Projected Perron Spectratope Analysis

• Projected Perron spectratope is a coordinate projection of a normalized Perron spectratope derived from an ideal Perron similarity constructed via scaled character tables.
• The method leverages representation theory to convert finite group character tables into polyhedral regions that address aspects of the nonnegative inverse eigenvalue problem.
• In real-valued cases, the projected spectratope forms an (n-1)-simplex with an explicit volume formula, illustrated by examples like S3 and Walsh matrices.

Searching arXiv for the cited papers to ground the article in current records. The projected Perron spectratope is the coordinate projection obtained from a normalized Perron spectratope associated with a Perron similarity, and in the representation-theoretic setting of finite groups it is constructed from a scaled character table. In the framework of "Character tables are ideal Perron similarities" (Gershnik et al., 4 Aug 2025), the central object is $\Pi_1(\mathcal P(D_{v^{-1}}Q))$, where $Q$ is the character table of a finite group, $v=Qe_1$ is the vector of irreducible character degrees, and $\Pi_1$ deletes the first coordinate. The construction is motivated by the nonnegative inverse eigenvalue problem, because ideal Perron similarities yield explicit polyhedral regions of realizable spectra; for real character tables, the projected Perron spectratope is an $(n-1)$-simplex with a closed-form group-theoretic volume formula (Gershnik et al., 4 Aug 2025).

1. Foundational framework

For $S\in GL_n(\mathbb C)$ and $x\in\mathbb C^n$, the 2025 paper defines

$M_x(S):=SD_xS^{-1},$

where $D_x$ is the diagonal matrix with diagonal entries $x_1,\dots,x_n$. A matrix $Q$0 is a Perron similarity if there exists a diagonal matrix $Q$1 such that $Q$2 is irreducible and entrywise nonnegative. Equivalently, $Q$3 is a Perron similarity if and only if there is a unique $Q$4 such that $Q$5 and $Q$6, where $Q$7 are complex numbers with $Q$8, and $Q$9 are positive vectors (Gershnik et al., 4 Aug 2025).

The associated spectral sets are the spectracone

$v=Qe_1$0

and the spectratope

$v=Qe_1$1

The affine normalization $v=Qe_1$2 means that the realizing matrices are row stochastic, and $v=Qe_1$3 because $v=Qe_1$4 (Gershnik et al., 4 Aug 2025).

Ideality is defined by comparison with the conical hull of the rows. If $v=Qe_1$5, then $v=Qe_1$6 is ideal if $v=Qe_1$7. The paper gives two equivalent criteria: $v=Qe_1$8 is ideal if and only if $v=Qe_1$9 and $\Pi_1$0 for all $\Pi_1$1, and also if and only if $\Pi_1$2 and $\Pi_1$3 has the row Hadamard conic property, meaning $\Pi_1$4 for all $\Pi_1$5 (Gershnik et al., 4 Aug 2025).

This formulation extends the earlier real-variable theory of Perron spectratopes. Johnson–Paparella define, for $\Pi_1$6,

$\Pi_1$7

and show that coordinate projections of Perron spectratopes are polytopes computable either from the linear inequality description or by projecting vertices (Johnson et al., 2015). The projected Perron spectratope in the character-table setting is a specialized realization of this general projection paradigm.

2. Character tables as ideal Perron similarities

Let $\Pi_1$8 be a finite group, let $\Pi_1$9 be irreducible representations, let $(n-1)$0 be the associated irreducible characters, and let $(n-1)$1 represent the conjugacy classes. The character table is

$(n-1)$2

The 2025 paper proves that $(n-1)$3 is a Perron similarity and, more strongly, an ideal Perron similarity (Gershnik et al., 4 Aug 2025).

The Perron-similarity property follows from character orthogonality. The columns of $(n-1)$4 are orthogonal with respect to the standard inner product, with

$(n-1)$5

so $(n-1)$6 is diagonal with diagonal entries $(n-1)$7 and

$(n-1)$8

Using the entrywise identity

$(n-1)$9

the paper computes

$S\in GL_n(\mathbb C)$0

Thus $S\in GL_n(\mathbb C)$1 diagonalizes a positive matrix (Gershnik et al., 4 Aug 2025).

