(e)\leq 1$ which yields that

[TABLE]

We then have that:

Proposition 2.2**.**

If $l_{\sigma}(A)\geq 0$ for all $\sigma\in S_{n}$, then $\bar{d}^{G}_{\chi}(A)\leq\operatorname{per}(A)$ for any character $\chi$ of $G$. In particular, if $A$ is a non-negative real matrix (including diagonal positive semi-definite matrix), then $A$ satisfies the permanent dominant conjecture.

We observe that if $A$ contains a zero row or zero column, then $l_{\sigma}(A)=0$ for all $\sigma\in S_{n}$ (because $\sigma:[n]\longrightarrow[n]$ is one to one and onto function, where $[n]:=\{1,2,\dots,n\}$). Also, by Frobenius-Konig theorem (see, for example, Theorem 5.20 in [9]), $l_{\sigma}(A)=0$ for all $\sigma\in S_{n}$ if and only if $A$ contains an $r\times s$ zero submatrix, where $r+s=n+1$. We then have that:

Proposition 2.3**.**

If $A$ has a zero row or zero column or $A$ contains an $r\times s$ zero submatrix, where $r+s=n+1$, then $A$ satisfies the permanent dominant conjecture.

Let $O_{n}$ be the set of all odd permutations in $S_{n}$. By Schur inequality, $\det(A)\leq\operatorname{per}(A)$; namely,

[TABLE]

which means that $\sum_{\sigma\in O_{n}}l_{\sigma}(A)\geq 0$. We can conclude that:

Proposition 2.4**.**

For the alternating group $A_{n}$ and the principle character $1$,

[TABLE]

Furthermore, if $\operatorname{rank}(A)=1$, then $A=vv^{*}$ for some $v=(a_{1},a_{2},\dots,a_{n})^{t}\in\mathbb{C}^{n}$ (since $A$ is a positive semi-definite matrix). Then, for each $\sigma\in S_{n}$,

[TABLE]

By Proposition 2.2, we can conclude that:

Proposition 2.5**.**

If positive semi-definite matrix $A\in M_{n}(\mathbb{C})$ has $\operatorname{rank}(A)=1$, then $A$ satisfies the permanent dominant conjecture.

Moreover, the above conclusion can be extended to any sum of rank one matrix and non-negative diagonal real matrix as the following proposition.

Proposition 2.6**.**

If $A=vv^{*}+\operatorname{diag}(d_{1},d_{2},\dots,d_{n})$, where $v\in\mathbb{C}^{n}$ and $d_{i}\geq 0$ for all $i$, then $A$ satisfies the permanent dominant conjecture.

Proof.

Let $v=(a_{1},a_{2},\dots,a_{n})^{t}$ and $\sigma\in S_{n}$. Then,

[TABLE]

The non-zero summand exists when the index set $I$ satisfies $\sigma(i)=i$ for all $i\notin I$; denote the family of all such index sets by $N^{c}$. Hence, $\sigma(I)=I$ for all $I\in N^{C}$ and thus

[TABLE]

By Proposition 2.2, the proof is completed. ∎

3. Classes of matrices affirming the conjecture

Throughout this section, let $G$ be a finite group embedded in the full symmetric group $S_{n}$ and let $\chi$ be an irreducible character (need not be linear) of $G$. We first recall the Cauchy-Binet Theorem for generalized matrix function:

Theorem 3.1** ([3], Theorem 7.34).**

If $\alpha,\beta\in\Omega:=\{\gamma\in\Gamma_{m,n}\;|\;(\chi,1)_{G_{\gamma}}\neq 0\}$, then

[TABLE]

for any $A,B\in M_{n}(\mathbb{C})$, where $\Gamma_{m,n}=\{\gamma:=(\gamma_{1},\gamma_{2},\dots,\gamma_{m})\in\mathbb{N}^{m}\;|\;1\leq\gamma_{i}\leq n\}$, $G_{\gamma}=\{g\in G\;|\;g\cdot\gamma:=(\gamma_{g(1)},\gamma_{g(2)},\dots,\gamma_{g(m)})=\gamma\}$, $(\chi,1)_{G_{\gamma}}=\frac{1}{|G_{\gamma}|}\sum_{g\in G_{\gamma}}\chi(g)$ and $AB[\alpha|\beta]$ is the matrix obtained from $AB$ in which the rows and columns are indexed by $\alpha$ and $\beta$, respectively.

