Second Immanant: Theory & Applications

• Second immanant is a matrix function defined by substituting standard sign or trivial weights in the determinant and permanent with an irreducible character of a symmetric group.
• It is derived via hook immanant notation and expressed through principal submatrix determinants, which provides concrete identities useful in graph theory and combinatorial analysis.
• Its unique position between determinants and permanents influences complexity classifications and enables quantum and interferometric applications in practical experiments.

The second immanant is a named special case of the immanant, the matrix function obtained by replacing the sign or trivial weights in the determinant and permanent expansions by an irreducible character of a symmetric group. The term is not fully uniform. In hook-immanant and graph-theoretic work it denotes the hook immanant $d_2$ attached to the partition $(2,1^{n-2})$, whereas some algebraic-complexity and quantum-statistical sources use it for the immanant indexed by $(n-1,1)$, the next-simplest irreducible representation after the trivial one. Several sources therefore treat “second immanant” as a contextual rather than universal label (Wu et al., 18 Feb 2025, Wu, 19 Jul 2025, Rugy-Altherre, 2013, Tichy et al., 2017, Cheraghpour et al., 2024).

1. Classical definition and hook-immanant form

For a partition $\lambda\vdash n$ and the corresponding irreducible character $\chi_\lambda$ of $S_n$, the immanant of an $n\times n$ matrix $M=(m_{ij})$ is

$$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$

This specializes to the determinant for $\lambda=(1^n)$ and to the permanent for $(2,1^{n-2})$0 (Wu et al., 18 Feb 2025, Wu, 19 Jul 2025).

A particularly important subfamily is the hook family

$(2,1^{n-2})$1

for $(2,1^{n-2})$2. In this notation,

$(2,1^{n-2})$3

and the second immanant in the hook-immanant sense is

$(2,1^{n-2})$4

For this hook character, one explicit formula is

$(2,1^{n-2})$5

where $(2,1^{n-2})$6 is the number of fixed points of $(2,1^{n-2})$7 (Wu, 19 Jul 2025, Campbell, 26 Jan 2025).

A basic working identity expresses $(2,1^{n-2})$8 through determinants of principal submatrices. If $(2,1^{n-2})$9 is $(n-1,1)$0 and $(n-1,1)$1 is obtained by deleting row $(n-1,1)$2 and column $(n-1,1)$3, then

$(n-1,1)$4

For $(n-1,1)$5 matrices this gives

$(n-1,1)$6

In the smallest nontrivial irreducible case, $(n-1,1)$7 and $(n-1,1)$8, one has

$(n-1,1)$9

which makes the intermediate character of the construction explicit (Wu et al., 18 Feb 2025, Tichy et al., 2017).

2. Competing conventions for the name

The principal ambiguity in the phrase “second immanant” is terminological rather than algebraic. In hook-immanant and graph-polynomial work, the name is fixed by the sequence

$\lambda\vdash n$0

so “second” means $\lambda\vdash n$1, namely $\lambda\vdash n$2 (Wu et al., 18 Feb 2025, Wu, 19 Jul 2025).

By contrast, some algebraic-complexity sources identify the second immanant with the representation $\lambda\vdash n$3, because it is the next-simplest irreducible representation after the trivial representation $\lambda\vdash n$4. In that convention, “second” refers to the first nontrivial step away from the permanent rather than the first nontrivial step away from the determinant (Rugy-Altherre, 2013, Tichy et al., 2017).

Several papers do not treat “second immanant” as a formal term at all. The terminology is explicitly described as nonstandard in work on vanishing immanants and in work on Temperley–Lieb and $\lambda\vdash n$5-immanants, where the phrase can instead denote the second term in a decomposition inside a different immanant family (Cheraghpour et al., 2024, Lu et al., 2023). This suggests that the nomenclature depends on the ambient ordering principle: hook index, simplicity relative to the permanent, or position inside a specialized combinatorial family.

3. Algebraic position between determinant and permanent

The second immanant occupies a distinguished position in inequalities for positive semidefinite matrices. Writing normalized immanants as $\lambda\vdash n$6, Heyfron’s theorem gives the hook chain

$\lambda\vdash n$7

for $\lambda\vdash n$8. In that ordering, the hook second immanant $\lambda\vdash n$9 is the first normalized immanant above the determinant (Huber et al., 2021).

This matrix-inequality viewpoint complements the hook-family definition $\chi_\lambda$0, $\chi_\lambda$1. It places $\chi_\lambda$2 as an intermediate invariant that preserves determinant-like sign structure while introducing class-function weights beyond $\chi_\lambda$3. The principal-minor identity

$\chi_\lambda$4

makes this intermediate status concrete: $\chi_\lambda$5 is still determinant-controlled, but not multiplicative in the determinant sense (Wu et al., 18 Feb 2025).

The same intermediate character appears in operator inequalities. In the hook chain, the second immanant is the first nontrivial normalized quantity between determinant and permanent; in low rank this produces explicit Löwner-order bounds. For $\chi_\lambda$6, the irreducible partition $\chi_\lambda$7 yields a two-sided inequality for the anticommutator of positive semidefinite trace-one matrices, reflecting the same “first nontrivial step beyond determinant” role (Huber et al., 2021).

4. Second immanantal polynomials in graph theory

A major modern use of the term is graph-theoretic. For a matrix $\chi_\lambda$8, the second immanantal polynomial is

$\chi_\lambda$9

In this setting the emphasis is not the bare matrix function $S_n$0 alone, but the polynomial invariant obtained by applying $S_n$1 to adjacency, Laplacian, and signless Laplacian matrices of graphs and digraphs (Wu et al., 18 Feb 2025, Wu, 19 Jul 2025).