Ideality is proved by showing that $S\in GL_n(\mathbb C)$2 satisfies the row Hadamard conic property. The first row is $S\in GL_n(\mathbb C)$3 for all $S\in GL_n(\mathbb C)$4, so $S\in GL_n(\mathbb C)$5 and therefore $S\in GL_n(\mathbb C)$6. For any $S\in GL_n(\mathbb C)$7,

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

Expanding $S\in GL_n(\mathbb C)$9 in the irreducible basis gives

$x\in\mathbb C^n$0

where $x\in\mathbb C^n$1 are multiplicities. Hence $x\in\mathbb C^n$2 for all $x\in\mathbb C^n$3, so $x\in\mathbb C^n$4 is RHC and therefore ideal (Gershnik et al., 4 Aug 2025).

A further consequence is a sharp reduction of the nonnegativity constraints. Instead of the $x\in\mathbb C^n$5 entrywise inequalities $x\in\mathbb C^n$6, the spectracone is characterized by the $x\in\mathbb C^n$7 inequalities

$x\in\mathbb C^n$8

When $x\in\mathbb C^n$9, the realizing matrices $M_x(S):=SD_xS^{-1},$0 are symmetric; when $M_x(S):=SD_xS^{-1},$1, they are normal. The paper therefore identifies group-theoretic sufficient conditions for the symmetric and normal versions of the nonnegative inverse eigenvalue problem (Gershnik et al., 4 Aug 2025).

3. Definition of the projected Perron spectratope

The projected Perron spectratope is defined only after a normalization step that uses the degree vector

$M_x(S):=SD_xS^{-1},$2

By a scaling lemma cited in the paper, if $M_x(S):=SD_xS^{-1},$3 is ideal and $M_x(S):=SD_xS^{-1},$4, then $M_x(S):=SD_xS^{-1},$5 is ideal. Since $M_x(S):=SD_xS^{-1},$6 is ideal, the scaled matrix

$M_x(S):=SD_xS^{-1},$7

is ideal and satisfies the normalization

$M_x(S):=SD_xS^{-1},$8

The paper also proves that $M_x(S):=SD_xS^{-1},$9 (Gershnik et al., 4 Aug 2025).

For $D_x$0, let $D_x$1 be the matrix obtained by deleting the $D_x$2-th row of $D_x$3, and define

$D_x$4

The projected Perron spectratope is then

$D_x$5

It is the orthogonal projection obtained by removing the first coordinate $D_x$6, identified in the paper as the Perron/Stochastic coordinate (Gershnik et al., 4 Aug 2025).

This construction is closely aligned with the coordinate-projection viewpoint developed earlier for real Perron spectratopes. Johnson–Paparella define

$D_x$7

prove that such projections are polytopes, and describe two standard computational routes: elimination from the inequality system or projection of a vertex description (Johnson et al., 2015). The later representation-theoretic construction specializes this to the scaled character-table setting.

4. Real character tables, simplex structure, and volume

Assume that $D_x$8, so that all irreducible characters are real-valued on the chosen conjugacy-class representatives. Under this realness hypothesis, the projected Perron spectratope has particularly rigid geometry. For an ideal matrix $D_x$9 with normalization $x_1,\dots,x_n$0, the paper invokes its general theory to assert that

$x_1,\dots,x_n$1

the convex hull of the rows of $x_1,\dots,x_n$2. Consequently,

$x_1,\dots,x_n$3

Because $x_1,\dots,x_n$4 scales the $x_1,\dots,x_n$5-th row by $x_1,\dots,x_n$6, the projected vertices are

$x_1,\dots,x_n$7

The rows are affinely independent, so $x_1,\dots,x_n$8 is an $x_1,\dots,x_n$9-simplex with vertices $Q$00 (Gershnik et al., 4 Aug 2025).

The paper derives an exact volume formula: $Q$01 Here $Q$02 is the size of the centralizer of $Q$03, and $Q$04 is the degree of the $Q$05-th irreducible representation. The derivation uses the orthogonality identity

$Q$06

so

$Q$07

together with

$Q$08

Substituting these into the simplex volume formula yields the stated expression (Gershnik et al., 4 Aug 2025).

The realness assumption is group-theoretic. A sufficient condition is that $Q$09 is a real, or ambivalent, group, meaning every element is conjugate to its inverse; the paper lists symmetric groups $Q$10, dihedral groups $Q$11, and many reflection/Coxeter groups as classical examples. For Abelian groups, realness means all irreducible characters take values in $Q$12, which occurs precisely for elementary Abelian $Q$13-groups. This is exactly the Walsh–Hadamard situation already prominent in the earlier spectratope literature (Gershnik et al., 4 Aug 2025, Johnson et al., 2015).

