---
title: Second Immanantal Polynomial
url: https://www.emergentmind.com/topics/second-immanantal-polynomial
type: topic
---

# Second Immanantal Polynomial

The second immanantal polynomial is an immanantal polynomial attached to a hook partition of \(n\), most commonly \((2,1^{n-2})\) in the graph, Laplacian, and recent hook-immanant literature, where one writes
\[
d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}
\]
for the corresponding immanant and \(d_2(xI-M)\) for the polynomial. The terminology is not completely uniform. In some representation-theoretic, hook-inequality, and Newton-polytope discussions, the adjacent hook \((n-1,1)\) is treated as the “second” immanant instead. Complexity-theoretic work on partition-indexed immanants often avoids the phrase entirely and works directly with \(\mathrm{imm}_\lambda\) for a partition \(\lambda\vdash n\). The resulting ambiguity is substantive rather than cosmetic, because the two hooks are conjugate but support different explicit character formulas, graph-theoretic interpretations, and complexity behavior [1710.02416] [2502.12781] [2307.15979] [2102.04340].

## 1. Terminology and defining formulas

For a partition \(\lambda\vdash n\) and an \(n\times n\) matrix \(A=(a_{ij})\), the immanant is
\[
d_\lambda(A)=\sum_{\psi\in S_n}\chi_\lambda(\psi)\prod_{i=1}^n a_{i,\psi(i)},
\]
and the corresponding immanantal polynomial is
\[
f_\lambda^A(x)=d_\lambda(xI-A).
\]
For \(\lambda=1^n\), this reduces to the characteristic polynomial, while for general \(\lambda\) it defines a partition-indexed family interpolating between determinant and permanent [1710.02416].

In the usage that is explicit in several graph-matrix papers, the second immanant is the hook case \(\lambda=(2,1^{n-2})\), and the second immanantal polynomial of a matrix \(M\) is \(d_2(xI-M)\). This convention is stated directly for general matrices, adjacency matrices, and linear-combination graph matrices, and it is the convention adopted in recent reconstruction and hook-immanant polynomial work [2502.12781] [2508.17743] [2604.04489].

A different convention appears in some literature organized from the permanent side of the hook chain. There the natural “second” object is the hook \((n-1,1)\), so that the relevant polynomial is \(\phi_{(n-1,1)}(L_G,x)=\mathrm{imm}_{(n-1,1)}(xI-L_G)\) in Laplacian settings, or \(\operatorname{Imm}_{\chi^{(n-1,1)}}(A)\) in character-immanant settings [2307.15979] [2510.00327]. This terminological split is one of the central points in any precise account of the subject.

## 2. Character-theoretic placement within the immanant family

Immanants sit between the determinant and permanent. For the sign character one has
\[
\det(A)=\mathrm{imm}_{(1^n)}(A),
\]
and for the trivial character one has
\[
\mathrm{per}(A)=\mathrm{imm}_{(n)}(A).
\]
Thus the hook cases immediately adjacent to these extremes are \((2,1^{n-2})\) and \((n-1,1)\), which are transposes of each other:
\[
(n-1,1)'=(2,1^{n-2}).
\]
This transpose relation is frequently the source of the competing “second” conventions [2102.04340].

For the determinant-side hook \((2,1^{n-2})\), a particularly useful explicit formula is
\[
\chi_{(2,1,\ldots,1)}(\pi)=\mathrm{sgn}(\pi)\cdot(\#\{\text{fixed points of }\pi\}-1),
\]
hence
\[
\mathrm{imm}_{(2,1^{n-2})}(A)
=\sum_{\pi\in S_n}\mathrm{sgn}(\pi)\bigl(\mathrm{fix}(\pi)-1\bigr)\prod_{i=1}^n A_{i,\pi(i)}.
\]
This gives the second immanant, in the \((2,1^{n-2})\) convention, as a sign-weighted deformation of the determinant by a fixed-point factor [2102.04340].

For the permanent-side hook \((n-1,1)\), the corresponding character is
\[
\chi^{(n-1,1)}(\pi)=fix(\pi)-1.
\]
In that convention the immanant is
\[
\operatorname{Imm}_{(n-1,1)}(A)=\sum_{\pi\in S_n}(fix(\pi)-1)\prod_{i=1}^n a_{i,\pi(i)},
\]
so the sign factor disappears and the weighting is governed directly by fixed points [2604.06615].

## 3. Laplacian second immanantal polynomials of trees

For a tree \(T\) on \(n\) vertices with Laplacian matrix \(L_T\), the Laplacian immanantal polynomial is
\[
f_\lambda^{L_T}(x)=d_\lambda(xI-L_T)=\sum_{r=0}^n (-1)^r c_{\lambda,r}^{L_T}\,x^{n-r}.
\]
The \(q\)-Laplacian is
\[
L_T^q = I + q^2(D-I) - qA,
\]
with \(L_G^q=L_G\) when \(q=1\), and the corresponding \(q\)-Laplacian immanantal polynomial is
\[
f_\lambda^{L_T^q}(x)=d_\lambda(xI-L_T^q)=\sum_{r=0}^n (-1)^r c_{\lambda,r}^{L_T^q}(q)\,x^{n-r}.
\]
In this setting, the phrase “second immanantal polynomial” is used explicitly for the partition
\[
\lambda=(2,1^{n-2}),
\]
following Merris’s usage [1710.02416].

