Character tables are ideal Perron similarities
David Z. Gershnik, Alexander J. Lewis, Pietro Paparella
TL;DR
The NIEP seeks spectra realizable by entrywise nonnegative matrices, and this paper advances the theory by framing realizability through Perron similarities, spectracones, and spectratopes. It proves that character tables of finite groups yield ideal Perron similarities, unifying past results under a single, constructive framework. For real character tables, it provides a closed-form volume formula for the projected spectratope and describes the spectracone via group-theoretic inequalities, enabling concrete realizability analysis. The results illuminate extremal structures in Abelian groups and suggest broad avenues for applying this framework to the NIEP and related inverse eigenvalue problems.
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.