5. Examples and explicit computation

The paper gives three representative examples. For the symmetric group $Q$14, the real character table is

$Q$15

The conjugacy class sizes are $Q$16, $Q$17, and $Q$18, while the centralizer sizes are $Q$19. The irreducible degrees are $Q$20, $Q$21, $Q$22. The spectracone inequalities become

$Q$23

With $Q$24,

$Q$25

After removing the first coordinate, the projected vertices are

$Q$26

and the volume is

$Q$27

The paper further notes that this projected simplex occupies $Q$28 of the trace-nonnegative feasible region in this case (Gershnik et al., 4 Aug 2025).

For the elementary Abelian group $Q$29, the character table is the Walsh matrix

$Q$30

All conjugacy classes have size $Q$31, all centralizers have size $Q$32, and all irreducible degrees are $Q$33. The inequalities are

$Q$34

Since $Q$35, the normalization does not change the matrix. The projected vertices are

$Q$36

and the volume is

$Q$37

This matches the Walsh–Hadamard case treated earlier as a spectratope equal to the convex hull of the rows (Gershnik et al., 4 Aug 2025, Johnson et al., 2015).

For a group with non-real character table, such as $Q$38, the character table is the discrete Fourier matrix

$Q$39

Here $Q$40, so the matrices $Q$41 are normal. The inequalities

$Q$42

still characterize the spectracone, but the projected Perron spectratope need not be a simplex, and the real-case volume formula does not apply (Gershnik et al., 4 Aug 2025).

The paper also gives an explicit computation procedure. One builds $Q$43 from conjugacy-class representatives and irreducible characters, records the degrees $Q$44 and centralizer sizes $Q$45, forms the linear forms

$Q$46

sets $Q$47, defines $Q$48, and projects by $Q$49. In the real case, the projected vertices are

$Q$50

and the volume is recovered from the determinant factorization above (Gershnik et al., 4 Aug 2025).

6. Position within the nonnegative inverse eigenvalue problem

The nonnegative inverse eigenvalue problem asks for a characterization of spectra of entrywise nonnegative matrices. In the Perron-similarity approach, the spectracone and spectratope collect vectors $Q$51 for which $Q$52 is nonnegative, and in the normalized case stochastic. Ideal Perron similarities are especially useful because the feasible region becomes explicitly polyhedral: if $Q$53 is ideal, then $Q$54, and under normalization $Q$55, the spectratope is the convex hull of the rows (Gershnik et al., 4 Aug 2025). In the earlier real theory, Johnson–Paparella state that the set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes (Johnson et al., 2015).

The 2025 paper places character tables into this framework by proving that the character table of every finite group is an ideal Perron similarity. This unifies previously studied families—Walsh–Hadamard matrices, discrete Fourier transforms, and their Kronecker products—inside a single representation-theoretic setting (Gershnik et al., 4 Aug 2025). A plausible implication is that phenomena earlier observed separately for Hadamard and Fourier matrices can be interpreted as manifestations of the same ideality mechanism coming from character theory.

This unification also connects with the Kronecker-product theory of Perron similarities. The 2021 paper proves that Kronecker products of Perron similarities are Perron similarities, that ideality is preserved under Kronecker products, and that for ideal strong matrices the spectratope is the convex hull of the rows; it also emphasizes tensor-product constructions with extremal rows (Dockter et al., 2021). The Abelian-group statement in the 2025 paper,

$Q$56

fits directly into that broader product framework (Gershnik et al., 4 Aug 2025).

Several limitations are explicit. The simplex geometry and the volume formula require $Q$57; for complex character tables, the projected set need not be a simplex, even though the linear inequalities still describe the spectracone (Gershnik et al., 4 Aug 2025). It is therefore inaccurate to treat simplex structure as a generic feature of projected Perron spectratopes. The paper also records a conjectural extremality statement: if $Q$58 is a normalized ideal Perron similarity that is totally extremal, then $Q$59 is the character table of a finite Abelian group (Gershnik et al., 4 Aug 2025). This suggests a boundary between general ideal Perron similarities and those arising from the most rigid representation-theoretic data.