Let $A$ be a positive semi-definite matrix in $M_{n}(\mathbb{C})$. Then, by the Cholesky decomposition, there exits a lower triangular matrix $L$ such that $A=LL^{*}$. We observe that, for $\gamma,(n):=(1,2,\dots,n)\in\Gamma_{n,n}$,

[TABLE]

By Theorem 3.1 with $G\leq S_{n}$ and $\alpha=\beta=(n)$, we now compute that

[TABLE]

namely,

[TABLE]

Note also that, for any $g\in S_{n}$, $L[(n)|g\cdot\gamma]=L[(n)|\gamma]P_{g}$, where $P_{g}=(\delta_{ig(j)})$ is the permutation matrix associated to $g\in S_{n}$. Then, by setting $L^{\omega}:=L[(n)|\omega]$ (for $\omega\in\Gamma_{n,n}$), we have that

[TABLE]

for any $g\in G$, where $g\cdot\gamma:=(\gamma_{g(1)},\gamma_{g(2)},\dots,\gamma_{g(n)})$ (the standard action of $G$ on $\Gamma_{n,n}$). Let $\Delta_{G}$ be the set of all representatives (first member of each orbit ordered by lexicographic ordering) of the action of $G$ on $\Gamma_{n,n}$ and let $\bar{\Delta}_{G}=\Delta_{G}\cap\Omega$. For each $\gamma\in\bar{\Delta}_{G}$ denote the set of all representatives of left cosets of $G_{\gamma}$ in $G$ by $S_{\gamma}$. Then, for each orbital $[\gamma]$,

[TABLE]

According to $\Omega=\cup_{\gamma\in\bar{\Delta}_{G}}[\gamma]$ (Lemma 6.22 in [3]), the relation (3.1) becomes

[TABLE]

Now, we consider the sequence $\gamma^{\circ}:=(n)$ which clearly belongs to $\bar{\Delta}_{G}$ and $S_{\gamma^{\circ}}=G$. Since $L$ is a lower triangular matrix,

[TABLE]

Since $\chi$ is an irreducible character, $(\chi,\chi)_{G}=1$ which means that

[TABLE]

So,

[TABLE]

Moreover, $\sigma\cdot\gamma^{\circ}$ for $\sigma\in S_{G}:=\{\sigma_{1},\dots,\sigma_{t}\}$, the set of all representatives for the right cosets of $S_{n}$ by $G$, are in different orbits and all entries of the sequence $\sigma\cdot\gamma^{\circ}$ are distinct. If $\sigma\in S_{G}\setminus G$, then

[TABLE]

because $\sigma gh\neq e$ for any $\sigma\notin G$ and $g,h\in G$.

Next, we consider the sequence $\gamma^{k}:=(k,k,\dots,k)$, for each $1\leq k\leq n$. This sequence belongs to $\bar{\Delta}_{G}$ if and only if $\chi=1$. Since $S_{\gamma^{k}}=\{e\}$, we get that

[TABLE]

Hence, by setting

[TABLE]

and using the above discussion together with relation (3.3), we can conclude that:

Theorem 3.2**.**

Let $G$ be a finite subgroup of $S_{n}$ and $\chi$ be an irreducible character of $G$. Let $A=LL^{*}$ be a positive semi-definite matrix in $M_{n}(\mathbb{C})$, where $L=(L_{ij})$ is a lower triangular matrix from a Cholesky decomposition of $A$. Then

[TABLE]

where $\delta_{\chi,1}$ is the Kronecker delta function.

This result immediately implies that:

Corollary 3.3** (Schur inequality).**

Let $\chi$ be an irreducible character of a finite subgroup $G$ of $S_{n}$. Then

[TABLE]

for any positive semi-definite matrix $A\in M_{n}(\mathbb{C})$.

In particular, when $\chi$ is linear, it turns out that

[TABLE]

Since $|\chi(g)|=1$ and $\chi(gh)=\chi(g)\chi(h)$ for all $g,h\in G$ (because $\chi$ is linear),

[TABLE]

By using Theorem 3.2 and using the same notations as above, we have that:

Corollary 3.4**.**

Let $\chi$ be a linear character of a finite subgroup $G$ of $S_{n}$. Then

[TABLE]

In particular,

[TABLE]

where $\bar{\Delta}^{**}_{S_{n}}=\bar{\Delta}^{*}_{S_{n}}-\{(1,1,\dots,1,n)\}$.

Proof.

