Hook Immanants

• Hook immanants are defined using hook partitions of the symmetric group, interpolating between the determinant (one-column) and the permanent (one-row).
• They exhibit distinct complexity regimes: bounded arm lengths allow polynomial-time computations while unbounded growth leads to VNP-completeness and #P-hardness.
• Applications include matrix positivity, algebraic graph theory, and operator inequalities, with explicit recurrences derived from graph-deletion formulas.

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 (Curticapean, 2021, Skandera, 30 Sep 2025).

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

$(k,1^{n-k})$0

with the conventions

$(k,1^{n-k})$1

Accordingly, hook immanants provide a canonical chain from determinant to permanent through intermediate irreducible characters (Dong et al., 25 Aug 2025).

The extreme cases are immediate. For the one-row partition $(k,1^{n-k})$2, the character is trivial and the immanant is the permanent. For the one-column partition $(k,1^{n-k})$3, the character is the sign and the immanant is the determinant. The hook family thus contains both endpoints and all single-hook intermediates (Curticapean, 2021, Skandera, 30 Sep 2025).

2. Representation-theoretic structure and normalization

For hook partitions, a central normalized quantity is

$(k,1^{n-k})$4

The normalization by $(k,1^{n-k})$5 is used because $(k,1^{n-k})$6 is the dimension of the irreducible representation $(k,1^{n-k})$7, and the dimensions vary strongly with $(k,1^{n-k})$8. For hooks, the dimension formula is especially simple: $(k,1^{n-k})$9 This gives the normalized hook chain a natural scale (Skandera, 30 Sep 2025).

A useful explicit small-hook character identity appears for the hook with one box to the right of the first column: $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$0 This places small hooks close to determinant-like behavior (Curticapean, 2021).

For positive semidefinite matrices, hook immanants admit an operator-theoretic realization. If $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$1 is written as a Gram matrix of vectors $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$2, then for the central idempotent associated with the hook representation one has

$$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$3

This shows directly that hook immanants are nonnegative on positive semidefinite input and identifies them with squared norms of isotypic projections in tensor space (Huber et al., 2021).

A further combinatorial fact, important in complexity theory, is that the number of Hamiltonian cycles in a directed $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$4-vertex graph can be written as a linear combination of the hook immanants

$$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$5

This situates hooks not as an incidental subclass, but as a natural expressive family (Curticapean, 2021).

3. Complexity classification

The complexity of immanant families in the cited dichotomy is governed by the partition parameter

$$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$6

where $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$7 is the number of parts of $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$8. Equivalently, $$\operatorname{imm}_{(n-k,1^k)}(A)=\sum_{\pi\in S_n}\chi_{(n-k,1^k)}(\pi)\prod_{i=1}^n A_{i,\pi(i)}.$$9 counts the boxes to the right of the first column in the Young diagram. For a hook partition

$\lambda\vdash n$0

the number of parts is $\lambda\vdash n$1, so

$\lambda\vdash n$2

For hooks, this is exactly the number of boxes to the right of the first column, equivalently the arm length minus $\lambda\vdash n$3 (Curticapean, 2021).

This parameter yields a full complexity dichotomy. If a hook family has bounded $\lambda\vdash n$4, then evaluation is polynomial-time computable: in the family notation of the paper, $\lambda\vdash n$5 and $\lambda\vdash n$6. This is the tractable regime, and older algorithms cited there, including Hartmann’s algorithm with running time

$\lambda\vdash n$7

are polynomial whenever $\lambda\vdash n$8 (Curticapean, 2021).

If the hook arm length is unbounded, then the complexity picture changes qualitatively. For a computationally reasonable family of hooks with

$\lambda\vdash n$9

the paper rules out polynomial-time algorithms unless

$$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$0

The algebraic analogue is that such a family is not in $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$1 unless

$$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$2

This covers even subpolynomial unbounded growth (Curticapean, 2021).

If the hook arm grows polynomially,

$$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$3

then stronger classical hardness applies: $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$4 is $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$5-complete; if the family is computationally supported, it is $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$6-hard; and there is no

$$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$7

time algorithm unless $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$8 fails. In the hook notation used in earlier work, this recovers Bürgisser’s hardness theorem for families $$\operatorname{imm}_{\lambda}(A)=\sum_{\pi\in S_n}\chi_{\lambda}(\pi)\prod_{i=1}^n A_{i,\pi(i)},$$9 with polynomially growing first row length (Curticapean, 2021).