A central theorem states that if \(T_2\) covers \(T_1\) in the generalized tree shift poset \(GTS_n\), then for every partition \(\lambda\vdash n\) and every \(0\le r\le n\),
\[
c_{\lambda,r}^{L_{T_1}^q}(q)-c_{\lambda,r}^{L_{T_2}^q}(q)\in \mathbb{R}^+[q^2].
\]
At \(q=1\), this yields coefficientwise monotonicity for ordinary Laplacian immanantal polynomials. Specializing to \((2,1^{n-2})\) shows that every coefficient of the second Laplacian immanantal polynomial decreases as one moves upward in \(GTS_n\) [1710.02416].

The second case is distinguished by an especially simple coefficient formula. If \(a_{r,i}^T(q)\) denotes the orientation statistic built from \(B\)-vertex orientations, then
\[
c_{(2,1^{n-2}),r}^{L_T^q}(q)=(n-1)a_{r,0}^T(q)+2a_{r,1}^T(q).
\]
For \(r=n-1\), the coefficient acquires a moment interpretation:
\[
c_{(2,1^{n-2}),\,n-1}^{L_T^q}(q)=\sum_{i=1}^n Moment_{q^2}^T(i).
\]
At \(q=1\), this recovers Merris’s theorem relating the second Laplacian immanantal polynomial to the sum of vertex moments and, consequently, to centroid-type structure [1710.02416].

## 4. Coefficients for graph linear-combination matrices

A broad recent framework studies immanantal polynomials of
\[
\beta D(G)+\gamma A(G),
\]
where \(D(G)\) is the degree matrix and \(A(G)\) the adjacency matrix of a graph \(G\). For hook partitions \((k,1^{n-k})\), the hook immanantal polynomial is
\[
\Phi_k(M,x)=d_{(k,1^{n-k})}(xI_n-M).
\]
Within this family, \(\Phi_2(M,x)\) is explicitly identified as the second immanant polynomial, while \(\Phi_1\) is the characteristic polynomial and \(\Phi_n\) the permanental polynomial [2508.17743].

For the second immanantal polynomial of the linear-combination matrix \(M=\beta D(G)+\gamma A(G)\),
\[
\operatorname{Imm}_{(2,1^{n-2})}(xI-\beta D(G)-\gamma A(G))
=\sum_{r=0}^n(-1)^r c_{2,r}\,x^{n-r},
\]
the first coefficients are
\[
c_{2,0}=n-1,\qquad c_{2,1}=2m\beta(n-1),
\]
\[
c_{2,2}=(n-1)\beta^2F_2(G)-(n-3)m\gamma^2,
\]
and
\[
c_{2,3}=(n-1)\beta^3F_3(G)-(n-3)\beta\gamma^2\mathcal M_3^1(G)-2(n-4)\gamma^3|\mathscr C_3(G)|.
\]
For the Laplacian specialization \(L(G)=D(G)-A(G)\), these become
\[
c_{(2,1^{n-2}),0}(L(G))=n-1,
\]
\[
c_{(2,1^{n-2}),1}(L(G))=2m(n-1),
\]
\[
c_{(2,1^{n-2}),2}(L(G))=(n-1)F_2(G)-m(n-3),
\]
and
\[
c_{(2,1^{n-2}),3}(L(G))=(n-1)F_3(G)-(n-3)\mathcal M_3^1(G)-2(n-4)|\mathscr C_3(G)|.
\]
These formulas place the second immanantal polynomial in direct contact with degree symmetric sums, matching statistics, and cycle counts [2604.04489].

The same framework yields structural equalities and recursions. In particular, for bipartite graphs,
\[
\Phi_2(L(G),x)=\Phi_2(Q(G),x),
\]
where \(Q(G)=D(G)+A(G)\) is the signless Laplacian. More generally, the \(k=2\) specialization of the hook recursions expresses \(\Phi_2\) in terms of deleted vertices, deleted edges, and, for graphs with cycles, deleted cycle-vertex sets; for trees the cycle terms vanish, leaving especially tractable recurrences [2508.17743].

## 5. Complexity-theoretic interpretation and the importance of convention

The complexity of “the” second immanantal polynomial depends decisively on which hook partition is meant. For a partition \(\lambda=(\lambda_1,\dots,\lambda_s)\vdash n\), let
\[
b(\lambda):=n-s,
\]
the number of boxes to the right of the first column of the Young diagram. For a family \(\Lambda\), define \(b(\Lambda)=\max_{\lambda\in\Lambda}b(\lambda)\), and let \(\mathrm{Imm}(\Lambda)\) denote the problem of evaluating \(\mathrm{imm}_\lambda(A)\) on input \(A\) and \(\lambda\in\Lambda\) [2102.04340].

