Pate's conjecture on the largest eigenvalue of the complementary-permanent matrix

About 6 years old · traced to 2020

Let $A$ be an $n\times n$ positive semi-definite Hermitian matrix, and let $1\leq k<n$ . For $k$ -element subsets $I,J\subset[n]$ , define $\mathscr{C}_k(A)$ by

(\mathscr{C}_k(A))_{I,J}=\operatorname{per}(A[I,J])\operatorname{per}(A[I^c,J^c]).

Pate's conjecture. The largest eigenvalue of $\mathscr{C}_k(A)$ is $\operatorname{per}(A)$ . This conjecture is refuted; the paper states that its $k=1$ case has an $8\times8$ counterexample and presents a new $5\times5$ rank- $2$ counterexample.

References

Primary source

Tran Hoang Anh, “A simple counterexample for the permanent-on-top conjecture”, arXiv:2101.03428 (2022).

Additional references

2 papers in this index state this conjecture (2020–2021). The statement above is taken from the most recent of them; the others are arXiv:2002.04149.

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A paper reports that the conjecture is false, including a smaller five-by-five example, but this scan found no independent verification.

Pate’s conjecture asserts that the complementary-permanent matrix has largest eigenvalue equal to the permanent for every positive-semidefinite Hermitian input and every permitted $k$ . It is weaker than the permanent-on-top conjecture.

Known results

Pate proved the $k=1$ case on a large subcone of Hermitian matrices and for all nonnegative real matrices.
Stephen W. Drury supplied an $8\times 8$ counterexample for the $k=1$ case.
Tran gave an explicit $5\times 5$ counterexample, recorded in a 2022 survey.

2021 explicit counterexample

The paper A simple counterexample for the permanent-on-top conjecture presents a positive-semidefinite Hermitian matrix $H$ of size $5\times 5$ and rank $2$ , with $\operatorname{per}(H)=504$ , for which the relevant eigenvalue exceeds the permanent; it states that the same construction refutes Pate’s conjecture for $k=2$ . The paper also records the earlier $k=1$ counterexample.

Current status (as of September 2026): The conjecture is reported refuted by explicit counterexamples, including a $5\times 5$ rank- $2$ example for $k=2$ ; the reported refutation remains unverified by this review.

Sources

No solutions have been posted yet.