Immanant inequalities and weight spaces

TL;DR

This paper develops a trace formula for immanants of submatrices using weight spaces, extends Kostant's formula, and establishes new inequalities and criteria for non-vanishing immanants in positive matrices.

Contribution

It introduces a generalized trace formula for immanants based on weight spaces, extending Kostant's work, and provides new non-vanishing criteria and inequalities for positive matrices.

Findings

Derived a trace formula for immanants on generalized principal submatrices.

Established a criterion for non-vanishing immanants in positive definite matrices.

Presented a unifying inequality encompassing Kostant, Schur, and Stembridge's inequalities.

Abstract

We first obtain a trace formula for immanants of generalized principal submatrix of any complex matrix based on any weight space for finite dimensional representations of the general linear group. Our trace formula contains Kostant's famous formula for immanants on $0$-weight spaces as special case. We then present a criterion for non-vanishing immanants for any generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices, which strengthened the well-known results of Schur and Stembridge. Furthermore, we present an inequality that contains Kostant, Schur and Stembridge's famous inequalities as special cases.

Equations

$${\mathrm{Imm}}_{{\chi }^{\lambda }}(A)=\sum _{\sigma \in {\mathfrak{S}}_n}{\chi }^{\lambda }(\sigma )\prod _{i=1}^n{a}_{\sigma (i),i}.$$

$${\chi }^{\lambda }(1)\det (A)\le {\mathrm{Imm}}_{{\chi }^{\lambda }}(A),$$

$${\mathrm{Imm}}_{{\varphi }^{\lambda }}(A)\ge 0,$$

$$H_{\mu /\nu }(X)=[h_{{\mu }_i-{\nu }_j}(X){]}_{i,j=1}^n.$$

$${\chi }^{\lambda }(1)\mathrm{per}(A)\ge {\mathrm{Imm}}_{{\chi }^{\lambda }}(A).$$

$$({\mathbb{C}}^n{)}^{\otimes m}\cong \underset{\genfrac{}{}{0ex}{}{\lambda ⊢m}{\text{depth}(\lambda )\le n}}{⨁}{U}^{\lambda }\otimes V^{\lambda },$$

$$⟨e_{i_1}\otimes \cdots \otimes e_{i_m},e_{j_1}\otimes \cdots \otimes e_{j_m}⟩$$

$$⟨x\cdot e_{i_1}\otimes \cdots \otimes e_{i_m},e_{j_1}\otimes \cdots \otimes e_{j_m}⟩$$

$$=⟨e_{i_1}\otimes \cdots \otimes e_{i_m},x^{\ast }\cdot e_{j_1}\otimes \cdots \otimes e_{j_m}⟩,$$

$${\lambda }_{ij}\ge {\lambda }_{i-1,j}\ge {\lambda }_{i,j+1},1\le j\le i\le n.$$

$$C_n(u)=\sum _{\sigma \in {\mathfrak{S}}_n}sgn(\sigma )(u+E{)}_{\sigma (1),1}\cdots (u+E-n+1{)}_{\sigma (n),n}.$$

$$C_k(u){\xi }_{\Lambda }=(u+{\lambda }_{k1})(u+{\lambda }_{k2}-1)\cdots (u+{\lambda }_{kk}-k+1){\xi }_{\Lambda },\text{for}1\le k\le n.$$

$$({\mathbb{C}}^n{)}^{\otimes m}\cong \sum _{\genfrac{}{}{0ex}{}{\lambda ⊢m,}{\text{depth}(\lambda )\le n}}\sum _{(\Lambda ,\mathcal{T})}\mathbb{C}{\xi }_{\Lambda }\otimes v_{\mathcal{T}}.$$

$$⟨{\xi }_{\Lambda }\otimes v_{\mathcal{T}},{\xi }_{{\Lambda }^{\prime }}\otimes v_{{\mathcal{T}}^{\prime }}⟩={\delta }_{\Lambda {\Lambda }^{\prime }}{\delta }_{\mathcal{T}{\mathcal{T}}^{\prime }}$$

$${\mathcal{P}}_{\mu }:({\mathbb{C}}^n{)}^{\otimes m}\to (({\mathbb{C}}^n{)}^{\otimes m}{)}_{\mu },$$

