Trace Formula for Immanants

• Immanants are generalized matrix invariants defined by weighted sums over permutations using symmetric group characters, subsuming determinants and permanents.
• Trace formulas for immanants unify classical invariant theory with quantum, combinatorial, and weight-space generalizations to derive practical matrix inequalities.
• Recent research uses representation theory, Capelli identities, and combinatorial techniques to enable explicit computations and complexity analysis of immanants.

Immanants are a broad generalization of the determinant and permanent, formulated as weighted sums over permutations via irreducible characters of the symmetric group. Trace formulas for immanants capture deep links among multilinear algebra, representation theory, combinatorics, and invariant theory. Recent research has developed a rich hierarchy of such formulas, ranging from classical invariance and stabilizer analyses to quantum, quasisymmetric, and weight-space generalizations that subsume many classical inequalities and identities.

1. Classical Definition and Trace Structure

Given an integer $n$ and a partition $T \vdash n$, the immanant associated to $T$ is defined for any $n\times n$ matrix $A = (x_{i,j})$ by

$$P_T(A) = \sum_{\sigma \in S_n} \chi_T(\sigma) \prod_{i=1}^n x_{i,\sigma(i)},$$

where $\chi_T$ is the character of $S_n$ corresponding to the partition $T$. For $T = (1,1,\ldots,1)$, $P_T$ is the determinant; for $T=(n)$, it is the permanent. Immanants thus generalize these classical matrix invariants.

Representation-theoretically, immanants are the homogeneous degree-$n$ polynomials on the space of $n\times n$ matrices that correspond to the unique trivial $S_n$-module inside the weight-zero subspace of $S^n(E \otimes F)$, for $n$-dimensional complex vector spaces $E$ and $F$ (Ye, 2011).

Any homogeneous degree-$n$ polynomial invariant under left and right action by diagonal matrices and conjugation by permutation matrices is necessarily a linear combination of immanants.

2. Trace Formula and Derivative Generalizations

A fundamental analytic result is the trace formula for the immanant's first derivative (Carvalho et al., 2013). Define the immanantal adjoint $adj_\chi(A)$ to have $(i,j)$ entry

$(adj_\chi(A))_{i,j} = d_\chi(A(i|j)),$

where $A(i|j)$ is the matrix obtained by zeroing out row $i$ and column $j$, except at the $(i,j)$-entry (set to $1$). The Jacobi-type trace formula then reads: $D d_\chi(A)(X) = \mathrm{tr}[ (adj_\chi(A))^T X ]$ for any direction $X$. Higher-order derivatives admit similar combinatorial Laplace expansions: $$D^k d_\chi(A)(X_1,\dots, X_k) = \sum_{a \in Q_{k,n}} d_\chi(A(a; X_1, \dots, X_k)),$$ where $Q_{k,n}$ indexes the sets of columns being replaced.

This framework generalizes to symmetric powers of operators, yielding trace-like formulas for induced $m$-th tensor symmetric powers, with explicit expressions controlled by representation-theoretic symmetrisers and immanantal minors.

3. Stabilizer Groups and Symmetry Analysis

The symmetry properties of immanants are precisely characterized (Ye, 2011). For generic partitions $T \neq (1,\ldots,1), (n), (4,1^3)$ and $n \geq 6$, the identity component of the stabilizer in $GL(E \otimes F)$ is

$$A(S_n) \ltimes T(GL_n \times GL_n) \times \mathbb{Z}_2$$

where $T(GL_n \times GL_n)$ is the torus of pairs of diagonal $n\times n$ matrices with product of determinants $=1$, $A(S_n)$ is the diagonal subgroup, and $\mathbb{Z}_2$ acts by transposition.

A key relation for stabilizer compatibility is

$\prod_{i=1}^n C_{i, \sigma(i)} = 1$

for permutations where $\chi_T(\sigma) \ne 0$.

These symmetry results underpin the invariance of immanants and their eligibility as building blocks for trace formulas, ensuring that traces over invariant subspaces pick out contributions exactly expressible in terms of immanants.

4. Quantum, Capelli, and Weight-Space Trace Formulas

The theory has been extended into noncommutative and quantum settings via Capelli identities and central elements in universal enveloping algebras (Zaitsev, 20 Nov 2024, Brini et al., 2018). The universal matrix Capelli identity in the enveloping algebra $U(\mathfrak{gl}_N)$ and its reflection equation algebra counterpart leverage Jucys–Murphy elements to encode Young diagram combinatorics and quantum deformation: $$L_1 (L_2 - j_2) \cdots (L_n - j_n) = X_1 \cdots X_n D_1 \cdots D_n$$ and its quantum deformation via $R$-matrices, with trace formulas for quantum immanants obtained by taking suitable $R$-traces.

