Hook Immanant Polynomial: Interpolation Theory

• The hook immanant polynomial is an immanantal polynomial defined via hook partitions, serving as a bridge between the determinant and the permanent.
• It employs representation theory and trace-polynomial methods to derive explicit formulas, monotonicity properties, and matrix inequalities on positive semidefinite and totally nonnegative matrices.
• The framework extends to graph theory by generating recurrences for graph matrices and extracting invariants such as edge, cycle, and matching counts.

A hook immanant polynomial is the immanantal polynomial attached to a hook partition $(k,1^{n-k})$. For an $n\times n$ matrix $M$, the hook immanant is $d_k(M):=d_{(k,1^{n-k})}(M)$, and the hook immanant polynomial is

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$

It interpolates between the characteristic polynomial and the permanental polynomial: $\Phi_1(M,x)=\varphi(M,x)$ and $\Phi_n(M,x)=\psi(M,x)$. A second, matrix-valued construction arises by specializing the trace-polynomial machinery for immanants to the hook character $\chi_{[k,1^{n-k}]}$, producing $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$. Both constructions are indexed by the irreducible character of $S_n$ corresponding to a hook-shaped partition, but they serve different purposes: the former is a scalar polynomial in $n\times n$0, while the latter is a matrix-valued trace polynomial used in Löwner-order inequalities (Dong et al., 25 Aug 2025, Huber et al., 2021).

1. Definition, notation, and endpoint cases

For a partition $n\times n$1 and an $n\times n$2 matrix $n\times n$3, the immanant is

$n\times n$4

When $n\times n$5, this becomes the hook immanant, denoted $n\times n$6. The convention used is that $n\times n$7 if $n\times n$8 or $n\times n$9 (Dong et al., 25 Aug 2025).

The extreme hook shapes recover the classical endpoint invariants. At $M$0, $M$1 is the sign character, so $M$2. At $M$3, $M$4 is the trivial character, so $M$5. Accordingly,

$M$6

The specialization $M$7 recovers the hook immanant itself: $M$8 Thus the hook family forms a one-parameter interpolation from determinant to permanent (Dong et al., 25 Aug 2025).

A parallel notation appears in the character-immanant literature. For an irreducible character $M$9 of $d_k(M):=d_{(k,1^{n-k})}(M)$0,

$d_k(M):=d_{(k,1^{n-k})}(M)$1

In particular, $d_k(M):=d_{(k,1^{n-k})}(M)$2 and $d_k(M):=d_{(k,1^{n-k})}(M)$3. For hooks, the normalized quantity

$d_k(M):=d_{(k,1^{n-k})}(M)$4

is especially important, with

$d_k(M):=d_{(k,1^{n-k})}(M)$5

This binomial dimension formula is one of the reasons hook immanants admit unusually explicit formulas (Skandera, 30 Sep 2025).

2. Representation-theoretic and trace-polynomial realizations

The representation-theoretic formulation starts from the general immanant

$d_k(M):=d_{(k,1^{n-k})}(M)$6

and specializes to hooks by taking $d_k(M):=d_{(k,1^{n-k})}(M)$7. For positive semidefinite $d_k(M):=d_{(k,1^{n-k})}(M)$8, the matrix can be written as a Gram matrix of vectors $d_k(M):=d_{(k,1^{n-k})}(M)$9, and the generalized matrix function has the tensor-trace form

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$0

where $\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$1. With the centrally primitive idempotent

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$2

one obtains

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$3

For hook immanants, $\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$4 and $\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$5, so the hook immanant is realized as a projected squared norm on the hook isotypic component (Huber et al., 2021).

The same framework defines trace polynomials $\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$6 and matrix-valued trace polynomials $\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$7, and hence

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$8

For a hook character this becomes

$\Phi_k(M,x):=d_k(xI_n-M)=d_{(k,1^{n-k})}(xI_n-M).$9

This is the direct hook specialization of the matrix-valued trace-polynomial construction. In this sense, “hook immanant polynomial” may denote not only $\Phi_1(M,x)=\varphi(M,x)$0 but also a noncommutative trace polynomial determined by the same hook character (Huber et al., 2021).