$$m(I)=m_1!m_2!\cdots m_n!.$$

$${\mathfrak{S}}_I={\mathfrak{S}}_{m_1}\times {\mathfrak{S}}_{m_2}\times \cdots \times {\mathfrak{S}}_{m_n}$$

$$s_i{v}_{\Lambda }=\frac{1}{c_{i+1}-c_i}{v}_{\Lambda }+\frac{\sqrt{(c_{i+1}-c_i{)}^2-1}}{c_{i+1}-c_i}{v}_{s_i\Lambda }$$

$${\mathcal{E}}_{\mathcal{T}}^{\lambda }=\frac{1}{{\prod }_{(i,j)\in \mathcal{T}}h(i,j)}\sum _{\sigma \in {\mathfrak{S}}_n}⟨{\sigma }^{-1}{v}_{\mathcal{T}},v_{\mathcal{T}}⟩\sigma ,$$

$$\frac{{\mathrm{Imm}}_{{\chi }^{\lambda }}(A_I)}{m(I)}=tr({\mathcal{P}}_{\mu }\otimes 1)\circ {\pi }_{\lambda }(A)\circ {\mathcal{P}}_{\mu }{|}_{U^{\lambda }},$$

$${\lambda }_{ij}$$

$${\mu }_i$$

$${\theta }_{\mu }:SYT(\lambda )\to YT(\lambda )$$

$$\displaystyle\sqrt{\frac{h_{\lambda}}{m(I)}}\mathcal{E}_{\mathcal{T}}^{\lambda}\cdot e_{i_{1}}\otimes\cdots\otimes e_{i_{m}}=\left\{\begin{array}[]{cc}c\cdot\xi_{\theta_{\mu}(\mathcal{T})}\otimes v_{\mathcal{T}}&\text{ if }\theta_{\mu}(\mathcal{T})\in SSYT(\lambda,n),\\ 0&\text{ if }\theta_{\mu}(\mathcal{T})\notin SSYT(\lambda,n),\end{array}\right.$$

$$\sqrt{h_{\lambda }}{\mathcal{E}}_{\mathcal{T}}^{\lambda }\cdot e_1\otimes \cdots \otimes e_m={\xi }_{\mathcal{T}}\otimes v_{\mathcal{T}}.$$

$$tr({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ {\mathcal{P}}_{\mu }{|}_{U^{\lambda }}$$

$$=⟨\sqrt{\frac{h_{\lambda }}{m(I)}}({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ \sum _{\mathcal{T}}{\mathcal{E}}_{\mathcal{T}}\cdot e_{i_1}\otimes \cdots \otimes e_{i_m},\sqrt{\frac{h_{\lambda }}{m(I)}}\sum _{\mathcal{T}}{\mathcal{E}}_{\mathcal{T}}\cdot e_{i_1}\otimes \cdots \otimes e_{i_m}⟩$$

$$=\frac{h_{\lambda }}{m(I)}⟨({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ \sum _{\mathcal{T}}{\mathcal{E}}_{\mathcal{T}}\cdot e_{i_1}\otimes \cdots \otimes e_{i_m},e_{i_1}\otimes \cdots \otimes e_{i_m}⟩.$$

$$\frac{h_{\lambda }}{m(I)}({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ \sum _{\mathcal{T}}{\mathcal{E}}_{\mathcal{T}}\cdot e_{i_1}\otimes \cdots \otimes e_{i_m}$$

$$=\frac{1}{m(I)}({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ \sum _{\genfrac{}{}{0ex}{}{\tau \in {\mathfrak{S}}_I,}{\mu \in \mathcal{M}({\mathfrak{S}}_m/{\mathfrak{S}}_I)}}{\chi }^{\lambda }({\mu }^{-1}\tau )P_{{\mu }^{-1}}{P}_{\tau }\cdot e_{i_1}\otimes \cdots \otimes e_{i_m}$$

$$=\frac{1}{m(I)}({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\sum _{\genfrac{}{}{0ex}{}{\tau \in {\mathfrak{S}}_I,}{\mu \in \mathcal{M}({\mathfrak{S}}_m/{\mathfrak{S}}_I)}}{\chi }^{\lambda }({\mu }^{-1}\tau )e_{i_{{\mu }_1}}\otimes \cdots \otimes e_{i_{{\mu }_m}}$$