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 $\chi_\lambda$0-hardness and $\chi_\lambda$1-completeness (Curticapean, 2021).

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: $\chi_\lambda$2 Here total nonnegativity means that every minor is nonnegative: $\chi_\lambda$3 This extends the hook inequalities from the Hermitian positive semidefinite setting to the non-Hermitian class of totally nonnegative matrices (Skandera, 30 Sep 2025).

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 $\chi_\lambda$4-tableaux. The other is algebraic, using the Frobenius characteristic map, Kostka numbers, and the expansion

$\chi_\lambda$5

where $\chi_\lambda$6 groups monomial-trace characters by number of parts (Skandera, 30 Sep 2025).

A key adjacent inequality at the poset level is

$\chi_\lambda$7

After normalization by hook dimensions, this becomes the monotonicity of adjacent hook immanants (Skandera, 30 Sep 2025).

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

$\chi_\lambda$8

for positive semidefinite $\chi_\lambda$9, hence in particular for all hooks (Huber et al., 2021).

The same matrix-inequality framework gives concrete low-rank examples. For the hook partition $S_n$0, the paper derives that if $S_n$1 and $S_n$2, then

$S_n$3

This identifies hook-immanantal inequalities with noncommutative operator inequalities rather than only scalar identities (Huber et al., 2021).

The broader ordering problem remains open beyond hooks. One open direction asks for all pairs $S_n$4 such that

$S_n$5

for all totally nonnegative matrices, or for all Hermitian positive semidefinite matrices (Skandera, 30 Sep 2025).

5. Structured matrices from graphs and digraphs

Hook immanants also admit explicit recurrences on graph-derived matrices. For an $S_n$6 matrix $S_n$7, the hook immanant polynomial is defined by

$S_n$8

Thus

$S_n$9

the characteristic and permanantal polynomials, respectively (Dong et al., 25 Aug 2025).

The graph-theoretic matrix families studied are

$\lambda$0

where $\lambda$1 is the degree or out-degree diagonal matrix and $\lambda$2 is the adjacency matrix. These include the standard specializations

$\lambda$3

and their digraph analogues (Dong et al., 25 Aug 2025).

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 $\lambda$4 and then, by evaluation at $\lambda$5, for the hook immanants $\lambda$6 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 (Dong et al., 25 Aug 2025).

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 (Dong et al., 25 Aug 2025).

For trees, all cycle terms vanish, leaving simpler recurrences. For adjacency matrices, the formulas become purely graph-deletion formulas. For Laplacian, signless Laplacian, and $\lambda$7 matrices, the paper obtains corresponding hook-immanantal recurrences. The cases $\lambda$8 and $\lambda$9 recover known formulas for determinants, characteristic polynomials, permanents, and permanantal polynomials, while intermediate $\lambda=(n-k,1^k),$0 produce genuinely hook-immanantal recurrences not previously available in general (Dong et al., 25 Aug 2025).

A notable consequence is that the theory unifies determinant-type and permanent-type graph recurrences inside a single hook-parameter family (Dong et al., 25 Aug 2025).

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

In the Jacobi–Trudi setting, the hook-shape problem concerns immanant characters $\lambda=(n-k,1^k),$1 attached to a skew shape $\lambda=(n-k,1^k),$2 and a hook partition

$\lambda=(n-k,1^k),$3

The cited hook theorem proves that for every skew shape $\lambda=(n-k,1^k),$4, the corresponding hook-shape immanant character is a finite nonnegative integer sum of Stanley–Stembridge characters. More precisely,

$\lambda=(n-k,1^k),$5

where $\lambda=(n-k,1^k),$6 is an explicitly modified Hessenberg function determined by the Jacobi–Trudi zero pattern (Lesnevich, 2023).

The underlying hook Kostka formula is

$\lambda=(n-k,1^k),$7

when the content $\lambda=(n-k,1^k),$8 has $\lambda=(n-k,1^k),$9 nonzero entries. This makes the hook case especially tractable. The decomposition implies that the coefficient of the