2. Preliminaries

In this section, we recall basic definitions and known results on symmetric functions and representation theory; see Stanley [Reference Stanley22] and Fulton [Reference Fulton and Harris7] for details.

We begin by recalling the definitions of symmetric functions. Given a partition $\lambda$, we identify it with the infinite sequence $(\lambda_1,\ldots,\lambda_n,0,\ldots,0)$. The monomial symmetric function $m_\lambda$ is defined as

(2.1)\begin{equation} m_\lambda = \sum_{\alpha} \mathbf{x}^\alpha, \end{equation}

where the sum runs over all distinct permutations $\alpha$ of $\lambda$, and $\mathbf{x}^\alpha = x_1^{\alpha_1}x_2^{\alpha_2}\cdots$. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the power sum symmetric function is defined by $p_\lambda = p_{\lambda_1}\cdots p_{\lambda_n}$, where for $k\ge 1$,

\begin{equation*}p_k =m_k= \sum_i x_i^k,\end{equation*}

The Schur function $s_{\lambda}$ can be defined via its expansion into the monomial symmetric functions, namely,

(2.2)\begin{equation} s_{\lambda} = \sum_{\mu} K_{\lambda,\mu} \, m_{\mu}. \end{equation}

Here, $K_{\lambda,\mu}$ denotes the Kostka number, which counts the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$. The complete homogeneous symmetric function $h_{\lambda}$ is defined by $h_\lambda = h_{\lambda_1}\cdots h_{\lambda_n}$, where

(2.3)\begin{equation} h_k = \sum_{\nu \vdash k} m_{\nu}=\sum_{i_1\le\cdots\le i_k} x_{i_1}\cdots x_{i_k}, \end{equation}

with the conventions $h_0 = 1$ and $h_k = 0$ for $k \lt 0$. It is well-known that these functions can be expressed in terms of Schur functions as

(2.4)\begin{equation} h_{\lambda} = \sum_{\mu} K_{\mu,\lambda} \, s_{\mu}. \end{equation}

It is important to note that the non-vanishing of terms in both expansions (2.2) and (2.4) is governed by the dominance order on partitions. Given two partitions $\lambda$ and $\mu$ of $d$, we say $\mu$ is dominated by $\lambda$, denoted $\mu \unlhd \lambda$, if

\begin{equation*} \mu_1 + \cdots + \mu_i \leq \lambda_1 + \cdots + \lambda_i \quad \text{for all } i \geq 1. \end{equation*}

The following well-known theorem characterizes the non-vanishing of Kostka numbers in terms of this order.

Next, turn to the introduction of representation theory, refer to [Reference Fulton and Harris7, Reference Sagan21] for details. Given a finite group $G$, we call a representation of $G$ on a finite-dimensional real vector space $V$ is a homomorphism $\rho:G\rightarrow {\rm GL}(V)$. Naturally, the trivial representation of $G$ is the one-dimensional representation $(\rho_{\text{triv}}, V)$ defined by

\begin{equation*} \rho_{\text{triv}}(g) = \operatorname{id}_V \quad \text{for all } g \in G. \end{equation*}

We sometimes call $V$ itself a representation of $G$. In this manner, if $V$ is a representation of $G$, the character of $V$, denoted by $\chi_V$, is the real-valued function on the group defined by $\chi_V(g)=\operatorname{tr} (g|_V)$, where $\operatorname{tr}$ denote the trace. Note that all characters are class functions, namely, they satisfy $\chi_V(hgh^{-1}) = \chi_V(g)$ for all $g, h \in G$. We call a representation $(\rho, V)$ is irreducible if $V$ contains no non-trivial proper $G$-invariant subspaces, and a character $\chi$ is said to be irreducible if it is the character of an irreducible representation. It is clear that the character of any trivial representation is

\begin{equation*} \chi^{\text{triv}}(g) = 1 \quad \text{for all } g \in G. \end{equation*}

In this paper, we focus on the case where $G$ is the symmetric group $\mathfrak{S}_n$. In this case, the irreducible characters of $\mathfrak{S}_n$ are in one-to-one correspondence with the partitions $\lambda$ of $n$. We denote the character associated to $\lambda$ by $\chi^\lambda$. They form an orthonormal basis under the inner product:

\begin{equation*} \langle \chi, \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \chi(g)\overline{\psi(g)}. \end{equation*}

