Hermitian finite-N convolution from precursor additivity

Determine whether the eigenvalues of the matrix C defined by the precursor equations K_n(C)=K_n(A)+K_n(B) for n=1,...,N are all real and, when they are real, whether they satisfy the Horn inequalities associated with the eigenvalues of the Hermitian matrices A and B.

Background

The paper considers defining a finite-N analogue of free convolution using the precursor polynomials K_n rather than the elementary symmetric polynomials e_n. For Hermitian matrices A and B, the proposed convolution would be represented by a matrix C whose precursor values satisfy K_n(C)=K_n(A)+K_n(B) for every n from 1 through N. These equations determine the eigenvalues of C up to permutation and always produce a complex matrix solution.

For N=3, the proposed precursor-based equations coincide with the equations defining the established finite free convolution, so the resulting eigenvalues are real and satisfy Horn’s inequalities. For N at least 4, the paper gives examples with nonreal eigenvalues and examples with real eigenvalues violating Horn’s inequalities; the authors therefore leave unresolved whether any general conditions ensure reality and Horn admissibility.

References

This condition can be rephrased as a system of $N$ algebraic equations on the eigenvalues $(\gamma_1,\dots,\gamma_N)$ of $C$ in terms of the eigenvalues $\alpha_i,\beta_i$ of $A$ and $B$. This system always admits a solution, unique up to permutation, so that the matrix $C$ exists, at least as a complex matrix. However, two important questions remain:\vspace{-2pt}

— Finite $N$ precursors of the free cumulants  (2508.21483 - Lacroix et al., 29 Aug 2025) in Section 6.1, “Finite N free convolution associated with precursors?” (Section \ref{sec:convolution})