Spectratope: Polyhedral Regions in NIEP

• Spectratope is the normalized polytope associated with an invertible matrix S that enforces entrywise nonnegativity in matrices of the form S Dₓ S⁻¹.
• It organizes realizable spectra of diagonalizable nonnegative matrices into explicit polyhedral regions, enhancing analysis in the nonnegative inverse eigenvalue problem.
• Special cases using Walsh–Hadamard matrices and group character tables yield tractable spectratopes with clear convex-hull descriptions and volume formulas.

A spectratope is the normalized polytope associated with an invertible matrix $S$ through the entrywise nonnegativity of matrices of the form $S D_x S^{-1}$, where $D_x=\operatorname{diag}(x_1,\dots,x_n)$. It is defined in tandem with the spectracone, the ambient polyhedral cone of admissible spectral data. In the nonnegative inverse eigenvalue problem (NIEP), these objects organize realizable spectra of diagonalizable nonnegative matrices into explicit polyhedral regions indexed by the similarity $S$. The modern literature treats both real and complex settings, and shows that special classes of matrices—most notably Walsh–Hadamard matrices and character tables of finite groups—produce particularly tractable spectratopes with strong row-hull descriptions (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

1. Definitions and normalizations

For $S\in GL_n(\mathbb F)$, with $\mathbb F=\mathbb R$ or $\mathbb C$, the basic construction begins from

$M_x(S)=S D_x S^{-1}.$

The associated spectracone is the set of vectors $x$ for which $M_x(S)$ is entrywise nonnegative. The spectratope is obtained by imposing a normalization condition on $S D_x S^{-1}$0 or, equivalently in the later formulation, on the row sums of $S D_x S^{-1}$1.

Source	Spectracone	Spectratope
Real RNIEP formulation	$S D_x S^{-1}$2	$S D_x S^{-1}$3
Character-table formulation	$S D_x S^{-1}$4	$S D_x S^{-1}$5

Here $S D_x S^{-1}$6 denotes the all-ones vector. In the real formulation, $S D_x S^{-1}$7 is a system of $S D_x S^{-1}$8 linear homogeneous inequalities in the coordinates of $S D_x S^{-1}$9, so $D_x=\operatorname{diag}(x_1,\dots,x_n)$0 is a polyhedral cone in $D_x=\operatorname{diag}(x_1,\dots,x_n)$1, and intersecting with $D_x=\operatorname{diag}(x_1,\dots,x_n)$2 produces an $D_x=\operatorname{diag}(x_1,\dots,x_n)$3-dimensional polytope. In the later formulation, $D_x=\operatorname{diag}(x_1,\dots,x_n)$4 is likewise a normalized slice of the cone, but expressed by the stochasticity condition $D_x=\operatorname{diag}(x_1,\dots,x_n)$5 (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

A basic structural point is that a spectratope is not a similarity-invariant object attached to a spectrum alone; it is indexed by the specific matrix $D_x=\operatorname{diag}(x_1,\dots,x_n)$6. Different choices of $D_x=\operatorname{diag}(x_1,\dots,x_n)$7 produce different polyhedral regions, even when they serve the same ambient spectral problem. This dependence is central to their use in the NIEP.

2. Perron similarities and the polyhedral geometry of spectratopes

The spectratope is meaningful only when $D_x=\operatorname{diag}(x_1,\dots,x_n)$8 is sufficiently compatible with nonnegative similarity classes. In the RNIEP formulation, $D_x=\operatorname{diag}(x_1,\dots,x_n)$9 is called a Perron-similarity if there exists a real non-scalar diagonal matrix $S$0 such that

$S$1

The paper gives two equivalent characterizations: there is an index $S$2 such that the $S$3th column of $S$4, namely $S$5, and the $S$6th row of $S$7, namely $S$8, are both entrywise nonnegative; equivalently, $S$9 lies in the conical hull of the rows of $S\in GL_n(\mathbb F)$0 and also in the conical hull of the rows of $S\in GL_n(\mathbb F)$1. It also proves

$S\in GL_n(\mathbb F)$2

In the later formulation, an invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix (Johnson et al., 2015, Gershnik et al., 4 Aug 2025).

When $S\in GL_n(\mathbb F)$3 is Perron-similar, $S\in GL_n(\mathbb F)$4 is full-dimensional, and $S\in GL_n(\mathbb F)$5 is an $S\in GL_n(\mathbb F)$6-polytope. Each entry of $S\in GL_n(\mathbb F)$7 yields a linear inequality

$S\in GL_n(\mathbb F)$8

so in general there are up to $S\in GL_n(\mathbb F)$9 facet-defining inequalities. The direct-sum behavior is also explicit: if $\mathbb F=\mathbb R$0, then

$\mathbb F=\mathbb R$1

with $\mathbb F=\mathbb R$2. These formulas show that spectratopes behave naturally under block decomposition and can be assembled from smaller building blocks (Johnson et al., 2015).

