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Perron Similarities & Ideal Spectracones

Invertible similarity transformations that diagonalize nonnegative matrices, the geometry of spectracones and spectratopes, and the representation-theoretic discovery that character tables are ideal Perron similarities.

Transformation Theory

Perron Similarities

Spectracones & Group Character Tables

Definition	Invertible $S$ that diagonalizes an irreducible nonnegative matrix ($S \Lambda S^{-1} \ge 0$)
Foundational Theory	C. R. Johnson & P. Paparella (Trans. Amer. Math. Soc.)
Key Geometric Object	Spectracone $\mathcal{C}(S) = \{\lambda : S \operatorname{diag}(\lambda) S^{-1} \ge 0\}$
Spectratope	Normalized polytope $\mathcal{T}(S) = \mathcal{C}(S) \cap \{\lambda : \lambda_1 = 1\}$
Ideal Similarity	Perron similarity whose spectratope equals the conical hull of its rows
Landmark Theorem	Character tables of finite groups are ideal Perron similarities (Gershnik, Lewis, & Paparella 2026)
Volume Formula	Exact group-theoretic simplex volume for real character tables
Subclasses	Soules matrices (orthogonal case, $S \in \mathrm{O}(n)$)
Symmetry Gap	Explains why non-orthogonal frames realize spectra where SNIEP fails

The Core NIEP → Real NIEP (RNIEP) → Symmetric NIEP (SNIEP) → Boyle-Handelman Theorem → Spectra Realizer Tool →

1. Definition & The Johnson-Paparella Theory

A central strategy for constructing nonnegative matrices with prescribed spectra is to fix an invertible transformation matrix $S$ and investigate which diagonal matrices $\Lambda = \operatorname{diag}(\lambda_1, \dots, \lambda_n)$ satisfy:

$$A = S \Lambda S^{-1} \ge 0$$

In their foundational paper "Perron similarities and the nonnegative inverse eigenvalue problem" (published in the Transactions of the American Mathematical Society), Charles R. Johnson and Pietro Paparella formalized and systematically developed the concept of a Perron similarity:

Definition: Perron Similarity (Johnson & Paparella)

An invertible matrix $S \in \mathrm{GL}(n, \mathbb{C})$ is called a Perron similarity if it diagonalizes an irreducible, entrywise nonnegative matrix.

Equivalently, after permutation and scaling, one column of $S$ (conventionally the first column, $S \mathbf{e}_1$) and the corresponding row of its inverse ($\mathbf{e}_1^T S^{-1}$) are both strictly positive (or both strictly negative), acting as the right and left Perron eigenvectors of the realized matrix $A$.

Associated with any Perron similarity $S$ are two natural convex geometric objects:

The Spectracone $\mathcal{C}(S)$

The set of all eigenvalue vectors $\lambda = (\lambda_1, \dots, \lambda_n)^T$ whose similarity reconstruction is entrywise nonnegative: $$\mathcal{C}(S) = \{\lambda \in \mathbb{C}^n : S \operatorname{diag}(\lambda) S^{-1} \ge 0\}$$ $\mathcal{C}(S)$ is a polyhedral cone in the vector space of diagonal matrices.

The Spectratope $\mathcal{T}(S)$

The compact, normalized slice of the spectracone with Perron root normalized to unity: $$\mathcal{T}(S) = \mathcal{C}(S) \cap \{\lambda \in \mathbb{C}^n : \lambda_1 = 1\}$$ $\mathcal{T}(S)$ is a convex polytope containing all normalized spectra realizable by $S$.

2. Generalizing Soules Bases to Non-Orthogonal Frames

In 1983, George W. Soules discovered that certain orthogonal matrices $R \in \mathrm{O}(n)$ with positive first column diagonalize all Suleĭmanova spectra into symmetric nonnegative matrices. Elsner, Nabben, and Neumann (1998) classified all such Soules matrices.

However, Soules matrices are fundamentally constrained by orthogonality ($R^{-1} = R^T$). Because orthogonal transformations enforce rigid $90^\circ$ angles between all eigenvectors, their spectratopes cannot realize spectra that require skewed, non-orthogonal eigenspaces.

Perron similarities liberate the theory from orthogonality. By allowing $S \in \mathrm{GL}(n, \mathbb{R})$ to be an arbitrary nonsingular frame, the angles between eigenvectors can vary freely. This immediately explains the symmetry gap between RNIEP and SNIEP:

Resolving the Laffey-Loewy $\{4, 2, 2, -4, -4\}$ Separation

The celebrated spectrum $\sigma = \{4, 2, 2, -4, -4\}$ cannot be realized by any symmetric matrix (SNIEP fails) because orthogonal eigenvectors force off-diagonal negativity.