$$=\frac{1}{m(I)}\sum _{(j_1,\dots ,j_m)}\sum _{\genfrac{}{}{0ex}{}{\tau \in {\mathfrak{S}}_I,}{\mu \in \mathcal{M}({\mathfrak{S}}_m/{\mathfrak{S}}_I)}}{\chi }^{\lambda }({\mu }^{-1}\tau )({\mathcal{P}}_{\mu }\otimes 1)a_{j_1,i_{{\mu }_1}}\cdots a_{j_m,i_{{\mu }_m}}\otimes e_{j_1}\otimes \cdots \otimes e_{j_m}$$

$$=\frac{1}{m(I)}\sum _{\sigma \in {\mathfrak{S}}_m}\sum _{\genfrac{}{}{0ex}{}{\tau \in {\mathfrak{S}}_I,}{\mu \in \mathcal{M}({\mathfrak{S}}_m/{\mathfrak{S}}_I)}}{\chi }^{\lambda }({\mu }^{-1}\tau )a_{i_{{\sigma }_1},i_{{\mu }_1}}\cdots a_{i_{{\sigma }_m},i_{{\mu }_m}}\otimes e_{i_{{\sigma }_1}}\otimes \cdots \otimes e_{i_{{\sigma }_m}},$$

$$tr({\mathcal{P}}_{\mu }\otimes 1)\circ \pi (A)\circ {\mathcal{P}}_{\mu }{|}_{U^{\lambda }}$$

Full text

MSC (2020): Primary: 20G05 Secondary: 15A15, 17B35, 05E10

Keywords: Immanant, weight spaces, Schur-Weyl duality, immanant inequality.

1. Introduction

In this paper, we will prove three results on the immanants of postive semidefinite matrices or totally nonnegative matrices which generalize and strengthen some well-known results of Schur, Stembridge and Kostant.

Immanants, a class of matrix functions generalizing determinants and permanents, were first formulated by Schur during his fundamental paper [13]. The term “immanant” was formally introduced by Littlewood and Richardson [7, 8] in their foundational study of representation theory of the symmetric groups.

Let $\mathfrak{S}_{n}$ be the symmetric group acting on the index set $\{1,\ldots,n\}$. For the complex character $\chi^{\lambda}:\mathfrak{S}_{n}\rightarrow\mathbb{C}$ of the irreducible representation parameterized by partition $\lambda$ of $n$ and $n\times n$ complex matrix $A=(a_{ij})$, the $\lambda$-immanant of $A$ is defined by

[TABLE]

For positive semidefinite Hermitian matrice $A$, Schur [13] established a fundamental inequality:

[TABLE]

demonstrating that the determinant (corresponding to $\lambda=(1^{n})$) is the minimal among all normalized immanants. Later, Stembridge [14] proved the same inequality for totally nonnegative matrices (real matrices with nonnegative minors). He conjectured a stronger inequality.

Conjecture 1.1**.**

[15]** If $A$ is a totally nonnegative $n\times n$ matrix and $\lambda\vdash n$, then

[TABLE]

where $\mathrm{Imm}_{\phi^{\lambda}}$ is the monomial immanant (i.e. the image of $\phi^{\lambda}$ is the monomial symmetric function $m_{\lambda}$, under the characteristic map from the algebra of class functions of $\mathfrak{S}_{n}$ to the algebra of symmetric functions).

Let $h_{n}$ denote the complete homogeneous symmetric function of degree $n$ in countably many variables $X=(x_{1},x_{2},\ldots)$, $\mu$ and $\nu$ be partitions with $n$ parts. The Jacobi–Trudi matrices are of the form

[TABLE]

Since the minors of a Jacobi-Trudi matrix are skew Schur functions, they are nonnegative in the sense of being nonnegative linear combinations of Schur functions. Haiman [3] showed that $\operatorname{Imm}_{\lambda}(H_{\mu/\nu})-\chi^{\lambda}(1)\det(H_{\mu/\nu})$ is a nonnegative linear combination of Schur functions.

A longstanding open problem is to determine the maximal immanant. In 1966, Lieb [6] conjectered that

Conjecture 1.2**.**

[6]** For any positive semidefinite Hermitian $A$ and partition $\lambda\vdash n$, the permanent dominates all $\lambda$-immanants:

[TABLE]

This conjecture has been proved for $n\leq 13$ [11, 12]. A comprehensive discussion of Lieb’s permanent dominace conjecture and related conjectures can be found in [16, 18].