In the representation-theoretic framework, trace formulas for immanants of generalized principal submatrices are given in terms of traces over weight spaces (Jing et al., 28 Aug 2025): $$\frac{Imm_{\chi^{\lambda}}(A_I)}{m(I)} = \operatorname{tr}(\mathcal{P}_\mu \circ \pi_\lambda(A) \circ \mathcal{P}_\mu |_{U^{(\lambda)}})$$ for partition $\lambda$, weight $\mu$, and multiplicity factor $m(I)$, extending Kostant's formula from the $0$-weight case to arbitrary weights.

5. Combinatorial and Quasisymmetric Generalizations

Combinatorial advances have generalized immanants using symmetric functions and, more broadly, quasisymmetric functions, which record cycle compositions of permutations (Campbell, 26 Jan 2025). Classical character coefficients $\chi^{\lambda}(\text{ctype}(\sigma))$ are replaced by those derived from quasisymmetric power sum bases (indexed by compositions rather than partitions), yielding the quasi-immanant: $$QImm^Q_\Psi(A) = \sum_{\sigma \in S_n} (\text{coefficient from } n! \cdot Q \text{ at } \Psi_{ccomp(\sigma)}) \prod_{i=1}^n a_{i,\sigma(i)}$$ where $ccomp(\sigma)$ is the cycle composition of $\sigma$.

When $Q$ is symmetric, the quasi-immanant coincides with the classical immanant, but for general $Q$, richer invariants arise. This refinement enables new combinatorial identities and trace formulas sensitive to more detailed permutation structures, e.g., for "second immanants" corresponding to hook shapes.

6. Inequalities and Matrix Positivity

Immanant trace formulas provide a unified approach to matrix positivity results and inequalities (Huber et al., 2021, Jing et al., 28 Aug 2025). Scalar inequalities (e.g., Schur’s, Stembridge’s, and Kostant’s) can be lifted to matrix inequalities in the Lӧwner order via trace polynomial formulations. The passage from scalar to operator inequalities is achieved by expressing immanants as norms or traces of projected Gram vectors, with positivity and non-vanishing criteria linked to representation-theoretic weight spaces.

For Hermitian positive semidefinite or totally nonnegative matrices, normalized immanants satisfy strengthened inequalities: $$\sum_{\nu \in O_\mu} \operatorname{tr}(\mathcal{P}_\nu \circ \pi_\lambda(A) \circ \mathcal{P}_\nu) \geq |O_\mu|\, \dim(U^{(\lambda)}_\mu) \det(A),$$ encompassing classical bounds as special cases, and characterizing equality in terms of weight space dimensions.

7. Structural, Computational, and Algebraic Impact

Trace formulas for immanants serve as a skeleton for analyzing polynomial invariants of matrices, centrality in enveloping algebras, and spectral properties in combinatorial representation theory. The detailed classification of stabilizers constrains the range of invariance, while explicit formulas for Capelli and quantum immanants supply concrete computational frameworks. Complexity-theoretic results demonstrate a dichotomy: immanants close to the sign character (small $b(\lambda)$) are tractable, those further away inherit the full $\#$P or VNP hardness of the permanent (Curticapean, 2021).

Combinatorial trace formulas—such as those expressing Temperley–Lieb immanants as sums over %-immanants indexed by pattern-avoiding permutations—enable non-recursive and algorithmic computation of matrix invariants, with explicit binomial coefficient expressions for coefficients (Lu et al., 2023).

Table: Landscape of Immanant Trace Formulas

Theory	Trace Formula Structure	Key Objects (Editor’s term)
Classical	Sum over $S_n$, characters	Immanants, weight-zero subspace
Quantum	Capelli identity, $R$-matrix traces	Quantum immanants, Jucys–Murphy
Weight-space	Trace on $U^{(\lambda)}_\mu$	Generalized principal submatrices
Quasisymmetric	Sum over cycle compositions	Quasi-immanants
Combinatorial	Explicit binomial sums over minors	TL immanants, %-immanants

The architecture of immanant trace formulas now spans classical invariant theory, symmetric function theory, quantum algebra, and combinatorial representation theory. This unified analytical machinery provides essential tools for studying matrix functions, their symmetries, inequalities, and computational complexity in algebraic settings.