For the special partition $(n-1,1)$, the corresponding representation is called the standard representation of $\mathfrak{S}_n$. It is realized on the subspace:

\begin{equation*} W = \left\{(x_1, \dots, x_n) \in \mathbb{R}^n\mid \sum_{i=1}^n x_i = 0 \right\}, \end{equation*}

where $\mathfrak{S}_n$ acts by permuting coordinates. It is well-known that the irreducible character of the standard representation is given by

(2.5)\begin{equation} \chi^{(n-1,1)}(\pi) = \operatorname{fix}(\pi) - 1, \end{equation}

where $\operatorname{fix}(\pi)$ is the number of fixed points of the permutation $\pi$, refer to [Reference Fulton and Harris7] for the details. Besides, we also need to consider a class of subgroup of $\mathfrak{S}_n$, called Young subgroup. Given a partition $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_{\ell})$ of $n$, the corresponding Young subgroup $\mathfrak{S}_\lambda$ of the symmetric group $\mathfrak{S}_n$ is defined as

\begin{equation*} \mathfrak{S}_\lambda = \mathfrak{S}_{\{1, \dots, \lambda_1\}} \times \mathfrak{S}_{\{\lambda_1+1, \dots, \lambda_1+\lambda_2\}} \times \dots \times \mathfrak{S}_{\{n-\lambda_{\ell}+1, \dots, n\}} \subset \mathfrak{S}_n., \end{equation*}

and it is isomorphic to the direct product $\mathfrak{S}_{\lambda_1} \times \mathfrak{S}_{\lambda_2} \times \dots \times \mathfrak{S}_{\lambda_{\ell}}$. It is clear to see that

(2.6)\begin{equation} |\mathfrak{S}_{\lambda}|= \lambda_1!\lambda_2!\cdots \lambda_{\ell}!. \end{equation}

For the irreducible character of the Young subgroup $\mathfrak{S}_{\lambda}$, we derive the following lemma.

Lemma 2.2. Given a partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_{n})\vdash d$, we have

\begin{equation*}\chi^{\lambda}\left(\mathfrak{S}_\lambda \right)=\lambda_1!\lambda_2!\cdots\lambda_{n}! , \end{equation*}

and $\chi^{\lambda}(\mathfrak{S}_{\mu})=0$ for any $\mu\trianglerighteq \lambda$ in dominance order.

Proof. By definition, we have

\begin{equation*}\chi^{\lambda}(\mathfrak{S}_{\mu})=|\mathfrak{S}_{\mu}|\cdot \langle \chi^{\lambda}|_{\mathfrak{S}_{\mu}}, \chi^{\rm triv} \rangle.\end{equation*}

Note that the inner product is preserved under the Frobenius map ${\rm ch}$, which maps class functions to symmetric functions, refer to [Reference Stanley22, Section 7.18] for the details. For the group $\mathfrak{S}_n$,

\begin{equation*}{\rm ch}(\chi^{\rm triv})=h_d,\, \text{and } {\rm ch}(\chi^{\lambda})=\mathfrak{S}_{\lambda}.\end{equation*}

By combining the equation (2.4), we can deduce that

\begin{equation*}\langle \chi^{\lambda}|_{\mathfrak{S}_{\mu}}, \chi^{\rm triv} \rangle =\langle \mathfrak{S}_{\mu}, h_{\lambda}\rangle =K_{\lambda,\mu}.\end{equation*}

Then the lemma holds by applying Theorem 2.1 and Equation (2.6).

Finally, we need to introduce the basic definitions of outside decompositions, cutting strips, and other relevant notations. Given a skew partition $\lambda/\mu$ of length $n$, we now identify it with a skew diagram defined as the set of boxes $\{(i,j) \in \mathbb{Z}^2 \mid \mu_i \lt j \le \lambda_i\, \text{and } 1 \le i \le n\}$ (using the English convention). The conjugate of $\lambda/\mu$, denoted $(\lambda/\mu)'$ (or $\lambda'/\mu'$), is the skew diagram obtained by transposing this set of boxes across the main diagonal. When $\mu=\emptyset$, the rank of $\lambda$ is defined as the integer $k$ such that $(k,k) \in \lambda$ and $(k+1,k+1) \notin \lambda$. Given a partition $\lambda$ of rank $k$, for $1 \le i \le k$, let $\alpha_i$ denote the number of cells to the right of $(i,i)$, and $\beta_i$ the number of cells below $(i,i)$. Then the Frobenius notation of $\lambda$ is