A structural vanishing theorem constrains these matrix-valued forms. If $\Phi_1(M,x)=\varphi(M,x)$1 corresponds to a Young diagram with more than $\Phi_1(M,x)=\varphi(M,x)$2 rows, then

$\Phi_1(M,x)=\varphi(M,x)$3

for all complex $\Phi_1(M,x)=\varphi(M,x)$4 matrices $\Phi_1(M,x)=\varphi(M,x)$5. For the hook $\Phi_1(M,x)=\varphi(M,x)$6, the number of rows is $\Phi_1(M,x)=\varphi(M,x)$7. Hence if $\Phi_1(M,x)=\varphi(M,x)$8, the hook matrix polynomial vanishes identically (Huber et al., 2021).

3. Positivity, monotonicity, and matrix inequalities

For Hermitian positive semidefinite matrices, the normalized hook immanants form a monotone chain from permanent down to determinant: $\Phi_1(M,x)=\varphi(M,x)$9 This is the hook chain conjectured by Merris and proved by Heyfron. Skandera proved that the same chain holds for every $\Phi_n(M,x)=\psi(M,x)$0 totally nonnegative matrix, so the normalized hook immanants are monotone not only on Hermitian positive semidefinite matrices but also on the distinct class of totally nonnegative matrices (Skandera, 30 Sep 2025).

The hook case also sits inside broader inequalities for immanants on positive semidefinite matrices. The appendix of the matrix-inequality framework records Schur’s inequality

$\Phi_n(M,x)=\psi(M,x)$1

which applies in particular to every hook $\Phi_n(M,x)=\psi(M,x)$2. It also records Watkins’s theorem in the form

$\Phi_n(M,x)=\psi(M,x)$3

and Heyfron’s theorem that the normalized hook immanants

$\Phi_n(M,x)=\psi(M,x)$4

form a chain between determinant and permanent on positive semidefinite matrices (Huber et al., 2021).

The matrix-valued trace-polynomial theory lifts such scalar inequalities to the Löwner order. If $\Phi_n(M,x)=\psi(M,x)$5 for all positive semidefinite $\Phi_n(M,x)=\psi(M,x)$6, then

$\Phi_n(M,x)=\psi(M,x)$7

for all positive semidefinite $\Phi_n(M,x)=\psi(M,x)$8 of equal size. Consequently, scalar hook inequalities yield matrix inequalities for the hook trace polynomial $\Phi_n(M,x)=\psi(M,x)$9. In degree $\chi_{[k,1^{n-k}]}$0, the hook partition $\chi_{[k,1^{n-k}]}$1 gives an explicit matrix inequality for trace-one positive semidefinite $\chi_{[k,1^{n-k}]}$2: $\chi_{[k,1^{n-k}]}$3 This exhibits a concrete low-dimensional hook trace-polynomial inequality (Huber et al., 2021).

4. Hook immanant polynomials of graph and digraph matrices

A major source of explicit hook immanant polynomials is the linear-combination matrix

$\chi_{[k,1^{n-k}]}$4

for a graph $\chi_{[k,1^{n-k}]}$5, and similarly

$\chi_{[k,1^{n-k}]}$6

for a digraph. Here $\chi_{[k,1^{n-k}]}$7 is the degree matrix and $\chi_{[k,1^{n-k}]}$8 the adjacency matrix. Specializations include adjacency $\chi_{[k,1^{n-k}]}$9, Laplacian $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$0, signless Laplacian $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$1, and $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$2 (Dong et al., 25 Aug 2025).

For graphs, the central vertex-deletion formula is

$\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$3

There is a parallel edge-deletion identity, and both formulas have digraph analogues in which graph cycles are replaced by consistently directed cycles. In the digraph case there is no $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$4-neighbor term, because a 2-cycle in the permutation expansion would require both directed arcs (Dong et al., 25 Aug 2025).

These recursions simplify sharply on trees, where all cycle sums vanish. For a tree $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$5,

$\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$6

For adjacency matrices this becomes

$\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$7

At the endpoints $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$8 and $\widetilde{\imm}_{\chi_{[k,1^{n-k}]}}$9, these identities recover classical characteristic-polynomial and permanental-polynomial recurrences (Dong et al., 25 Aug 2025).