Based on the inequalities established by Schur and Stembridge, it is known that the immanant of a singular matrix (whether positive semidefinite or totally nonnegative) is nonnegative. Knowing whether they are nontrivial or not will provide further insight to Lieb’s conjecture. In this work, we first provide a sufficient condition for the immanant of such a singular matrix to be strictly positive, which then strengthens Schur and Stembridge’s results.

On the other hand, Kostant [5] has focused on an interpretation of immanants based on 0-weight spaces for finite dimensional representations of $\mathrm{SL}(n,\mathbb{C})$ and gives a different proof of the inequalities of Schur and Stembridge [13, 14] as an application. Kostant’s formulation also provides a trace formula for the $\lambda$-immanants without repeating indices. Inspired by the work of Kostant, we recently generalized the trace formula to quantum coordinate algebra in [4].

In the second part, with the help of the Schur-Weyl duality, we will give a weight space interpretation for immanants of any generalized principal submatrix of complex matrix in the general situation, i.e. for all weight space situation. The new trace formula leads to a new criterion on the nonvanishing of $\mathrm{Imm}_{\chi^{\lambda}}(A_{I})$, and we show $\mathrm{Imm}_{\chi^{\lambda}}(A_{I})=0$ or not only depends on whether the dimension of the weight space corresponding to $I$ in the highest weight representation $U^{\lambda}$ of $\mathrm{GL}_{n}$ is nonzero. In other words, $\mathrm{Imm}_{\chi^{\lambda}}(A_{I})>0$ if and only if $\lambda$ dominates the weight corresponding to $I$. In view of Stembridge’s conjecture, this also provides a deeper understanding of immanants for totally nonnegative matrices.

Our last main result provides an immanant inequality for either positive semidefinite or totally nonnegative matrix, which generalized the Kostant inequality in [math]-weight space to the orbit $O_{\mu}$ of any weight space $\mu$ under the action of Weyl group. In particular, Schur’s and Stembridge’s inequalities can be acquired as special cases.

The paper is organized as follows. In section 2, we obtain a trace formula for the immanants of any generalized principal submatrix. In section 3, we give a criterion for the nonvanishing of immanants of certain positive semidefinite Hermitian or totally nonnegative matrices. Finally, we prove an immanant inequality for either positive semidefinite or totally nonnegative matrix.

2. Immanants of generalized submatrix

2.1. Schur-Weyl duality

By the classical Schur-Weyl duality, $(\mathbb{C}^{n})^{\otimes m}$ is a multiplicity free $\mathrm{GL}_{n}\times\mathfrak{S}_{m}$-module and there exists a decomposition of the space of tensors

[TABLE]

where $(\pi_{\lambda},U^{\lambda})$ is the simple left $\mathrm{GL}_{n}$-module of highest weight $\lambda$ and $(\rho_{\lambda},V^{\lambda})$ is the simple left $\mathfrak{S}_{m}$-module indexed by $\lambda$.

Let $\{e_{i}\}$ be the basis of $\mathbb{C}^{n}$. Define an inner product $\langle\cdot|\cdot\rangle$ on $(\mathbb{C}^{n})^{\otimes m}$ by

[TABLE]

where $i_{1},\ldots,i_{m},j_{1},\ldots,j_{m}\in\{1,2,\ldots,n\}$. We introduce $*$-operations for $\mathfrak{S}_{m}$ and $\mathrm{GL}_{n}$ respectively. Let $\sigma^{*}=\sigma^{-1}\in\mathfrak{S}_{m}$. And denote by $A^{*}$ the conjugate transpose of matrix $A$ in $\mathrm{GL}_{n}$. They are involutive anti-automorphisms.

Lemma 2.1**.**

The $*$-operations for $\mathfrak{S}_{m}$ and $\mathrm{GL}_{n}$ on the inner product space $(\mathbb{C}^{n})^{\otimes m}$ afford a contravariant inner product, i.e.

[TABLE]

where $x$ is any element in $\mathfrak{S}_{m}$ or $\mathrm{GL}_{n}$.