(2.7)\begin{equation} (\alpha \mid \beta) = (\alpha_1, \ldots, \alpha_k \mid \beta_1, \ldots, \beta_k). \end{equation}

Given a skew diagram $\lambda/\mu$, the content of a box $(i,j)$ is defined as $c = j - i$, and the $c$-th diagonal consists of all boxes in $\lambda/\mu$ having content $c$. Then a (border) strip can also be defined as a connected skew diagram whose cells have distinct contents.

An outside decomposition of $\lambda/\mu$ is a partition of the boxes of $\lambda/\mu$ into pairwise disjoint strips such that every strip in the decomposition has a starting box on the left or bottom perimeter of the diagram and an ending box on the right or top perimeter of the diagram. In a strip, any two adjacent boxes share exactly one common edge, which naturally gives rise to a well-defined direction from the starting box to the ending box. Specifically, for each box that is not the ending box, we say that it goes right if the next box is to its right; otherwise, we say that it goes up. Based on Hamel and Goulden’s observation [Reference Hamel and Goulden12], the contents of the ending (or starting) boxes of the strips in an outside decomposition are distinct. Thus, we may use the contents of the ending boxes to order the strips in an outside decomposition. For an outside decomposition $\Pi$ with $\ell$ strips, we denote it by $\Pi=(\theta_1,\theta_2,\ldots,\theta_{\ell})$, where $\theta_1$ is the strip whose ending box has the largest content. For example, let $\lambda/\mu = (6,5,5,4,3)/(3,3,1,1)$, an outside decomposition

(2.8)\begin{equation} \Pi=((3),(2),(2,1),(2,2)/(1),(1),(3)) \end{equation}

of this skew shape is depicted on the left side of Figure 1, and the directions of boxes in $\theta_3$ are: up, right, right.

Let $\Pi$ be an outside decomposition of $\lambda/\mu$, and suppose $\lambda/\mu$ has $\ell$ diagonals. The cutting strip $T$ of $\Pi$ is constructed as follows: for $1\le i\le \ell-1$, the $i$-th box in $T$ goes right or goes up, matching the direction of the boxes in the $i$-th diagonal of $\lambda/\mu$ with respect to the decomposition $\Pi$. The corresponding cutting strip (with contents) for the outside decomposition in (2.8) is illustrated on the right side of Figure 1. Given an outside decomposition $\Pi$ of shape $\lambda/\mu$ and a strip in $\Pi$. Let $\phi$ be the corresponding cutting strip of $\Pi$. For $p\le q$, let $[p,q]$ denote a segment of the cutting strip $\phi$ starting with the box having content $p$ and ending with the box having content $q$. Chen, Yan, and Yang [Reference Chen, Yan and Yang2] gave the correspondence of the strip $\theta_i \# \theta_j$ defined in [Reference Hamel and Goulden12] with $[p(\theta_j),q(\theta_i)]$, where $p(\theta)$ (respectively, $q(\theta)$) denotes the content of the starting box (respectively, ending box) of a strip $\theta$.

For example, for the outside decomposition in (2.8), some strips obtained by the operation $\#$ are given as follows:

\begin{equation*} \begin{array}{ll} \theta_1\#\theta_5=[-2,4]=(5,1,1), & \theta_5\#\theta_1=[2,-2]= \text{undefined} \\ \theta_2\#\theta_3=[-1,2]=(3,1), & \theta_3\#\theta_2=[1,1]=(1) \\ \theta_4\#\theta_6=[-5,-1]=(4,4)/(3), & \theta_6\#\theta_4=[-3,-3]=(1). \end{array} \end{equation*}

For a partition $\lambda$ of rank $k$ with Frobenius notation $(\alpha\mid\beta)=(\alpha_1,\ldots,\alpha_k\mid\beta_1,\ldots,\beta_k)$, the Giambelli matrix $G_{\lambda}$ is a special case of the Hamel–Goulden matrix [Reference Hamel and Goulden12]. Indeed, for $1\le i,j\le k$ we have $\theta_i\#\theta_j = [j-\lambda'_j,\lambda_i-i] = (\alpha_i\mid\alpha_j)$, so the outside decomposition of $\lambda$ consists precisely of the hooks on the diagonal. For example, let $\lambda = (6,6,4,4,1,1)$, as shown in Figure 2, the outside decomposition of $\lambda$ is $\Pi = (\theta_1, \theta_2, \theta_3, \theta_4)$, where

\begin{equation*} \theta_1 = (6,1^5),\quad \theta_2 = (5,1^2),\quad \theta_3 = (2,1),\quad \theta_4 = (1), \end{equation*}

and the corresponding cutting strip is $(6,1^5)$ as shown on the right side.