The same framework produces direct formulas for hook immanants by setting $S_n$0. It also yields special identities such as

$S_n$1

for bipartite graphs, where $S_n$2 and $S_n$3. The reason given is that bipartite graphs have no odd cycles, so the cycle terms in the Laplacian and signless Laplacian recursions match (Dong et al., 25 Aug 2025).

5. Coefficients, graph invariants, and regular graphs

For the hook partition $S_n$4, the immanantal polynomial of

$S_n$5

is expanded as

$S_n$6

An explicit theorem gives the first six coefficients $S_n$7 for $S_n$8 (Dong et al., 6 Apr 2026).

The first coefficients are

$S_n$9

$n\times n$00

and

$n\times n$01

For $n\times n$02, the formula adds a triangle term

$n\times n$03

alongside the degree-symmetric and matching-type terms. The formulas for $n\times n$04 and $n\times n$05 involve $n\times n$06, $n\times n$07, weighted cycle sums $n\times n$08, weighted matching sums $n\times n$09, and the triangle-degree quantity $n\times n$10 (Dong et al., 6 Apr 2026).

The adjacency specialization is especially transparent. With $n\times n$11 and $n\times n$12,

$n\times n$13

$n\times n$14

$n\times n$15

The subsequent coefficients show that, for adjacency matrices, the first hook coefficients detect the number of edges, triangles, 2-matchings and 4-cycles, and 5-cycles and triangle-plus-edge configurations (Dong et al., 6 Apr 2026).

A rigidity theorem holds for regular graphs. If $n\times n$16 and $n\times n$17 are nonzero real numbers and $n\times n$18 are regular graphs, then

$n\times n$19

if and only if

$n\times n$20

For regular graphs, the hook immanantal polynomial of $n\times n$21 therefore carries exactly the same information as that of the adjacency matrix (Dong et al., 6 Apr 2026).

6. Complexity and adjacent hook-indexed constructions

The computational complexity of hook immanants is controlled by the Young-diagram parameter

$n\times n$22

where $n\times n$23 is the number of parts of $n\times n$24. For the hook $n\times n$25,

$n\times n$26

and in the notation $n\times n$27,

$n\times n$28

Thus the governing parameter is the arm length beyond the first column (Curticapean, 2021).

The resulting dichotomy is sharp. If $n\times n$29, then $n\times n$30 and $n\times n$31. For hooks this means bounded arm length, equivalently constant $n\times n$32 in $n\times n$33. Hartmann’s algorithm runs in

$n\times n$34

so hooks of bounded arm length are polynomial-time computable. If $n\times n$35, then for computationally reasonable families $n\times n$36, polynomial-time computability is ruled out unless $n\times n$37, and the algebraic analogue is ruled out unless $n\times n$38. If $n\times n$39 grows polynomially, then the corresponding hook family is $n\times n$40-hard and $n\times n$41-complete (Curticapean, 2021).

Several neighboring theories use the same hook partitions without defining the classical hook immanant polynomial itself. In chromatic symmetric function theory, the hook coefficient of $n\times n$42 is

$n\times n$43

where $n\times n$44 counts acyclic orientations with $n\times n$45 sinks; this gives positivity of all hook Schur coefficients for arbitrary graphs, but not a matrix-immanant formula (Kaliszewski, 2014). In the restriction problem for $n\times n$46 to $n\times n$47, hook highest weights $n\times n$48 lead to positive tableau formulas for restriction coefficients $n\times n$49, again through hook-indexed character theory rather than direct immanant evaluation (Narayanan, 2024). In a different direction, the twisted immanant is defined only for self-conjugate partitions; among hooks, this leaves exactly $n\times n$50, and for matrices with anticommuting entries one has

$n\times n$51

That identity belongs to a distinct, twisted theory rather than the ordinary hook immanant polynomial (Itoh, 2015).

These neighboring constructions underscore a recurring fact: the hook partition $n\times n$52 is unusually tractable across representation theory, symmetric functions, graph polynomials, and positivity theory. For ordinary immanants, that tractability appears most concretely in the binomial dimension formula, the determinant-to-permanent hook inequalities, the explicit graph recursions for $n\times n$53, and the complete complexity transition governed by hook arm length (Skandera, 30 Sep 2025, Dong et al., 25 Aug 2025, Curticapean, 2021).