For a simple graph $S_n$2, the basic object is

$S_n$3

For a digraph $S_n$4, one studies

$S_n$5

The reconstruction theory parallels older determinant- and permanent-based results, but with identities specific to $S_n$6.

Setting	Polynomial	Reconstruction data when $S_n$7
Undirected graph	$S_n$8	$S_n$9
Digraph, adjacency	$n\times n$0	$n\times n$1
Digraph, Laplacian/signless Laplacian	$n\times n$2	$n\times n$3

For undirected graphs, the central differential identity is

$n\times n$4

For digraphs, the corresponding identity is

$n\times n$5

These formulas imply reconstructibility when $n\times n$6. For simple graphs, the adjacency second immanantal polynomial is reconstructible from edge-deleted and vertex-pair-deleted subgraphs; for digraphs, the adjacency, Laplacian, and signless Laplacian second immanantal polynomials are reconstructible from arc-deleted subdigraphs. By contrast, analogous reconstruction results for graph Laplacian and signless Laplacian hook immanantal polynomials with $n\times n$7, including the second case $n\times n$8, remain open (Wu et al., 18 Feb 2025, Wu, 19 Jul 2025).

5. Complexity-theoretic interpretations

The complexity status of the “second immanant” depends sharply on which convention is intended. For the family indexed by $n\times n$9, algebraic-complexity results place it on the hard side: the family $M=(m_{ij})$0 is VNP-complete for c-reductions. The reason stated in the complexity analysis is that $M=(m_{ij})$1 has bounded width $M=(m_{ij})$2 but $M=(m_{ij})$3 boxes to the right of the first column, placing it squarely in the hard regime of bounded-width immanants (Rugy-Altherre, 2013).

A later full dichotomy for immanant families organizes the picture by

$M=(m_{ij})$4

where $M=(m_{ij})$5 is the number of parts of $M=(m_{ij})$6. If $M=(m_{ij})$7 for a family $M=(m_{ij})$8, then $M=(m_{ij})$9 is polynomial-time computable and lies in $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$0. If $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$1 is unbounded for a computationally reasonable family, then polynomial-time computation is ruled out under standard parameterized assumptions; if $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$2 grows polynomially, then $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$3 is #P-hard and VNP-complete (Curticapean, 2021).

This suggests that the terminological ambiguity is also a complexity-theoretic one. Using $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$4, the hook family $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$5 stays at bounded distance from the determinant side, while $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$6 moves linearly away from it. Accordingly, the two conventional meanings of “second immanant” land on opposite sides of the known complexity boundary: the $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$7 family is explicitly VNP-complete, whereas the hook convention aligns with the bounded-$$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$8 side of the dichotomy (Rugy-Altherre, 2013, Curticapean, 2021).

6. Generalizations, variants, and specialized uses

The term also appears indirectly in several extensions of immanant theory. In the classification of immanants that vanish on alternating matrices, the phrase “second immanant” is not formal, but the results furnish a complete test for any candidate: for odd $$d_\lambda(M)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}.$$9, every immanant vanishes on alternating matrices; for even $\lambda=(1^n)$0, vanishing is equivalent to the Young diagram being indestructible under recursive domino rim-hook removal (Cheraghpour et al., 2024). This suggests that a candidate indexed by $\lambda=(1^n)$1 is typically nonvanishing on $\lambda=(1^n)$2, because that diagram is generally destructible.

In Temperley–Lieb theory, “second immanant” can describe the second $\lambda=(1^n)$3-immanant in a minimal decomposition. When $\lambda=(1^n)$4 is $\lambda=(1^n)$5-avoiding and avoids $\lambda=(1^n)$6 but contains $\lambda=(1^n)$7, the signed Temperley–Lieb immanant satisfies

$\lambda=(1^n)$8

Here $\lambda=(1^n)$9 is literally the second $(2,1^{n-2})$00-immanant in the decomposition, obtained from a modified skew Ferrers region (Lu et al., 2023).

A further generalization replaces symmetric functions by quasisymmetric functions. In the theory of quasi-immanants, the quasisymmetric Schur function $(2,1^{n-2})$01 yields a quasisymmetric analogue of the second immanant. Its coefficients are indexed by cycle compositions rather than cycle types, and Campbell proves an explicit coefficient formula depending on the first part of the cycle composition and the number of permutations of a given cycle type (Campbell, 26 Jan 2025).

7. Quantum and interferometric realizations

The second immanant has also acquired a concrete quantum interpretation. In the immanon framework, many-body states with exchange symmetry labeled by a partition $(2,1^{n-2})$02 have scalar products proportional to $(2,1^{n-2})$03, where $(2,1^{n-2})$04 is the overlap matrix of one-particle states. For $(2,1^{n-2})$05, the partition $(2,1^{n-2})$06 is the unique nontrivial, nonalternating irreducible representation, and it plays the role of the simplest non-bosonic, non-fermionic species. Its occupation rule satisfies a partial Pauli principle: the allowed multiplicity patterns are $(2,1^{n-2})$07 and $(2,1^{n-2})$08, while $(2,1^{n-2})$09 is forbidden. The corresponding bunching factor is the normalized immanant of the distinguishability matrix (Tichy et al., 2017).

Multiphoton interferometry provides a second experimental route. Character-weighted time-bin entangled inputs can be designed so that the measured coincidence probability is a quadratic form in $(2,1^{n-2})$10 and its row permutations. For $(2,1^{n-2})$11, the natural nontrivial case is precisely $(2,1^{n-2})$12; for $(2,1^{n-2})$13, the analogous cases are $(2,1^{n-2})$14, $(2,1^{n-2})$15, and $(2,1^{n-2})$16. In this sense, the second immanant is not merely a formal interpolation between determinant and permanent, but a measurable symmetry sector in linear-optical interference (Khalique et al., 2020).