  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.
