---
title: Hook Immanants
url: https://www.emergentmind.com/topics/hook-immanant
type: topic
---

# Hook Immanants

A hook immanant is the immanant attached to an irreducible character of the symmetric group indexed by a hook partition. For a partition of the form $(n-k,1^k)$, or equivalently $(k,1^{n-k})$ under the alternate parameter convention used in part of the literature, the associated matrix function is
\[
\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.
\]
It is a representation-theoretic interpolation between the permanent and the determinant: the one-row partition gives the permanent, and the one-column partition gives the determinant. Hook immanants recur in algebraic complexity, matrix positivity, algebraic graph theory, and the theory of immanant characters of Jacobi–Trudi matrices [2102.04340, 2510.00327].

## 1. Definition and basic examples

The general immanant attached to a partition $\lambda\vdash n$ is
\[
\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},
\]
where $\chi_\lambda$ is the irreducible character of $S_n$ indexed by $\lambda$. A hook partition is a partition whose Young diagram has one long first row and one long first column. In the notation used in complexity-theoretic work it is written
\[
\lambda=(n-k,1^k),
\]
while in several matrix-theoretic papers it is written
\[
\lambda=(k,1^{\,n-k}).
\]
These are the same family, with different choices of the parameter.

The hook immanant is therefore the immanant corresponding to such a hook shape. In the notation of graph-theoretic work, one often writes
\[
d_k(M):=d_{(k,1^{n-k})}(M),
\]
with the conventions
\[
d_1(M)=\det M,\qquad d_n(M)=\operatorname{per} M.
\]
Accordingly, hook immanants provide a canonical chain from determinant to permanent through intermediate irreducible characters [2508.17743].

The extreme cases are immediate. For the one-row partition $(n)$, the character is trivial and the immanant is the permanent. For the one-column partition $(1^n)$, the character is the sign and the immanant is the determinant. The hook family thus contains both endpoints and all single-hook intermediates [2102.04340, 2510.00327].

## 2. Representation-theoretic structure and normalization

For hook partitions, a central normalized quantity is
\[
\frac{\mathrm{Imm}_{\chi^{(k,1^{n-k})}}(A)}{\chi^{(k,1^{n-k})}(e)}.
\]
The normalization by $\chi^\lambda(e)$ is used because $\chi^\lambda(e)$ is the dimension of the irreducible representation $V^\lambda$, and the dimensions vary strongly with $\lambda$. For hooks, the dimension formula is especially simple:
\[
\chi^{(k,1^{n-k})}(e)=\binom{n-1}{k-1}.
\]
This gives the normalized hook chain a natural scale [2510.00327].

A useful explicit small-hook character identity appears for the hook with one box to the right of the first column:
\[
\chi_{(2,1,\ldots,1)}(\pi)=\mathrm{sgn}(\pi)\cdot(\#\{\text{fixed points of }\pi\}-1).
\]
This places small hooks close to determinant-like behavior [2102.04340].

For positive semidefinite matrices, hook immanants admit an operator-theoretic realization. If $A\ge 0$ is written as a Gram matrix of vectors $v_1,\dots,v_n$, then for the central idempotent associated with the hook representation one has
\[
\imm_{(k,1^{n-k})}(A)
=
\frac{n!}{\chi_{(k,1^{n-k})}(e)}
\|p_{(k,1^{n-k})}(v_1\otimes\cdots\otimes v_n)\|^2.
\]
This shows directly that hook immanants are nonnegative on positive semidefinite input and identifies them with squared norms of isotypic projections in tensor space [2103.04317].

A further combinatorial fact, important in complexity theory, is that the number of Hamiltonian cycles in a directed $n$-vertex graph can be written as a linear combination of the hook immanants
\[
\operatorname{imm}_{(r,1^{n-r})},\qquad 1\le r\le n.
\]
This situates hooks not as an incidental subclass, but as a natural expressive family [2102.04340].

## 3. Complexity classification

The complexity of immanant families in the cited dichotomy is governed by the partition parameter
\[
b(\lambda)=n-s,
\]
where $s$ is the number of parts of $\lambda$. Equivalently, $b(\lambda)$ counts the boxes to the right of the first column in the Young diagram. For a hook partition
\[
\lambda=(n-k,1^k),
\]
the number of parts is $k+1$, so
\[
b((n-k,1^k))=n-(k+1)=n-k-1.
\]
For hooks, this is exactly the number of boxes to the right of the first column, equivalently the arm length minus $1$ [2102.04340].

