Character tables are ideal Perron similarities
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918346664968192 |
|---|---|
| author | Gershnik, David Z. Lewis, Alexander J. Paparella, Pietro |
| author_facet | Gershnik, David Z. Lewis, Alexander J. Paparella, Pietro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02830 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Character tables are ideal Perron similarities Gershnik, David Z. Lewis, Alexander J. Paparella, Pietro Spectral Theory Representation Theory 15A29, 20C15, 20C99, 15B51, 52B99 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. |
| title | Character tables are ideal Perron similarities |
| topic | Spectral Theory Representation Theory 15A29, 20C15, 20C99, 15B51, 52B99 |
| url | https://arxiv.org/abs/2508.02830 |