Lieb's Permanental-Dominance Conjecture

• Lieb's Permanental-Dominance Conjecture is a hypothesis stating that the permanent of a positive semidefinite Hermitian matrix dominates its corresponding immanants derived from subgroup characters.
• The conjecture bridges matrix analysis, representation theory, and probabilistic combinatorics by generalizing classical determinant and permanent inequalities.
• Partial results and explicit criteria in quantum information and algebra have verified the conjecture for specific matrix classes, spurring further interdisciplinary research.

Lieb's Permanental-Dominance Conjecture is a central open problem in matrix analysis, algebraic combinatorics, and quantum physics. It asserts that, for any positive semidefinite Hermitian matrix and any subgroup character, the permanent function dominates the corresponding generalized matrix function (immanant) arising from group character averages. This dominance is conjectured to hold universally, extending deep analogies between determinant and permanent inequalities and shaping key methodologies in representation theory, geometric complexity theory, probabilistic combinatorics, and experimental quantum information.

1. Statement and Mathematical Formulation

Lieb's Permanental-Dominance Conjecture posits that for any $n \times n$ positive semidefinite Hermitian matrix $A \in H_n$ and any subgroup $G$ of the symmetric group $S_n$ with character $x$, the inequality

$\operatorname{per} A > f_x(A)$

holds, where

$$f_x(A) = \frac{1}{|G|}\sum_{\sigma \in G} x(\sigma) \prod_{i=1}^n a_{i, \sigma(i)}.$$

The special case for irreducible characters yields normalized immanants. For the partition $\lambda \vdash n$ and corresponding irreducible character $\chi_\lambda$, the normalized immanant is

$$\overline{\operatorname{Imm}}_\lambda(A) = \frac{1}{\chi_\lambda(e)} \sum_{\sigma \in S_n} \chi_\lambda(\sigma) \prod_{i=1}^n a_{i,\sigma(i)},$$

and the conjecture claims $$\overline{\operatorname{Imm}}_\lambda(A) \leq \operatorname{per} A$$ for all $\lambda$ and all $A \geq 0$.

This formalism generalizes both the comparison between permanent and determinant and the well-known chain of classical inequalities

$\det A \leq h(A) \leq \operatorname{per} A,$

where $h(A)$ is the product of the diagonal elements.

2. Historical Context and Evolution

The conjecture originated in Lieb's influential 1966 paper, where analogies were drawn between determinant and permanent inequalities, inverting and extending results such as Hadamard's inequality and its permanent analogues. Several further conjectures in the same spirit followed, including block inequalities and eigenvalue dominance statements (e.g., Marcus-Newman conjecture, "permanent on top" conjecture).

Many related conjectures have since been disproven — notably, explicit counterexamples have refuted the stronger "permanent on top" conjecture, which claimed the permanent is the maximum eigenvalue of the Schur power matrix $T(A)$, and several block/permanent inequalities for $n \geq 5$ matrices (Zhang, 2016, Wanless, 2022). Despite this, Lieb's Permanental-Dominance Conjecture itself remains unchallenged in general, verified for certain classes (e.g., all immanants for $n \leq 13$) and an ongoing target for research (Zhang, 2016, Wanless, 2022, Rodtes, 2023).

3. Connections to Representation Theory and Geometric Complexity

Representation-theoretic approaches, notably the geometric complexity theory (GCT) program, investigate the permanent-versus-determinant problem via orbit closures and module decompositions. In this setting, irreducible $GL_{n^2}(\mathbb{C})$-modules occurring in the coordinate ring of the determinant polynomial closure form the monoid $S(\mathrm{Det}_n)$, with irreducibles labeled by partitions $\lambda$. Saturation results (Bürgisser et al., 2015) guarantee that all partitions $\lambda$ with $\ell(\lambda) \leq n$ and $|\lambda| \equiv 0 \pmod n$ lie in the saturation of $S(\mathrm{Det}_n)$, and any separation between permanent and determinant must occur in the set-theoretic "holes" of $S(\mathrm{Det}_n)$.

This is analogous to the analytic setting: generic behavior of matrix functions is "saturated," and true separation occurs only in delicate exceptional cases. Thus, both domains highlight that obstructions — generic or representation-theoretic — are central to understanding functional dominance.

