A simple counterexample for the permanent-on-top conjecture Fizika, Földtudományok és Matematika

35 OTDK, Fizika, Földtudományok és Matematika Szekció, Algebra és számelmélet Tagozat.

A simple counterexample for the permanent-on-top conjecture

Helyezés: 2

Hallgató: Tran Hoang Anh
Szak: Matematika, Képzés típusa: msc, Intézmény: Eötvös Loránd Tudományegyetem, Kar: Természettudományi Kar

Témavazető: Frenkel Péter - egyetemi adjunktus, Eötvös Loránd Tudományegyetem Természettudományi Kar

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 permanental compound matrices. By that, we are able to study the properties of the spectrum of the Schur power matrix through the 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.