This parameter yields a full complexity dichotomy. If a hook family has bounded $b(\lambda)$, then evaluation is polynomial-time computable: in the family notation of the paper, $Imm(\Lambda)\in FP$ and $Imm(\Lambda)\in VP$. This is the tractable regime, and older algorithms cited there, including Hartmann’s algorithm with running time
\[
O(n^{6b(\lambda)+4}),
\]
are polynomial whenever $b(\lambda)=O(1)$ [2102.04340].

If the hook arm length is unbounded, then the complexity picture changes qualitatively. For a computationally reasonable family of hooks with
\[
n-k_n-1\to\infty,
\]
the paper rules out polynomial-time algorithms unless
\[
FPT=\#W[1].
\]
The algebraic analogue is that such a family is not in $VP$ unless
\[
VFPT=VW[1].
\]
This covers even subpolynomial unbounded growth [2102.04340].

If the hook arm grows polynomially,
\[
b(\lambda^{(n)})=\Omega(n^\alpha)
\quad\text{for some }\alpha>0,
\]
then stronger classical hardness applies: $Imm(\Lambda)$ is $VNP$-complete; if the family is computationally supported, it is $\#P$-hard; and there is no
\[
\exp(o(n^\alpha))
\]
time algorithm unless $\#ETH$ fails. In the hook notation used in earlier work, this recovers Bürgisser’s hardness theorem for families $(t(n),1^{n-t(n)})$ with polynomially growing first row length [2102.04340].

In the hook case, the resulting classification is exact: bounded arm length gives polynomial-time computability, unbounded arm length rules out polynomial time under parameterized assumptions, and polynomially growing arm length yields $\#P$-hardness and $VNP$-completeness [2102.04340].

## 4. Positivity, dominance, and monotone hook chains

For Hermitian positive semidefinite matrices, a classical theorem of Heyfron gives a monotone chain of normalized hook immanants. For totally nonnegative matrices, the same chain now holds:
\[
\operatorname{per}(A)=\frac{\mathrm{Imm}_{\chi^{(n)}}(A)}{\chi^{(n)}(e)}
\ge
\frac{\mathrm{Imm}_{\chi^{(n-1,1)}}(A)}{\chi^{(n-1,1)}(e)}
\ge
\cdots
\ge
\frac{\mathrm{Imm}_{\chi^{(1^n)}}(A)}{\chi^{(1^n)}(e)}
=
\det(A).
\]
Here total nonnegativity means that every minor is nonnegative:
\[
\det(A_{I,J})\ge 0
\qquad
\text{for all }I,J\subseteq[n],\ |I|=|J|.
\]
This extends the hook inequalities from the Hermitian positive semidefinite setting to the non-Hermitian class of totally nonnegative matrices [2510.00327].

The proof on totally nonnegative matrices is given in two complementary forms. One is combinatorial, using planar networks, a Lindström–Karlin–McGregor realization of totally nonnegative matrices, and Kaliszewski’s theorem identifying hook character values on incomparability graphs with counts of standard $P$-tableaux. The other is algebraic, using the Frobenius characteristic map, Kostka numbers, and the expansion
\[
\chi^{(k,1^{n-k})}
=
\sum_{\ell=n-k+1}^n
\binom{\ell-1}{n-k}\,\theta^\ell,
\]
where $\theta^\ell$ groups monomial-trace characters by number of parts [2510.00327].

A key adjacent inequality at the poset level is
\[
(k-1)\chi^{(k,1^{n-k})}(\operatorname{inc}(P))
\ge
(n-k+1)\chi^{(k-1,1^{n-k+1})}(\operatorname{inc}(P)).
\]
After normalization by hook dimensions, this becomes the monotonicity of adjacent hook immanants [2510.00327].

For positive semidefinite matrices, general immanant inequalities can be transferred to matrix inequalities in the Löwner order. In particular, Heyfron’s scalar hook chain yields Löwner-order inequalities for the corresponding trace-polynomial matrix forms, and the paper also records the general Schur inequality
\[
\det(A)\le \frac{\imm_\lambda(A)}{\chi_\lambda(e)}
\]
for positive semidefinite $A$, hence in particular for all hooks [2103.04317].