Let $\Lambda=(\lambda_{ij})_{1\leq i\leq n\atop 1\leq j\leq i}$ be the Gelfand-Tsetlin patterns satisfying the relations [2]:

[TABLE]

Denote by $\{\xi_{\Lambda}\}$ the Gelfand-Tsetlin basis of the $\mathfrak{gl}(n)$-module $U^{\lambda}$. Let $E_{ij}$, $1\leq i,j\leq n$, be the basis elements of $\mathfrak{gl}(n)$ and $E=(E_{ij})_{n\times n}$. Recall the central element $C_{n}(u)$ of $U(\mathfrak{gl}_{n})$ introduced by Capelli [1]:

[TABLE]

These Capelli elements act on the Gelfand-Tsetlin basis of $U^{\lambda}$ as scalar multiplications, see [10].

Lemma 2.2**.**

Let $\{\xi_{\Lambda}\}$ be the Gelfand-Tsetlin basis of $U^{\lambda}$. Then

[TABLE]

Let $\{v_{\mathcal{T}}|sh(\mathcal{T})=\lambda\}$ be Young’s orthonormal basis of $V^{\lambda}$, where $\mathcal{T}$ runs through standard Young tableaux of shape $\lambda$. We can decompose (5) into basis vectors:

[TABLE]

By Lemma 2.1, these basis vectors in decomposition (11) are normalized orthogonal, i.e.

[TABLE]

for any Gelfand-Tsetlin patterns $\Lambda,\Lambda^{\prime}$ and standard Young tableaux $\mathcal{T},\mathcal{T}^{\prime}$.

An $n$-tuple $(a_{1},\ldots,a_{n})$ in $\mathbb{Z}_{\geq 0}^{n}$ such that $a_{1}+\cdots+a_{n}=m$ is called a weak composition of $m$ into $n$ parts or a weak $n$-composition of $m$. It’s clear that the weights of $\mathrm{GL}_{n}$ in $(\mathbb{C}^{n})^{\otimes m}$ are exactly weak $n$-compositions of $m$. For any weak $n$-composition $\mu$ of $m$, define the projection operator

[TABLE]

then $\mathcal{P}_{\mu}:U^{\lambda}\rightarrow(U^{\lambda})_{\mu}$ where $((\mathbb{C}^{n})^{\otimes m})_{\mu}$ and $(U^{\lambda})_{\mu}$ are the subspaces with weight $\mu$ as a $\mathrm{GL}_{n}$-module.

It is convenient to write $I=(1^{m_{1}},\ldots,n^{m_{n}})$ to specify the multiplicity $m_{i}$ of $i$ in the multiset $I=(i_{1},\ldots,i_{m})$. Here $m_{i}=m_{i}(I)=\mathrm{Card}\{j\in I|j=i\}$. Let

[TABLE]

Denote by

[TABLE]

the Young subgroup of $\mathfrak{S}_{m}$, and $\mathfrak{S}_{m_{j}}=1$ if $m_{j}=0$.

Let $\lambda\vdash n$, the irreducible representation $V^{\lambda}=\{v_{\Lambda}|sh(\mathcal{T})=\lambda\}$ of the symmetric group $\mathfrak{S}_{n}$ is realized by the following action:

[TABLE]

where $c_{i}$ is the content of node $i$ in $\Lambda$. The primitive idempotents can be constructed by semistandard tableaux. Let $\mathcal{T}$ be a semistandard tableau of shape $\lambda$, the idempotent

[TABLE]

where $h(i,j)$ are the hook length of the $(ij)$-box in the Young tableau $\mathcal{T}$ of shape $\lambda$.

We extend $\pi_{\lambda}$ to all $n\times n$ matrices $\mathrm{Mat}_{n}(\mathbb{C})=\mathfrak{gl}(n)$, then $(\mathbb{C}^{n})^{\otimes m}$ has a $\mathrm{Mat}_{n}(\mathbb{C})$-module structure. The following is a generalization of Kostant’s Theorem [5, Thm.3] and gives a representation theoretic interpretation of the immanant.

Theorem 2.3**.**

Let $\lambda\vdash m$, $\mu$ be a weak $n$-composition of $m$, and let the multiset $I=(1^{\mu_{1}},2^{\mu_{2}},\ldots,n^{\mu_{n}})$ of $[n]$. Then

[TABLE]

for any matrix $A\in\mathrm{Mat}_{n}(\mathbb{C})$.

Remark 2.4*.*

For $\mu=(1^{n})$, we obtain Kostant’s theorem in [5, Thm. 3].