Figure 2.

An outside decomposition and the cutting strip of $\lambda=(6,6,4,4,1,1)$.

3. Immanants of Jacobi–Trudi matrices

In this section, we first recall some related results concerning the immanants of Jacobi–Trudi matrices presented by Greene [Reference Greene10], including the lattice path construction of Gessel and Viennot [Reference Gessel and Viennot8] and the reduction of Goulden and Jackson [Reference Goulden and Jackson9]. Based on these, we provide a proof of Theorem 1.3. Furthermore, we also prove the SNP property for the polynomials defined by Stanley and Stembridge [Reference Stanley and Stembridge23] in the border strip case.

Greene’s lattice path construction [Reference Greene10] is inherited from Gessel and Viennot’s approach [Reference Gessel and Viennot8], which was used to give a combinatorial interpretation of the Jacobi–Trudi identity (1.1). The crucial idea is to interpret the complete homogeneous symmetric function $h_i(x_1, \ldots, x_{\ell})$ as the generating function for certain lattice paths. These paths start at $(0, 1)$ and end at $(i, \ell)$, where for $1\le j\le \ell$, a horizontal step at height $j$ is weighted by $x_j$, a vertical step is weighted by $1$, and the weight of a path is the product of the weights of its steps. Then the product

\begin{equation*}\prod_{i=1}^n h_{(\lambda_i-i)-(\mu_{\pi_i}-\pi_i)}\end{equation*}

can be interpreted as the generating function for $n$ (not necessarily disjoint) lattice paths, where the $i$-th path runs from the point $(\mu_{\pi_i} - \pi_i, 1)$ to the point $(\lambda_i - i, \ell)$. The weight of a path is the product of the weights of its steps, as defined previously.

For $i \in [n]$, let $P_i = (\mu_i - i, 1)$ and $Q_i = (\lambda_i - i, \ell)$, and let $\mathcal{F}$ be the collection of all lattice paths connecting ${P_1, \dots, P_n}$ to ${Q_1, \dots, Q_n}$ according to some permutation $\pi\in\mathfrak{S}_n$. Greene [Reference Greene10] defined the skeleton of $\mathcal{F}$ as

\begin{equation*}\alpha=\alpha_{\mathcal{F}}=\bigcup_{P\in \mathcal{F}} \{\text{multiset of edges in some path }P\}.\end{equation*}

In the same manner, let $\textbf{x}^{\alpha}=\prod_{P\in\mathcal{F}}\textbf{x}^{{\rm weight}(P)}$. Then each immanant of the Jacobi–Trudi matrix has the following expansion:

(3.1)\begin{align}{\rm Imm}_\nu H(\lambda,\mu) &= \sum_{\pi\in \mathfrak{S}_{n}}\chi^{\nu}(\pi)\prod_{i=1}^{n}h_{(\lambda_i-i)-(\mu_{\pi_i}-\pi_i)} \nonumber \\ &= \sum_{\pi\in\mathfrak{S}_n}\chi^{\nu}(\pi)\sum_{\mathcal{F}}\textbf{x}^{\alpha_{\mathcal{F}}} \nonumber \\ &= \sum_{\alpha}\textbf{x}^{\alpha}\sum_{\mathcal{F}}\chi^{\nu}(\pi_{\mathcal{F}}) \nonumber \\ &= \sum_{\alpha}\textbf{x}^{\alpha}\chi^{\nu}\left(\sum_{\mathcal{F}}\pi_{\mathcal{F}}\right),\end{align}

where the inner sum is over all families of paths $\mathcal{F}$ having skeleton $\alpha$, and $\pi_{\mathcal{F}}$ denotes the permutation of endpoints induced by $\mathcal{F}$.

