A simple counterexample for the permanent-on-top conjecture

TL;DR

This paper presents a new, hand-checkable counterexample to the permanent-on-top conjecture, using group representations to analyze spectra of matrices, and also refutes a related weaker conjecture with a smaller counterexample.

Contribution

It introduces a novel method using group representations to find a smaller counterexample to the permanent-on-top conjecture and a related weaker conjecture.

Findings

Discovered a new counterexample to the permanent-on-top conjecture that can be verified by hand.

Provided a smaller counterexample to a weaker related conjecture.

Connected spectra of Schur power matrices with entrywise product matrices of permanental compounds.

Abstract

The permanent-on-top conjecture states that the largest eigenvalue of the Schur power matrix of a positive semi-definite Hermitian matrix H is per(H). A counterexample has been found with the help of computers, but here, I present another counterexample that can be checked by hand. My method is to use linear representations of groups to connect the spectrum of the Schur power matrix with the spectra of the entrywise(Hadamard) product matrices of permanental compound matrices. By that, we are able to study the properties of the spectrum of the Schur power matrix through the entrywise(Hadamard) product matrices of permanental compound matrices. The counterexample we find is in fact also a counterexample to a weaker conjecture related to permanental compound matrices. This conjecture was also known to be false, but the new counterexample is smaller than the known one.