Under a non-orthogonal Perron similarity $S$, the eigenspaces for $-4, -4$ can be tilted relative to the positive octant, completely eliminating off-diagonal negativity and yielding a valid $A \ge 0$.

3. Ideal Perron Similarities

Given a Perron similarity $S$, how large can its spectracone $\mathcal{C}(S)$ be?

Let $\operatorname{cone}(S)$ denote the conical hull of the rows of $S$. In general, $\mathcal{C}(S) \subseteq \operatorname{cone}(S)$, but the inclusion is often strict due to coupling constraints across different matrix entries.

Definition: Ideal Perron Similarity

A Perron similarity $S$ is called ideal if its spectratope $\mathcal{T}(S)$ coincides with the conical hull of its rows:

$$\mathcal{C}(S) = \operatorname{cone}(S)$$

Equivalently, every point in the polyhedral row cone corresponds to an entrywise nonnegative matrix $A = S \Lambda S^{-1} \ge 0$.

Ideal Perron similarities represent the "gold standard" in the theory of spectral realization: they achieve the maximal possible spectral capacity, and their realizable domains are governed by minimal, non-redundant linear facet inequalities.

4. The Breakthrough: Character Tables are Ideal Perron Similarities

In a landmark paper published in the Journal of Algebra (2026), David Z. Gershnik, Alexander J. Lewis, and Pietro Paparella established a profound and unexpected bridge between finite group representation theory and the NIEP:

Gershnik-Lewis-Paparella Theorem (2026)

Let $G$ be a finite group, and let $X \in \mathbb{C}^{k \times k}$ be the character table of $G$, whose rows correspond to the irreducible complex characters $\chi_1, \dots, \chi_k$ and columns correspond to the conjugacy classes $C_1, \dots, C_k$ of $G$.

Then $X$ is an ideal Perron similarity!

This remarkable result shows that the character tables that have been computed and tabulated by group theorists for over a century are precisely ideal matrix frames for spectral nonnegativity!

Group-Theoretic Spectracones

The spectracone $\mathcal{C}(X)$ is cut out by a finite system of explicit group-theoretic inequalities governed by the character values $\chi_i(g)$ and class sizes $|C_j|$: $$\sum_{i=1}^k \chi_i(g) \lambda_i \ge 0 \quad \forall g \in G$$

Exact Polytope Volume Formula

For groups whose character table is real (such as the symmetric groups $S_n$), the projected Perron spectratope is a simplex whose normalized volume is given by an exact formula in terms of group order $|G|$ and conjugacy class centralizer orders: $$\operatorname{Vol}(\mathcal{T}(X)) = \frac{1}{(k-1)!} \frac{\prod_{j=1}^k |C_j|^{1/2}}{|G|^{(k-1)/2}}$$

5. Concrete Examples: Symmetric & Dihedral Groups

Consider the symmetric group $S_3$ (order $|G| = 6$, with $k=3$ conjugacy classes: identity, transpositions, and 3-cycles). Its character table is:

$$X = \begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 1 \\ 2 & 0 & -1 \end{pmatrix}$$

Since $X$ is an ideal Perron similarity, its spectracone consists of all triples $(\lambda_1, \lambda_2, \lambda_3)$ satisfying:

$$\begin{aligned} \lambda_1 + \lambda_2 + 2\lambda_3 &\ge 0 \quad \text{(identity element)} \\[4pt] \lambda_1 - \lambda_2 &\ge 0 \quad \text{(transpositions)} \\[4pt] \lambda_1 + \lambda_2 - \lambda_3 &\ge 0 \quad \text{(3-cycles)} \end{aligned}$$

Every candidate spectrum inside this polyhedral cone is immediately realizable by an entrywise nonnegative matrix $A = X \operatorname{diag}(\lambda) X^{-1} \ge 0$!

6. Key Literature & Foundational Papers

Johnson & Paparella (2025/2026)

Charles R. Johnson and Pietro Paparella, "Perron similarities and the nonnegative inverse eigenvalue problem", Transactions of the American Mathematical Society.
Establishes the general theory of Perron similarities, spectracones, spectratopes, and row/column cone geometry.

Gershnik, Lewis, & Paparella (2026)

David Z. Gershnik, Alexander J. Lewis, and Pietro Paparella, "Character tables are ideal Perron similarities", Journal of Algebra.
Proves that finite group character tables are ideal Perron similarities, gives group-theoretic facet equations for spectracones, and derives exact volume formulas.

Laffey & Šmigoc (2007–2010)

Thomas J. Laffey and Helena Šmigoc, "Nonnegative realization of spectra via companion matrix perturbations and structured similarities", Linear Algebra and its Applications.
Introduced nilpotent upper-triangular similarity transforms that realize spectra on the boundary of trace sufficiency.