Character tables are ideal Perron similarities

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.

💡 Research Summary

The paper addresses a long‑standing challenge in matrix theory, the nonnegative inverse eigenvalue problem (NIEP), by investigating a special class of similarity transformations called Perron similarities. An invertible matrix S is a Perron similarity if there exists a diagonal matrix D such that A = S D S⁻¹ is a nonnegative, irreducible matrix. For any such S one defines the “spectracone” C(S) = {x ∈ ℂⁿ | S diag(x) S⁻¹ ≥ 0} and the associated “spectratope” P(S) = {x ∈ C(S) | S diag(x) S⁻¹ e = e}. The rows of S generate a conical hull Cr(S); when C(S) coincides with Cr(S) the similarity is called ideal. Ideal Perron similarities are valuable because they give a complete polyhedral description of all spectra that can be realized by nonnegative matrices via the similarity S.

The authors prove that the character table Q of any finite group G is an ideal Perron similarity. A character table is the n × n matrix whose (i, j) entry is the value of the i‑th irreducible character χ_i on a representative of the j‑th conjugacy class. Classical orthogonality relations give Q Q* = |G| I, showing that Q is a normalized complex Hadamard matrix. The paper introduces the notion of “row‑Hadamard conic” (RHC): a matrix S is RHC if the Hadamard (entrywise) product of any two rows lies in the conical hull of the rows. Using the orthogonality of characters, the authors demonstrate that every row of Q is a non‑negative scalar multiple of the all‑ones vector and that any entrywise product of two rows is a non‑negative linear combination of the rows. By Theorem 3.5 these two facts are precisely the criteria for S to be ideal, establishing that Q is an ideal Perron similarity (Theorem 4.1).

Having identified Q as ideal, the spectracone admits a clean group‑theoretic description. For a vector x ∈ ℂⁿ, the condition x ∈ C(Q) is equivalent to the finite family of linear inequalities
 ∑_{k=1}^n x_k χ_i(g_k) ≥ 0 for every irreducible character χ_i.
Thus the spectracone is the intersection of finitely many half‑spaces determined by the character values, providing an explicit polyhedral model.

When the character table is real (e.g., for groups whose irreducible characters are all real), the spectratope P(Q) becomes a simplex. The authors compute its volume in closed form:
 Vol(P(Q)) = |G| · ∏_{i=1}^n (dim χ_i)⁻¹,
where dim χ_i is the degree of the i‑th irreducible character. This formula follows from the fact that the rows of Q, after normalisation, form the vertices of a regular simplex in the hyperplane ∑ x_i = 1.

The paper also shows that many previously known ideal Perron similarities are special cases of the main theorem. The Walsh matrices, normalized Hadamard matrices, and the discrete Fourier transform matrix are precisely the character tables of the elementary abelian 2‑group, cyclic groups, and direct products thereof. Consequently, earlier results on these matrices are unified under the single framework of group character theory.

From a constructive perspective, the results give a method to generate realizable spectra: given any finite group, the rows of its character table provide the extreme points of a polytope of spectra closed under the Hadamard product. In particular, for abelian groups the extreme points lie on the boundary of the set of all realizable spectra, offering new insight into the geometry of the NIEP solution set.

In summary, the authors expand the class of ideal Perron similarities dramatically by proving that every finite‑group character table possesses this property. They translate the abstract Perron similarity conditions into concrete group‑theoretic inequalities, derive a volume formula for the associated spectratope, and show how this unifies and extends earlier isolated examples. The work supplies both theoretical depth and practical tools for tackling the nonnegative inverse eigenvalue problem.