4. Probabilistic and Combinatorial Perspectives

Significant connections have been drawn between the conjecture and probabilistic characterizations of partition dominance. For Ferrers diagrams of partitions, probabilistic inequalities (involving randomly filling cells) mirror classical dominance order. The total non-negativity of certain infinite matrices built from probability generating functions implies coefficient inequalities (Smyth, 2015), such as

$(p^A)_b (p^B)_a \leq (p^A)_a (p^B)_b,$

which structurally resemble those predicted by the permanental-dominance conjecture.

Key results show that if these probabilistic (and combinatorial) injection conditions hold for matrix sequences, then the dominance order on partitions is "shadowed" by permanent-type inequalities. Combinatorial conjectures about injections among compositions of integers directly encode the structure needed for permanental inequalities to hold (Smyth, 2015).

5. Structural Rigidity, Invariance, and Algebraic Properties

Research into linear preservers of permanental rank has delineated the rigidity of the permanent as a matrix function (Guterman et al., 2023). The only linear maps preserving bounded permanental rank are compositions of row/column permutations, nonzero diagonal scalings, and transposition. The Zariski density of matrices of exact permanental rank within the closure implies that no exotic invariances exist for permanental structures, constraining the set of transformations admissible in any proof of permanental dominance.

Loop-soup constructions for Markov processes (Fitzsimmons et al., 2012) provide a probabilistic origin for general permanental processes, where joint moments obey a permanental formula and invariance properties of loop measures (such as rotation and space-time transform invariance) are crucial to the geometric control of permanental fields. These invariance and moment structures are speculated to be key ingredients in any future progress on the conjecture.

6. Partial Results, Criteria, and Classes Affirming the Conjecture

Novel algebraic identities linking determinants and generalized matrix functions have been established via Cholesky decompositions, offering explicit representations for entire infinite families of matrices for which the conjecture holds (Rodtes, 2023). Specifically, given a matrix $A = LL^*$, if off-first-column entries of $L$ are uniformly bounded, all associated immanants are controlled by the permanent. This yields concrete, checkable criteria (based on matrix factorization and entry bounds) under which the conjecture is validated, and allows the generation of infinite matrix classes through diagonal scaling transformations that all satisfy the permanent-dominance inequality.

7. Quantum Information, Bosonic Interference, and Physical Applications

The conjecture underpins monotonicity principles in bosonic linear optics and quantum information (Geller et al., 4 Sep 2025). In interferometric experiments, the generalized bunching probability — the chance that all bosons exit in a designated output mode subset — can be shown, contingent on Lieb's conjecture, to be maximized for perfectly indistinguishable bosons. Under the refinement partial order for partially-labelled states, the generalized bunching probability becomes monotonic: more indistinguishability equates to higher bunching probability.

Further, the Haar-average of the bunching probability is Schur convex with respect to the eigenvalues of the single-particle density matrix. When this density matrix is Gibbsian, the average bunching probability scales with inverse temperature, offering a framework for thermometer functions in quantum experiments.

8. Logical Implications among Permanental Inequalities

Recent results have illuminated logical relationships among classical permanental conjectures (Pioge et al., 31 Jul 2025). For instance, Hadamard-type product inequalities for positive semidefinite matrices (now known to be false in general) directly imply related eigenvalue bounds on derived matrices constructed from permanents of minors. Explicit counterexamples have demonstrated that violations in the eigenvalue domain automatically produce new counterexamples to Hadamard permanent inequalities via tailored matrix constructions.

While these logical chains do not yet impinge on the status of Lieb's original conjecture, they map the landscape of condition hierarchies among permanental inequalities and may inform the discovery of structures critical for broader dominance statements.

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In summary, Lieb's Permanental-Dominance Conjecture integrates foundational ideas from matrix analysis, representation theory, combinatorial probability, and quantum physics. It remains an unresolved but guiding principle, with partial results, explicit criteria, computational evidence, and implications for experimental systems continuing to shape research examinations. Its resolution is anticipated to unlock new connections in both pure mathematics and physical applications, and to characterize precisely the regime where symmetry and positivity enforce optimal functional dominance of the permanent.