Within the RNIEP, the set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes. This makes enumeration and structural classification of such polytopes a natural research objective.

3. Ideality and hull-of-rows descriptions

A particularly important situation occurs when the spectracone is generated directly by the rows of the diagonalizing matrix. If $\mathbb F=\mathbb R$3 are the rows of $\mathbb F=\mathbb R$4, viewed as column vectors, the row cone is

$\mathbb F=\mathbb R$5

A matrix $\mathbb F=\mathbb R$6 is called ideal if

$\mathbb F=\mathbb R$7

The later work records the equivalent criterion that $\mathbb F=\mathbb R$8 is ideal if and only if $\mathbb F=\mathbb R$9 and $\mathbb C$0 for $\mathbb C$1 (Gershnik et al., 4 Aug 2025).

Ideality has two immediate consequences. First, the ambient realization cone becomes explicit: its extreme rays are precisely the rows of $\mathbb C$2. Second, the normalized slice inherits a convex-hull description. In the real RNIEP treatment, if $\mathbb C$3 is “strong” in the sense that all row-vectors of $\mathbb C$4 are affinely independent and $\mathbb C$5 is exactly the convex hull of those rows, then

$\mathbb C$6

General lower and upper bounds on $\mathbb C$7 then follow from Hadamard’s inequality after scaling rows to lie in $\mathbb C$8 (Johnson et al., 2015).

The canonical Walsh matrices $\mathbb C$9 furnish the basic example. For the $M_x(S)=S D_x S^{-1}.$0 Sylvester matrix,

$M_x(S)=S D_x S^{-1}.$1

the RNIEP paper proves

$M_x(S)=S D_x S^{-1}.$2

The proof uses the fact that the rows of $M_x(S)=S D_x S^{-1}.$3 form an elementary Abelian $M_x(S)=S D_x S^{-1}.$4-group under Hadamard product. This establishes a prototype for later ideality results: algebraic closure properties among rows collapse the nonnegativity constraints to a finite generating family (Johnson et al., 2015).

4. Character tables and group-theoretic spectratopes

The principal extension in the later paper is from Hadamard matrices to character tables of finite groups. Let $M_x(S)=S D_x S^{-1}.$5 be a finite group with conjugacy classes $M_x(S)=S D_x S^{-1}.$6, and let $M_x(S)=S D_x S^{-1}.$7 be the distinct irreducible complex representations with characters $M_x(S)=S D_x S^{-1}.$8. Ordered so that $M_x(S)=S D_x S^{-1}.$9 is the identity and $x$0 is the trivial character, the character table is

$x$1

The main theorem states that $x$2 is a Perron similarity, and in fact is ideal. The proof begins from column orthogonality,

$x$3

which yields the explicit inverse

$x$4

Evaluating $x$5 shows that every entry is positive, so $x$6 diagonalizes a positive matrix. Ideality then follows from two observations: the first row of $x$7 is $x$8, hence $x$9; and for each pair of rows $M_x(S)$0, the Hadamard product $M_x(S)$1 is realized as a nonnegative combination of rows of $M_x(S)$2 coming from the character of $M_x(S)$3. Thus $M_x(S)$4 for all $M_x(S)$5, and consequently $M_x(S)$6 (Gershnik et al., 4 Aug 2025).

The same paper gives a group-theoretic half-space description of the cone. Since

$M_x(S)$7

it proves that

$M_x(S)$8

Because $M_x(S)$9, these become finitely many real linear inequalities on the coordinates of $S D_x S^{-1}$00. This makes the spectracone—and hence its normalized spectratope slice—computable directly from character-theoretic data (Gershnik et al., 4 Aug 2025).

5. Real character tables, projection, and volume

When the character table $S D_x S^{-1}$01 is real, the projected spectratope acquires a simplex structure with a closed-form volume. The normalization uses the first column

$S D_x S^{-1}$02

whose entries are the positive degrees $S D_x S^{-1}$03. Setting

$S D_x S^{-1}$04

one obtains a matrix satisfying $S D_x S^{-1}$05, and the paper states that $S D_x S^{-1}$06 remains ideal with $S D_x S^{-1}$07. Writing

$S D_x S^{-1}$08

the projection $S D_x S^{-1}$09 is an $S D_x S^{-1}$10-simplex whose vertices are the projections of the row-vectors of $S D_x S^{-1}$11 (Gershnik et al., 4 Aug 2025).