Goulden and Jackson showed the following result, which was also noted in Greene’s paper.

Now we are prepared to present the proof of Theorem 1.3.

Proof of Theorem 1.3.

Given partitions $\lambda=(\lambda_{1},\ldots,\lambda_n)$ and $\mu=(\mu_1,\ldots,\mu_n)$ with $\mu_i\le \lambda_i$, it is clear that the degree of ${\rm Imm}_{\nu}H(\lambda,\mu)$ is $d=|\lambda_1-1|+|\mu_n-n|$.

By definition, for the skeleton $\alpha = (d)$, the lattice family $\mathcal{F}$ corresponding to $\alpha$ must have the form depicted in Figure 3.

Figure 3.

The skeleton $\alpha=(d)$ with weight $\textbf{x}^{\alpha}=x_j^d$.

Therefore, by applying Proposition 3.1, we have

\begin{align*} \chi^{\nu}(S_J)=\chi^{\nu}\left(\mathfrak{S}_{\{1,2\}}\times \mathfrak{S}_{\{2,3\}}\times \cdots\times \mathfrak{S}_{\{n-1,n\}}\right) \end{align*}

If we set $\nu=(n-1,1)$, then $V_{\nu}$ is the standard representation

\begin{equation*}V=\{(x_1,\ldots,x_n)\mid x_1+x_2+\cdots+x_n=0\},\end{equation*}

and as stated in (2.5), for each $g\in \mathfrak{S}_n$, its irreducible character has the explicit explanation

\begin{equation*}\chi_V(g)=\operatorname{fix}(g)-1. \end{equation*}

Let $f_k(n)$ denote the number of permutations in $\mathfrak{S}_{\{1,2\}}\times \mathfrak{S}_{\{2,3\}}\times\cdots\times \mathfrak{S}_{\{n-1,n\}}$ with $k$ fixed points. Then we have

\begin{equation*}\chi^{(n-1,1)}\left(\mathfrak{S}_{\{1,2\}}\times \mathfrak{S}_{\{2,3\}}\times \cdots\times \mathfrak{S}_{\{n-1,n\}} \right) =\sum_{k=0}^nf_k(n)\cdot (k-1).\end{equation*}

For all $n\ge 1$, we have $f_k(n)\ge 0$ and $f_n(n)=1$. Moreover, for $n\ge 3$, we have $f_0(n)\ge n-2$ since the permutation $(123\cdots n)$ and those permutations of the form $(1\cdots i)(i+1\cdots n)$ (of which there are $n-3$ choices for $i$) all have zero fixed points. Thus, the coefficient of $x_j^d$ must be non-zero for each $1\le j\le k$. Specifically, let $[x_{\alpha}]f$ denote the coefficient of the monomial $x_{\alpha}$ in $f$, by the expression illustrated in (3.1), we have

\begin{align*} [x_j^d]\,{\rm Imm}_{(n-1,1)}(\lambda,\mu)&=\chi^{(n-1,1)}\left(\mathfrak{S}_{\{1,2\}}\mathfrak{S}_{\{2,3\}}\cdots \mathfrak{S}_{\{n-1,n\}}\right)\\ &\ge(n-2)(-1)+\sum_{k=1}^{n-1}f_k(n)(k-1)+n-1 \gt 0 \end{align*}

Based on the symmetric property, the Schur expansion of ${\rm Imm}^{(n-1,1)}H(\lambda,\mu)$ must have $s_{(d)}$ as the maximal term in the dominance order if $\lambda/\mu$ is a border strip of size $d$. Consequently,

\begin{equation*}{\rm Newton}\left({\rm Imm}_{(n-1,1)}H(\lambda,\mu) \right) =\mathcal{P}_{(d)},\end{equation*}

which follows by combining Theorem 1.2 with the Schur-positivity of ${\rm Imm}^{(n-1,1)}H(\lambda,\mu)$.

Now, for $\nu=(n)$, the immanant specializes to the permanent, and we have

\begin{equation*}{\rm supp}\left( {\rm per}\, H(\lambda,\mu)\right)= {\rm supp}\left(s_{(d)} \right)\end{equation*}

for any skew partition $\lambda/\mu \vdash d$. This follows from Equation (2.4), Theorem 1.1 and the non-negativity of those coefficients in the $h$-expansion of ${\rm per}\, H(\lambda,\mu)$. This completes the proof of Theorem 1.3.

