On a conjecture concerning totally extremal ideal Perron similarities

Abstract

Identifying ideal Perron similarities is a problem of central interest in the longstanding nonnegative inverse eigenvalue problem (NIEP). A normalized ideal Perron similarity is called totally extremal if every entry has modulus one. Recently, Gershnik et al. [J. Algebra 694 (2026), 782--800] proved that the character table of a finite Abelian group is totally extremal and conjectured the converse. In this paper, we settle this conjecture in the affirmative by first showing that the rows of a totally extremal normalized ideal Perron similarity form a group under the Hadamard product. Then, it is shown that the rows of a nonsingular matrix form a group under the Hadamard product if and only if it is the character table of a finite Abelian group, and we further show that this group is isomorphic to the underlying group. These results extend the classical theorem due to Romanovsky and Karpelevi\v{c} on the unimodular eigenvalues of stochastic matrices in the complex unit disk to the setting of spectratopes in the unit ball of complex Euclidean space.