An update on a few permanent conjectures

Abstract: We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture${}^\dagger$, Lieb permanent dominance conjecture, Bapat and Sunder conjecture${}^\dagger$ on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open. We also raise a few new questions for future study. (${}^\dagger$conjectures have been recently settled negatively.)

• The paper refutes the Permanent-on-Top conjecture with explicit computational counterexamples.
• It demonstrates the limitations of Schur power matrices in reliably capturing largest eigenvalues for positive semidefinite matrices.
• The study outlines open questions, particularly the unresolved Lieb permanent dominance conjecture and related matrix challenges.

Summary of "An Update on a Few Permanent Conjectures" (1608.02844)

Introduction to the Concept of Permanents

The paper begins by situating the permanent as a central matrix function analogous to the determinant, with significant roles in symmetric tensor theory and combinatorics. Historically introduced alongside the determinant in the 1800s, the concept of permanents remains vital in current mathematical research. The paper discusses the prominence of the van der Waerden conjecture, resolved in 1981, and moves towards exploring other central conjectures around permanents, potentially offering a modern perspective on these mathematical challenges.

Prominent Conjectures in Permanent Theory

Permanent-on-Top Conjecture

The Permanent-on-Top (POT) conjecture, formulated in the mid-1960s, proposed that the permanent of a positive semidefinite matrix was the largest eigenvalue of its Schur power matrix. This conjecture was refuted by computational examples, revealing that while $A$ is an eigenvalue of the Schur power matrix $\pi(A)$, it is not necessarily the largest one. Notably, Shchesnovich's example provided a counterexample with a $5 \times 5$ matrix, challenging the previously assumed validity of the POT conjecture for all matrix sizes.

Figure 1: Resolution of the POT conjecture showcases that permanents are not the largest eigenvalues of their Schur power matrices.

Lieb Permanent Dominance Conjecture

Stemming from numerical range discussions in the POT conjecture, the Lieb permanent dominance conjecture remains unresolved. It suggests that for all positive semidefinite matrices, specific inequalities involving a character-dependent group of permutations should hold, exhibiting a tighter bound than initially perceived. The resolution of this conjecture could potentially revolutionize the understanding of the permanent dominance within matrix theory.

New Questions and Implications

The recent insights and unresolved conjectures prompt several questions:

1. Existence of Smaller Counterexamples: Can counterexamples to the POT conjecture be identified for $n=4$ matrices?
2. Precise Numerical Range Determination: What delineates the extremal values within the numerical range of Schur power matrices when applied to positive semidefinite matrices?
3. Comprehensive Characterization: A broader question is how the largest eigenvalue of $\pi(A)$ compares universally across various classes of matrices beyond correlation and hermitian matrices.

Open Conjectures

Continuing the tradition of exploring permanents, unresolved conjectures such as Marcus per-in-per, Foregger's condition on doubly stochastic matrices, and Marcus-Minc max-per-$U$ problem, among others, showcase areas where further computational and theoretical exploration could yield new mathematical insights.

Conclusion

The paper emphasizes the importance of updated computational tools and modern perceptions in revisiting longstanding mathematical conjectures related to matrix permanents. While some conjectures have recently been proven false, others remain open areas of research, with their resolution offering potentially profound implications for matrix analysis, combinatorial math, and related fields. Addressing these conjectures could redefine the boundaries of current mathematical theory surrounding permanents and their applications.