Next, to illustrate the second main result of this section, we first recall some basic notations. Given a composition $\alpha=(\alpha_1,\ldots,\alpha_{\ell})$ of positive integer $d$. We call a refinement of composition $\alpha$ is the replacement of $\alpha_i$ by a composition of $\alpha_i$. If $\beta$ is a refinement of $\alpha$, then conversely, $\alpha$ is a meld of $\beta$. Note that the compositions of $\ell$ (the length of $\alpha$) can be used to index the melds of $\alpha$. Let $\alpha|_{\gamma}$ denote the meld of $\alpha$ indexed by $\gamma$.

Recall that $E_{\lambda/\mu}^{\theta}(\textbf{y})$ is the coefficient of $s_{\theta}(\textbf{x})$ in $F_{\lambda/ \mu}(\textbf{x},\textbf{y})$ (refer to ). Stanley and Stembridge [Reference Stanley and Stembridge23] gave the $p$-expansion of $E_{\lambda/\mu}^{\theta}$ for the case when $\lambda/\mu$ is a border strip.

Proposition 3.2. ([Reference Stanley and Stembridge23, Lemma 2.1])

If $\lambda/\mu$ is a border strip, then

\begin{equation*}E_{\lambda/\mu}^{\theta}=\sum_{\alpha}K_{\theta,\,(\lambda-\mu)|_{\alpha}}p_{\alpha_1}\cdots p_{\alpha_{\ell}}, \end{equation*}

where the summation is over all compositions of $n$, i.e., the length of $\lambda/\mu$.

We find that the Kostka numbers satisfy the following refinement condition.

Proof. Given a tableaux of shape $\theta$, and type $(\lambda-\mu)|_{\alpha}$. Set $\lambda-\mu=\gamma$, then, by definition, we have

\begin{align*} \gamma|_{\alpha}=(\gamma_1+\cdots+\gamma_{\alpha_1},\gamma_{\alpha_1+1}+\cdots+\gamma_{\alpha_1+\alpha_2},\ldots)\\ \gamma|_{\beta}=(\gamma_1+\cdots+\gamma_{\beta_1},\gamma_{\beta_1+1}+\cdots+\gamma_{\beta_1+\beta_2},\ldots) \end{align*}

Since $\beta$ is a refinement of $\alpha$, we have $\gamma|_{\beta}\trianglelefteq \gamma|_{\alpha}$ in dominance order. Thus, by applying Theorem 2.1, $K_{\theta,\,\gamma|_{\alpha}}\neq 0$ implies that $\gamma|_{\beta}\trianglelefteq \gamma|_{\alpha}\trianglelefteq \theta$, and therefore $K_{\theta, \, \gamma|_{\beta}}\neq 0$. This completes the proof.

Combining the above results, we can finally present the proof of Theorem 1.4.

4. Immanants of Giambelli matrices

In this section, we present the proof of Theorem 1.5. Our approach relies heavily on the planar network construction introduced by Hamel and Goulden [Reference Hamel and Goulden12].

Given a partition $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_{n})$ of rank $k$, as illustrated at the end of Section 2, the outside decomposition is $\Pi = (\theta_1, \theta_2, \ldots, \theta_k)$, where

\begin{equation*} \theta_i = (\lambda_i - i + 1, 1^{\lambda'_i - i}), \end{equation*}

and the cutting strip of the Giambelli matrix $G_{\lambda}$ is the partition $\phi = (\lambda_1, 1^{n-1})$.

Assume that the contents in $\phi$ range from $c_1$ to $c_2$, and for $1 \le i \le k$, the contents of the starting box and the ending box of $\theta_i$ are $c_i'$ and $c_i^{\prime\prime}$, respectively. Consider the directions of boxes in $\phi$, if a box with content $c$ (where $c_1 \lt c \leq c_2$) comes from the left (go right), then we set the direction of $x = c$ to be downward; conversely, if the box comes from the bottom (go up), then we set the direction of $x = c$ to be upward. Additionally, the directions of $x = c_1$ and $x = c_2 + 1$ can be chosen arbitrarily, as these choices will not affect the construction of our planar network.