The term $(n-1)!\prod_{i=1}^{n-1}|L_{i1}|^{2}|L_{nn}|^{2}$ comes from the evaluating $\frac{1}{|(S_{n})_{\gamma}|}|\operatorname{per}(L^{\gamma})|^{2}$ with $\gamma=(1,1,\dots,1,n)$ directly. ∎

In order to bound the value of $\bar{d}^{G}_{\chi}(A)$, by Theorem 3.2, it is reasonable to concentrate only on $\gamma\in\bar{\Delta}^{*}_{G}$ for which $d^{G}_{\chi}(L^{\gamma}P_{g})\neq 0$ for some $g\in S_{\gamma}$; denote the set of all such sequences by $\hat{\Delta}^{*}_{G}$. We have that:

Lemma 3.5**.**

By using the same notions as above and $n>3$,

[TABLE]

Proof.

Let $A:=\{\sigma\cdot(1,2,\dots,n)\;|\;\sigma\in S_{n}\}$ and $B:=\{\omega:=(\omega_{1},\omega_{2},\dots,\omega_{n})\in\Gamma_{n,n}\;|\;2\leq\omega_{i}\leq n,\forall i=1,2,\dots,n\}$. If $n>3$ is even, we define $C^{e}$ to be the set of all sequences $\omega:=(\omega_{1},\omega_{2},\dots,\omega_{n})\in\Gamma_{n,n}$ satisfying $\omega_{j}=1$ for some $j\in[n]$ and there exist $J\subseteq[n]$ with $|J|=\frac{n}{2}+1$ such that $\omega_{k}\in\{\frac{n}{2}+1,\frac{n}{2}+2,\dots,n\}$ for all $k\in J$. If $n>3$ is odd, we define $C^{o}$ to be the set of all sequences $\omega:=(\omega_{1},\omega_{2},\dots,\omega_{n})\in\Gamma_{n,n}$ satisfying $\omega_{j}=1$ for some $j\in[n]$ and there exist $J\subseteq[n]$ with $|J|=\frac{n+1}{2}$ such that $\omega_{k}\in\{\frac{n+1}{2},\frac{n+1}{2}+1,\dots,n\}$ for all $k\in J$. Since $L$ is a lower triangular matrix, for any sequence $\gamma$ in $B$ or $C^{e}$ or $C^{o}$, it turns out that $d^{G}_{\chi}(L^{\gamma}P_{g})=0$ for all $g\in G$. By setting $\Omega^{*}:=\Omega-A-B-C^{\delta(n)}$ ($C^{\delta(n)}=C^{e}$ if $n$ is even, and $C^{\delta(n)}=C^{o}$ if $n$ is odd), we have that $\hat{\Delta}^{*}_{G}\subseteq\Omega^{*}$. By direct calculation, we have $|\Omega|\leq n^{n}$, $|A|=n!$, $|B|=(n-1)^{n}$, $|C^{e}|\geq n^{\frac{n-4}{2}}(\frac{n-2}{2})^{{\frac{n+2}{2}}}(\frac{n+4}{2})$ and $|C^{o}|\geq n^{\frac{n-5}{2}}(\frac{n-1}{2})^{{\frac{n+1}{2}}}(\frac{n+3}{2})$. Due to the sets $A,B$, $C^{\delta(n)}$, $\{(1,1,\dots,1)\}$ are pairwise disjoint and $(1,1,\dots,1)\notin\hat{\Delta}^{*}_{G}$, the result follows. ∎

For each positive semi-definite matrix $A=LL^{*}$, denote $\alpha=\prod_{i=1}^{n}|L_{i1}|$ and $\alpha_{0}=\max_{1\leq i\leq n}|L_{i1}|$. For each finite group $G\leq S_{n}$, denote

[TABLE]

Let $\bar{\epsilon}(n,G):=\min\{\frac{1}{\alpha_{0}^{n-1}}\epsilon(n,G),1\}$. By using the above notations, the following theorem holds:

Theorem 3.6**.**

If $A=LL^{*}$ with $|L_{kj}|\leq\bar{\epsilon}(n,G)$ for all $k,j$ with $j\neq 1$, then

[TABLE]

for any irreducible character $\chi$ of $G\leq S_{n}$ with $n>3$.

Proof.

Let $\gamma=(\gamma_{1},\dots,\gamma_{n})\in\hat{\Delta}^{*}_{G}$. Because of $(1,1,\dots,1)\notin\hat{\Delta}^{*}_{G}$, there exists $s$ such that $\gamma_{s}\neq 1$. Since $|L_{kj}|\leq\epsilon(n,G)$ and $|L_{kj}|\leq 1$ for all $k,j$ such that $j\neq 1$, we compute that, for any $g\in G$,

[TABLE]

Then, by power mean inequality (or by Cauchy-Schwartz inequality),

[TABLE]

Thus, for each $\gamma\in\hat{\Delta}^{*}_{G}$,

[TABLE]

which also yields that, by Lemma 3.5,

[TABLE]

By Theorem 3.2, the proof is completed. ∎

Now, by setting, for $n>3$,

[TABLE]

we see that $\epsilon_{n}\leq\bar{\epsilon}(n,G)$ for any finite group $G\leq S_{n}$. Hence, the following holds:

Corollary 3.7**.**

Let $A=LL^{*}$ with $|L_{kj}|\leq\epsilon_{n}$ for all $k,j$ with $j\neq 1$. Then

[TABLE]

for any group $G\leq S_{n}$ and any irreducible character $\chi$ of $G$.