Its volume is obtained from the determinant formula for simplices. In the group-theoretic setting, the resulting expression is

$S D_x S^{-1}$12

The proof uses $S D_x S^{-1}$13 together with $S D_x S^{-1}$14, interpreted as the Jacobian determinant of the affine map sending the standard simplex to $S D_x S^{-1}$15. In this setting, the spectratope is not merely polyhedral but affinely equivalent to a simplex defined by representation-theoretic data (Gershnik et al., 4 Aug 2025).

This result also clarifies the geometric role of ideality. Once the cone is generated by the rows, the normalized slice is governed by the affine geometry of those generators, and group-theoretic invariants become volume data.

6. Examples, applications to the NIEP, and open directions

Concrete computations illustrate the range of the construction.

Group	Data	Consequence
$S D_x S^{-1}$16	$S D_x S^{-1}$17, with inequalities $S D_x S^{-1}$18, $S D_x S^{-1}$19	$S D_x S^{-1}$20, length $S D_x S^{-1}$21
$S D_x S^{-1}$22	$S D_x S^{-1}$23, with $S D_x S^{-1}$24, $S D_x S^{-1}$25, $S D_x S^{-1}$26	$S D_x S^{-1}$27 is cut out by $S D_x S^{-1}$28, $S D_x S^{-1}$29, $S D_x S^{-1}$30, and $S D_x S^{-1}$31
$S D_x S^{-1}$32	$S D_x S^{-1}$33	$S D_x S^{-1}$34

These examples show how spectratopes interpolate between elementary polyhedral sets and group-controlled simplices (Gershnik et al., 4 Aug 2025).

For the NIEP, each ideal Perron similarity $S D_x S^{-1}$35 yields an explicit polyhedral cone $S D_x S^{-1}$36 of realizable spectra and a compact polytope $S D_x S^{-1}$37 of stochastic spectra. The character-table theorem therefore supplies an infinite family of constructive realization regions, one for each finite group. In the abelian case, the extreme rays of $S D_x S^{-1}$38 are precisely the rows of $S D_x S^{-1}$39, and those rows are comprised of roots of unity and hence lie on the boundary of the general Karpelevič region. This suggests a direct interface between polyhedral realizability and extremal spectral geometry (Gershnik et al., 4 Aug 2025).

The RNIEP paper develops a complementary perspective. It proves that every normalized real spectrum $S D_x S^{-1}$40 of a diagonalizable nonnegative matrix lies in at least one projected spectratope $S D_x S^{-1}$41. In dimensions $S D_x S^{-1}$42, the choices

$S D_x S^{-1}$43

suffice, together with row and column permutations, to cover the entire trace-nonnegative polytope $S D_x S^{-1}$44. At the same time, even for $S D_x S^{-1}$45, uncountably many distinct spectratopes are required to cover the trace-nonnegative triangle, so no finite list of “master similarities” suffices in dimension $S D_x S^{-1}$46 (Johnson et al., 2015).

The same paper also gives constructive realizations of classical RNIEP results for Suleĭmanova spectra in the Hadamard setting. If $S D_x S^{-1}$47 is any normalized Hadamard matrix of order $S D_x S^{-1}$48, and $S D_x S^{-1}$49 is a normalized Suleĭmanova spectrum, then

$S D_x S^{-1}$50

so

$S D_x S^{-1}$51

is symmetric, entrywise nonnegative, and has spectrum $S D_x S^{-1}$52. By adjoining enough zero eigenvalues so that the new size is a Hadamard order, one obtains a doubly stochastic, symmetric realization; in powers of $S D_x S^{-1}$53, it is trisymmetric (Johnson et al., 2015).

Several open directions are explicit in the character-table paper. One may ask which normalized ideal Perron similarities arise as character tables of finite groups. A natural conjecture is that these are exactly those ideal similarities whose extreme-ray lists are themselves totally extremal points of the stochastic spectral region, meaning they lie on the maximal-roots-of-unity arcs. More generally, one can try to extend the inequalities

$S D_x S^{-1}$54

to non-diagonalizable spectra, or incorporate the JLL-inequalities into a unified polyhedral description. Additional questions concern the union of all cones $S D_x S^{-1}$55 as $S D_x S^{-1}$56 ranges over finite groups, and the identification of other natural families of ideal Perron similarities, such as those arising from association schemes or from highly symmetric combinatorial designs (Gershnik et al., 4 Aug 2025).