For $1\le i\le k$, set $P_i = (c_i', -\infty)$ and $Q_i = (c_i^{\prime\prime} + 1, -\infty)$ to be the starting points and ending points, respectively. The weights of the planar network are assigned according to the following rules:

• • All vertical (up and down) steps have weight 1.

• • The horizontal step from $(c_1, -j)$ to $(c_1 + 1, -j)$ has weight $x_j$.

• • For $c_1 \lt c \le c_2$, suppose the horizontal step from $(c-1, j)$ to $(c, j)$ has weight $x_t$.

  • • If the direction at $x = c$ is upward, then the horizontal step from $(c, j)$ to $(c+1, j)$ has weight $x_{t - 1}$.

  • • If the direction at $x = c$ is downward, then the horizontal step from $(c, j)$ to $(c+1, j)$ has weight $x_t$.

Following the aforementioned example, let $\lambda = (6,6,4,4,1,1)$, refer to Figure 2, the starting points and ending points are as follows:

\begin{align*} P_1 &= (-5, -\infty), & P_2 &= (-2, -\infty), & P_3 &= (-1, -\infty), & P_4 &= (0, -\infty), \\ Q_1 &= (6, -\infty), & Q_2 &= (5, -\infty), & Q_3 &= (2, -\infty), & Q_4 &= (1, -\infty). \end{align*}

In Figure 4, we present all possible weights and those starting and ending points for the planar network associated with the Giambelli matrix $G_{\lambda}$, omitting the labels of steps with weight $1$. Besides, we also present a family of lattice paths corresponding to the monomial $\textbf{x}^{\alpha}=(x_1^6x_2x_3x_4x_5x_6)\cdot(x_1^5x_2x_3)\cdot(x_2^2x_3)\cdot x_3=x_1^{11}x_2^4x_3^4x_4x_5x_6$.

Figure 4.

A planar network of $G_{\lambda}$.

Now we can give the proof of Theorem 1.5.

Proof of Theorem 1.5.

Since the cutting strip of the Giambelli matrix $G_{\lambda} $ is the largest hook of $\lambda$, the boxes from the starting box to the ending box must first go up, then go right. According to the rules of the planar network construction, the directions of all vertical steps must first be all upward, then all downward, from left to right. Specifically, let $c_1$ and $c_2$ be the contents of the starting box and the ending box, respectively; then the direction of each $x=c$ is upward if $c_1\le c\le0$, and is downward if $0 \lt c \le c_2$. Thus, on the one hand, when considered horizontally, the weights of the horizontal steps to the left of the line $x=0$ decrease step by step from right to left, while those to the right of $x=0$ remain constant on each level. On the other hand, when viewed vertically, the weights of the horizontal steps form a sequence that decreases step by step from top to bottom.

Suppose that $\nu = (\nu_1, \nu_2, \ldots, \nu_t)$. By the construction of the planar network for $G_{\lambda}$ described before, we observe that the lattice paths corresponding to the largest term (in dominance order) in the monomial expansion of $\mathrm{Imm}_{\nu}G_{\lambda}$ are distributed as high as possible and have type $\nu$. That is, the lattice paths $\{P_1 \rightarrow Q_1, \ldots, P_{\nu_1} \rightarrow Q_{\nu_1} \}$ overlap on the row containing the most steps weighted by $x_1$, and for $2 \le i \le t$, the lattice paths

\begin{equation*} \left\{P_{\sum_{j=1}^{i-1}\nu_j + 1} \rightarrow Q_{\sum_{j=1}^{i-1}\nu_j + 1}, \ldots, P_{\sum_{j=1}^{i}\nu_j} \rightarrow Q_{\sum_{j=1}^{i}\nu_j} \right\} \end{equation*}

overlap on the row containing the most steps weighted by $x_i$, respectively.

Similar to the proof of Theorem 1.3, by applying the expression in (3.1) and Proposition 3.1, it suffices to prove that

\begin{equation*} \chi^{\nu}\left(\mathfrak{S}_{\{1,2,\ldots,\nu_1\}} \mathfrak{S}_{\{\nu_1+1,\ldots,\nu_1+\nu_2\}} \cdots \mathfrak{S}_{\{\sum_{j=1}^{t-1}\nu_j+1,\ldots,k\}} \right) \neq 0, \end{equation*}

and $\chi^{\nu}(\mathfrak{S}_{\mu}) = 0$ for any $\mu \trianglerighteq \nu$ in the dominance order. This now follows from Lemma 2.2, which completes the proof.