If one interprets the second immanant as the determinant-side hook \((2,1^{n-2})\), then \(s=n-1\) and
\[
b(2,1^{n-2})=1.
\]
For the family \(\Lambda=\{(2,1^{n-2}):n\ge 2\}\), one has \(b(\Lambda)=1<\infty\), so
\[
\mathrm{Imm}(\Lambda)\in FP\qquad\text{and}\qquad \mathrm{Imm}(\Lambda)\in VP.
\]
In this interpretation, the second immanant lies on the polynomial-time side of the dichotomy [2102.04340].

If instead one interprets the second immanant as the permanent-side hook \((n-1,1)\), then \(s=2\) and
\[
b(n-1,1)=n-2.
\]
For the family \(\Lambda=\{(n-1,1):n\ge 2\}\), \(b(\Lambda)=\infty\), indeed linearly growing. The dichotomy then yields much stronger hardness: this family is \(\#P\)-hard and \(VNP\)-complete, and it admits no \(\exp(o(n))\)-time algorithm unless \(\#ETH\) fails [2102.04340].

The complexity paper itself does not use the phrase “second immanantal polynomial”; the specialization to either hook is an inference from partition indexing. That inference, however, is exact enough to show that the terminology is computationally nontrivial: one natural “second” hook is easy, the other is hard [2102.04340].

## 6. Reconstruction, inequalities, and later generalizations

In reconstruction theory, the second immanantal polynomial in the \((2,1^{n-2})\) convention satisfies deletion identities analogous to those known for characteristic and permanental polynomials. For a simple graph \(G\) with
\[
T(G;x)=d_2(xI_n-A(G)),
\]
one has
\[
(m-n)T(G;x)+xT'(G;x)
=
\sum_{uv\in E(G)}\bigl[T(G-uv;x)+T(G-u-v;x)\bigr].
\]
Hence, when \(m\neq n\), \(T(G;x)\) can be reconstructed from the second immanantal polynomials of the graphs in
\[
\{G-uv,\;G-u-v\mid uv\in E(G)\}.
\]
For a digraph \(\overrightarrow G\), the analogous identity is
\[
(m-n)g(\overrightarrow G;x)+xg'(\overrightarrow G;x)
=
\sum_{e\in E(\overrightarrow G)} g(\overrightarrow G-e;x),
\]
which yields reconstruction of
\[
d_2(xI-A(\overrightarrow G)),\qquad
d_2(xI-D(\overrightarrow G)+A(\overrightarrow G)),\qquad
d_2(xI-D(\overrightarrow G)-A(\overrightarrow G))
\]
from the deleted-edge deck whenever \(m\neq n\) [2502.12781].

Under the alternate \((n-1,1)\) convention, a different line of work studies normalized hook immanants on totally nonnegative matrices. There the principal theorem gives the hook chain
\[
\operatorname{per}(A)
=
\frac{\operatorname{Imm}_{\chi^{(n)}}(A)}{\chi^{(n)}(e)}
\ge
\frac{\operatorname{Imm}_{\chi^{(n-1,1)}}(A)}{\chi^{(n-1,1)}(e)}
\ge
\cdots
\ge
\frac{\operatorname{Imm}_{\chi^{(1^n)}}(A)}{\chi^{(1^n)}(e)}
=
\det(A)
\]
for every totally nonnegative matrix \(A\). Since \(\chi^{(n-1,1)}(e)=n-1\), the normalized \((n-1,1)\)-immanant lies immediately below the permanent and above the later hook immanants, and the difference
\[
\operatorname{per}(x)-\frac{\operatorname{Imm}_{\chi^{(n-1,1)}}(x)}{n-1}
\]
is itself totally nonnegative [2510.00327].

Several recent generalizations move beyond the classical symmetric-function setting. Quasi-immanants replace cycle type by cycle composition and symmetric functions by quasisymmetric functions. The quasi-immanant attached to the quasisymmetric Schur function \(\mathcal S_{(2,1^{n-2})}\) is presented as an analogue of the second immanant rather than as the classical second immanant itself; its coefficients depend on the first part of \(\operatorname{ccomp}(\sigma)\) and vanish when \(\operatorname{ccomp}(\sigma)_1>2\) [2501.15667]. In another direction, Newton-polytope work treats \((n-1,1)\) as the relevant second immanant: for Jacobi–Trudi matrices \(\operatorname{Imm}_{(n-1,1)}H(\lambda,\mu)\) is SNP when \(\lambda/\mu\) is a border strip, and for Giambelli matrices \(\operatorname{Imm}_\nu G_\lambda\) is SNP for every \(\nu\vdash k\), hence in particular for \((k-1,1)\) [2604.06615].

The cumulative picture is therefore bifurcated but coherent. In graph-theoretic and Laplacian applications, the second immanantal polynomial is usually \(d_{(2,1^{n-2})}(xI-M)\); in some hook-chain, representation-theoretic, and Newton-polytope settings, the label shifts to \((n-1,1)\). The two conventions are linked by conjugation of partitions, yet they lead to different explicit formulas, different monotonicity and positivity statements, and, in families, different computational complexity classes.

Source: https://www.emergentmind.com/topics/second-immanantal-polynomial