Character tables are ideal Perron similarities

TL;DR

This paper proves that character tables of finite groups are a special class of matrices called ideal Perron similarities, linking group theory with spectral properties and advancing the understanding of the nonnegative inverse eigenvalue problem.

Contribution

It establishes that character tables are ideal Perron similarities, unifying previous results and providing group-theoretic descriptions of spectracones and spectratopes.

Findings

Character tables are ideal Perron similarities.

Spectracone described by finitely-many group inequalities.

Derived a formula for the volume of the projected spectratope.

Abstract

An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Each Perron similarity gives a nontrivial polyhedral cone, called the spectracone, and polytope, called the spectratope, of realizable spectra (thought of as vectors in complex Euclidean space). A Perron similarity is called ideal if its spectratope coincides with the conical hulls of its rows. Identifying ideal Perron similarities is of great interest in the pursuit of the longstanding nonnegative inverse eigenvalue problem. In this work, it is shown that the character table of a finite group is an ideal Perron similarity. In addition to expanding ideal Perron similarities to include a broad class of matrices, the results unify previous works into a single, theoretical framework. It is demonstrated that the spectracone can be described by finitely-many group-theoretic inequalities. When the character table is real, we derive a group-theoretic formula for the volume of the projected Perron spectratope, which is a simplex. Finally, an implication for further research is given.