Proof.

Since $\bar{d}^{G}_{\chi}(A)\leq per(A)$ for any positive semi-definite matrix $A\in M_{n}(\mathbb{C})$ with $n\leq 3$, it suffices to consider the cases $n>3$. Thus, by Theorem 3.6, we have

[TABLE]

When $|G|=n!$ and $\chi=1$, $\bar{d}^{G}_{\chi}(A)$ becomes $\operatorname{per}(A)$ where the inequality is obviously true. Also, when $|G|=n!$ but $\chi\neq 1$, $\operatorname{per}(A)-\bar{d}^{G}_{\chi}(A)\geq(\frac{1}{3}n!)\prod_{i=1}^{n}|L_{i1}|^{2}\geq 0$. Furthermore, when $|G|=\frac{n!}{2}$, it is well known that $G$ must be the alternating group $A_{n}$, which is proved that $d^{A_{n}}_{1}(A)\leq\operatorname{per}(A)$ in Proposition 2.4. If, however, $\chi\neq 1$, then

[TABLE]

Hence, by Lagrange’s Theorem ($|G|$ divides $|S_{n}|$), it remains only to consider groups $G$ with $|G|\leq\frac{n!}{3}$ and this is done by (3.4). ∎

The criterion in Corollary 3.7 provides us infinitely many infinite classes of positive semi-definite matrix affirming the conjecture.

Theorem 3.8**.**

Let $A=LL^{*}\in M_{n}(\mathbb{C})$ with $m:=\max\{|L_{ij}|\;|\;2\leq i,j\leq n\}$ and $a:=\frac{\epsilon_{n}}{m}$. If each entry in the first column of $A$ is not zero, then

[TABLE]

is an infinite class of positive semi-definite matrix satisfying the permanent dominant conjecture.

Proof.

Let $\lambda_{2},\dots,\lambda_{n}$ be positive real numbers lying in $\{r\in\mathbb{R}\;|\;0<r\leq a\}$ and let $D:=\operatorname{diag}(1,\lambda_{2},\dots,\lambda_{n})$. Then, the first columns of the lower triangular matrices $L$ and $L_{D}:=LD$ are identical. This yields that $\epsilon_{n}(L_{D})=\epsilon_{n}(L):=\epsilon_{n}$. Since each entry in the first column of $A$ is not zero, so is $L$. Thus $\epsilon_{n}>0$; namely $[A]$ is infinite. Also, for $2\leq i,j\leq n$,

[TABLE]

By Corollary 3.7, $L_{D}L_{D}^{*}=LD^{2}L^{*}=L\operatorname{diag}(1,\lambda^{2}_{2},\dots,\lambda^{2}_{n})L^{*}$ affirms the conjecture. ∎

Note that if $a\geq 1$, then such positive semi-definite $A$ satisfies the conjecture. Furthermore, for any complex diagonal matrix $D:=\operatorname{diag}(c_{1},\dots,c_{n})$ and $\sigma\in G$,

[TABLE]

which yields that $\bar{d}^{G}_{\chi}(DAD^{*})=|\prod_{i=1}^{n}c_{i}|^{2}\bar{d}^{G}_{\chi}(A)$. Hence, for $\prod_{i=1}^{n}c_{i}\neq 0$, we have that $\bar{d}^{G}_{\chi}(DAD^{*})\leq\operatorname{per}(DAD^{*})$ if and only if $\bar{d}^{G}_{\chi}(A)\leq\operatorname{per}(A)$. In particular, for non-zero complex numbers $c_{1},\dots,c_{n}$, any matrix in the form

[TABLE]

where $X\in[A]$ (in Theorem 3.8), satisfies the permanent dominant conjecture.

Acknowledgments

The author would like to thank anonymous referee(s) for reviewing this manuscript. He also would like to thank Naresuan University (NU), and National Science, Research and Innovation Fund (NSRF): Grant NO. FRB660001/0179, for financial support.

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyses during the current study.

Conflict of interest

The author has no relevant financial or non-financial interests to disclose.