5. Further work

In this paper, for immanants of Jacobi–Trudi matrices, we have only proved their SNP property for several special partitions (namely $\nu=(n)$ and $\nu=(n-1,1)$ under a border strip condition). It is natural to ask whether this property holds in general. Based on computer experiments conducted in Sage [25], we propose the following conjecture.

For now, we have established the SNP property for several families of immanant-derived polynomials and shown that their Newton polytopes are special classes of generalized permutohedra. In fact, these polynomials satisfy a stronger property than SNP, known as M-convexity.

A set $J \subset \mathbb{N}^n$ is called M-convex if for any $\alpha,\beta \in J$ and any index $i$ with $\alpha_i \gt \beta_i$, there exists $j$ such that $\alpha_j \lt \beta_j$ and $\alpha - e_i + e_j \in J$. A polynomial is M-convex if its support is M-convex. For a homogeneous polynomial, M-convexity is equivalent to the SNP property together with the condition that its Newton polytope is a generalized permutahedron [Reference Dizier3, Remark 4.1.1]. For further background, see [Reference Murota18, Reference Postnikov19]. For more background, see [Reference Murota18, Reference Postnikov19]. Moreover, there is a close connection between M-convex polynomials and Lorentzian polynomials, the latter of which was introduced by Brändén and Huh [Reference Brändén and Huh1] as a generalization of volume polynomials and stable polynomials. Specifically, we say that a homogeneous polynomial $f(x_1,\ldots,x_n)$ of degree $d$ with non-negative coefficients is a Lorentzian polynomial if and only if it satisfies:

• • The support of $f$ forms an M-convex set.

• • The quadratic form $\partial_{i_1} \cdots \partial_{i_{d-2}}f $ has at most one positive eigenvalue for any index $i_1,\ldots, i_{d-2}\in[n]$.

Brändén and Huh proved that if ${\rm supp}(f)$ is M-convex then the polynomial

\begin{equation*}\sum_{\alpha\in{\rm supp}(f)}\frac{x_1^{\alpha_1}}{\alpha_1!}\cdots\frac{x_{n}^{\alpha_{n}}}{\alpha_{n}!}\end{equation*}

is Lorentzian; see [Reference Brändén and Huh1, Theorem 3.10] for more information. Inspired by the above result, they defined the normalization of a polynomial $f=\sum_{\alpha=(\alpha_1,\ldots,\alpha_n)}c_{\alpha}x^{\alpha}$ as

\begin{equation*}N(f)=\sum_{\alpha\in{\rm supp}(f)}c_{\alpha}\frac{x_1^{\alpha_1}}{\alpha_1!}\cdots\frac{x_{n}^{\alpha_{n}}}{\alpha_{n}!}.\end{equation*}

Later, Huh, Matherne, Mészáros, and St. Dizier [Reference Huh, Matherne, Mészáros and Dizier.13] conjectured that $N(s_{\lambda/\mu})$ is Lorentzian for any skew partition $\lambda/\mu$ and proved this conjecture when $\mu=\emptyset$. Motivated by their conjecture and Theorem 1.3, we consider the following problem.

Unfortunately, the answer is negative in general. For example, take the skew partition $\lambda/\mu=(3,2)/(1)$ and the partition $\nu=(2)$, one can compute that

\begin{equation*}{\rm Imm}_{\nu}H(\lambda,\mu)=7m_{(1,1,1,1)}+5m_{(2,1,1)}+4m_{(2,2)}+3m_{(3,1)}+2m_{(4)},\end{equation*}

and the quadratic form $\partial_i^2 N({\rm Imm}_{\nu}H(\lambda,\mu)(x_1,x_2,x_3,x_4))$ has two positive eigenvalue for $i\in[4]$.

Nevertheless, for the polynomials $E_{\lambda/\mu}^{\theta}$ associated to the Jacobi–Trudi matrix $H(\lambda,\mu)$, we conjecture that their normalizations are Lorentzian. We have verified this conjecture for all border strips with at most 10 boxes.

For immanants of Giambelli matrices, based on Theorem 1.5, we also propose the following conjecture and verify it for all possible partitions of size less than or equal to $9$ by using SageMath [25].