2.2. Proof of Theorem 2.3

First we recall some notations. For any partition $\lambda$, the set of all Young tableaux of shape $\lambda$ is denoted as $YT(\lambda)$. The set of semistandard Young tableaux of shape $\lambda$ with entries at most $n$ is denoted by $SSYT(\lambda,n).$ The set of standard Young tableaux of shape $\lambda$ is denoted as $SYT(\lambda).$

Let $\lambda$ be a partition of $m$ and $\mu$ be a weak composition of $m$. It is well-known that the semi-standard Young tableau $\mathcal{T}_{\lambda\mu}$ of shape $\lambda$ and weight $\mu$, where $\mu_{i}$ equals the number of $i$’s in $\mathcal{T}_{\lambda\mu}$, is in one-to-one correspondence to a Gelfand-Tsetlin pattern $\Lambda=(\lambda_{ij})_{1\leq i\leq m\atop 1\leq j\leq i}$ such that

[TABLE]

We will briefly write $\mathcal{T}_{\mu}=\mathcal{T}_{\lambda\mu}$ if the shape of $\mathcal{T}$ is clear from the context. By the correspondence (15), the dimension of $(U^{\lambda})_{\mu}$ is the number of the semi-standard tableaux $\mathcal{T}_{\lambda\mu}$. We also let $\{\xi_{\mathcal{T}_{\mu}}\}$ be the Gelfand-Tsetlin basis vectors in $U^{\lambda}$.

Given any weight $\mu=(\mu_{1},\ldots,\mu_{n})$, which can be written as a multiset $I=(1^{\mu_{1}},\ldots,n^{\mu_{n}})=(i_{1},\ldots,i_{m})$ or a weak $n$-composition of $m$. We define the following map

[TABLE]

which maps any standard Young tableau $\mathcal{T}$ to a Young tableau $\theta_{\mu}(\mathcal{T})$ that replaces each node $r$ in $\mathcal{T}$ by $i_{r}$ for $1\leq r\leq m$. Note that $SSYT(\lambda,n)\subset SYT(\lambda)$ is contained in $YT(\lambda)$.

Lemma 2.5** ([4, Lemma 3.7]).**

Let $\lambda\vdash m$ and $\mu$ be a weak $n$-composition of $m$ written as the multiset $I=(1^{\mu_{1}},2^{\mu_{2}},\ldots,n^{\mu_{n}})=(i_{1}\leq\ldots\leq i_{m})$ of $[n]$, then as a $U(\mathfrak{gl}_{n})\times\mathfrak{S}_{m}$-module

[TABLE]

where $h_{\lambda}=\prod_{(ij)\in\mathcal{T}}h(i,j)$ and $c$ is a nonzero constant. Explicitly, $c=1$, if $\mathcal{T}$ is the unique pre-image of $\theta_{\mu}(\mathcal{T})$. And $\sum_{i=1}^{s}||c_{i}||^{2}=1$ for $\theta_{\mu}(\mathcal{T}_{1})=\cdots=\theta_{\mu}(\mathcal{T}_{s})$ ($s>1$). In particular, assume $m\leq n$, then

[TABLE]

Proof of theorem 2.3.

By Lemma 2.5, the right side of (14) equals to

[TABLE]

By the relation of characters and primitive idempotents of the symmetric group, one has that

[TABLE]

where $\mathcal{M}(\mathfrak{S}_{m}/\mathfrak{S}_{I})$ refers to the coset representatives in the quotient group $\mathfrak{S}_{m}/\mathfrak{S}_{I}$. The above expression then equals to

[TABLE]

where the first sum is over all sequence $(j_{1}\ldots,j_{m})$ of $[n]$.

As the action of $\mathrm{Mat}_{n}(\mathbb{C})$ commutes with that of $\mathfrak{S}_{m}$ on the tensor space $(\mathbb{C}^{n})^{\otimes m}$, so

[TABLE]

∎

3. Immanant inequalities and nonzero criterion

Let $\mu$ be a weak $n$-composition of $m$, then $\mu$ is a weight of $\mathrm{GL}_{n}$. We now present a criterion for non-vanishing of immanants of generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices based on the dimensions of the weight spaces.

Schur [13] and Stembridge [14] have shown that the immanant of a singular matrix (whether positive semidefinite or totally nonnegative) is nonnegative. Our criterion further strengthened their important results.