The same matrix-inequality framework gives concrete low-rank examples. For the hook partition $(2,1)$, the paper derives that if $X,Y\ge 0$ and $\tr(X)=\tr(Y)=1$, then
\[
X+Y+[\tr(XY)-1]1 \le XY+YX \le \frac23\,[X+Y+\tr(XY)]1.
\]
This identifies hook-immanantal inequalities with noncommutative operator inequalities rather than only scalar identities [2103.04317].

The broader ordering problem remains open beyond hooks. One open direction asks for all pairs $(\lambda,\mu)$ such that
\[
\frac{\mathrm{Imm}_{\chi^\lambda}(A)}{\chi^\lambda(e)}
\ge
\frac{\mathrm{Imm}_{\chi^\mu}(A)}{\chi^\mu(e)}
\]
for all totally nonnegative matrices, or for all Hermitian positive semidefinite matrices [2510.00327].

## 5. Structured matrices from graphs and digraphs

Hook immanants also admit explicit recurrences on graph-derived matrices. For an $n\times n$ matrix $M$, the hook immanant polynomial is defined by
\[
\Phi_k(M,x)=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).
\]
Thus
\[
\Phi_1(M,x)=\varphi(M,x),
\qquad
\Phi_n(M,x)=\psi(M,x),
\]
the characteristic and permanantal polynomials, respectively [2508.17743].

The graph-theoretic matrix families studied are
\[
H(G)=\beta D(G)+\gamma A(G),
\qquad
H(\overrightarrow G)=\beta D(\overrightarrow G)+\gamma A(\overrightarrow G),
\]
where $D$ is the degree or out-degree diagonal matrix and $A$ is the adjacency matrix. These include the standard specializations
\[
A(G),\quad L(G)=D(G)-A(G),\quad Q(G)=D(G)+A(G),\quad A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)
\]
and their digraph analogues [2508.17743].

The main theorems characterize two hook immanantal equalities in both the graph and digraph settings: a vertex-deletion formula and an edge- or arc-deletion formula, first for the hook immanant polynomials $\Phi_k$ and then, by evaluation at $x=0$, for the hook immanants $d_k$ themselves. These decompositions are governed by fixed points, transpositions along edges in the undirected case, and longer cycles corresponding to graph cycles or consistently directed cycles [2508.17743].

The proofs combine the Murnaghan–Nakayama rule specialized to hook partitions with a decomposition of contributing permutations by the cycle containing a distinguished vertex. In this framework, a permutation term is nonzero only when each vertex is either fixed or mapped along an edge or arc, so permutation-cycle structure is matched directly to graph-cycle structure [2508.17743].

For trees, all cycle terms vanish, leaving simpler recurrences. For adjacency matrices, the formulas become purely graph-deletion formulas. For Laplacian, signless Laplacian, and $A_\alpha$ matrices, the paper obtains corresponding hook-immanantal recurrences. The cases $k=1$ and $k=n$ recover known formulas for determinants, characteristic polynomials, permanents, and permanantal polynomials, while intermediate $k$ produce genuinely hook-immanantal recurrences not previously available in general [2508.17743].

A notable consequence is that the theory unifies determinant-type and permanent-type graph recurrences inside a single hook-parameter family [2508.17743].

## 6. Immanant characters, self-conjugate hooks, and related distinctions

In the Jacobi–Trudi setting, the hook-shape problem concerns immanant characters $\Gamma^\theta_{\mu/\nu}$ attached to a skew shape $\mu/\nu$ and a hook partition
\[
\theta=(N-k,1^k).
\]
The cited hook theorem proves that for every skew shape $\mu/\nu$, the corresponding hook-shape immanant character is a finite nonnegative integer sum of Stanley–Stembridge characters. More precisely,
\[
\Gamma^\theta_{\mu/\nu}=\sum_{\substack{J\subset[n-1]\\ |J|=k}}\Gamma_{h^J},
\]
where $h^J$ is an explicitly modified Hessenberg function determined by the Jacobi–Trudi zero pattern [2304.05285].

The underlying hook Kostka formula is
\[
K_{(N-k,1^k),c}=\binom{r-1}{k},
\]
when the content $c$ has $r$ nonzero entries. This makes the hook case especially tractable. The decomposition implies that the coefficient of the

Source: https://www.emergentmind.com/topics/hook-immanant