Theorem 3.1**.**

Let $A$ be any an $n\times n$ either positive definite or nonsingular totally nonnegative matrix, $\lambda\vdash m$ and $\mu$ is a weak $n$-composition of $m$. Given the multiset $I=(1^{\mu_{1}},2^{\mu_{2}},\ldots,n^{\mu_{n}})$ of $[n]$, then

[TABLE]

if and only if $\dim(U^{\lambda})_{\mu}\neq 0$.

Proof.

It follows from Theorem 2.3 that the immanant of $A_{I}$ is zero if $\dim(U^{\lambda})_{\mu}=0$.

Conversely, for $\dim(U^{\lambda})_{\mu}\neq 0$, we show that $\mathrm{Imm}_{\chi^{\lambda}}(A_{I})>0$. We simply write $e_{I_{\mu}}=e_{i_{1}}\otimes\cdots\otimes e_{i_{m}}$. By Theorem 2.3, we have that

[TABLE]

where the sum is taken over all standard Young tableaux $\mathcal{T}$ with $\theta_{\mu}(\mathcal{T})\in SSYT(\lambda,n)$.

If $A$ is positive definite, by the Cholesky decomposition, we may write $A=\overline{U}^{t}DU$, where $U$ is upper triangular and unipotent, $D=\mbox{diag}(d_{1},\ldots,d_{n})\in\mathrm{GL}_{n}$ is a diagonal matrix with positive entries. Thus

[TABLE]

If $A$ is nonsingular totally nonnegative, then by the Whitney theorem in [17, 9, 5], we can write $A=U_{-}U_{+}D$, where $U_{+}$ is a product of elements of the form $\exp tE_{i,i+1}$ with $t>0$, $U_{-}$ is a product of elements of the form $\exp tE_{i+1,i}$ with $t>0$, and $D=\mbox{diag}(d_{1},\ldots,d_{n})\in\mathrm{GL}_{n}$ is a diagonal matrix with positive entries. Thus if $\dim(U^{\lambda})_{\mu}\neq 0$,

[TABLE]

∎

Based on the inequalities established by Schur and Stembridge, the immanant of a singular matrix (whether positive semidefite or totally nonnegative) is nonnegative. Theorem 3.1 provides a sufficient condition for the immanant of such a singular matrix to be positive.

Denote by $O_{\mu}$ the orbit of weight $\mu$ under the action of the Weyl group $\mathfrak{S}_{n}$. Let $I=(1^{\mu_{1}},2^{\mu_{2}},\ldots,n^{\mu_{n}})$ be the multiset of $[n]$ corresponding to $\mu$. Finally, we have the following generalization of Kostant’s Theorem 4 in [5].

Theorem 3.2**.**

Let $\lambda\vdash m$, $\mu$ is a weak $n$-composition of $m$, then

[TABLE]

whenever $A$ is either positive semidefinite or totally nonnegative.

Proof.

It suffices to show that

[TABLE]

From the proof in theorem 3.1, we know that

[TABLE]

where $D=\mbox{diag}(d_{1},\ldots,d_{n})$ is a diagonal matrix. For any $\nu_{1},\nu_{2}\in O_{\mu}$, there exists an element $w\in\mathfrak{S}_{n}$ such that $w\cdot\nu_{1}=\nu_{2}$ and $\dim(U^{\lambda})_{\nu_{1}}=\dim(U^{\lambda})_{w\cdot\nu_{1}}=\dim(U^{\lambda})_{\nu_{2}}$. By the inequality of arithmetic and geometric means,

[TABLE]

Note that $\prod_{\nu\in O_{\mu}}(d_{i_{1}}\cdots d_{i_{m}})$ is a symmetric polynomial in the variables $d_{1},\ldots,d_{n}$. Hence $\prod_{\nu\in O_{\mu}}(d_{i_{1}}\cdots d_{i_{m}})^{\frac{1}{|O_{\mu}|}}=d_{1}\cdots d_{n}=\det(A)$. Moreover,

[TABLE]

∎

Acknowledgments

The work is supported in part by the National Natural Science Foundation of China grant nos. 12171303 and 12001218, the Humboldt Foundation, the Simons Foundation grant no. 523868, and the Fundamental Research Funds for the Central Universities grant nos. CCNU24JC001, CCNU25JC025 and CCNU25JCPT031.