Symmetrization of Principal Minors and Cycle-Sums
Huajun Huang, Luke Oeding
TL;DR
The paper solves the Symmetrized Principal Minor Assignment Problem by moving to cycle-sum coordinates, enabling a precise classification of matrices with symmetrized principal minors (SCS) across symmetric, skew-symmetric, and general cases. It provides explicit parametrizations, normal forms, and a detailed description of the associated algebraic varieties, including their connections to tangential and secant varieties of Veronese and Segre-Veronese embeddings and to Eulerian polynomials. The main contributions are (i) complete structure theorems for SCS matrices, (ii) explicit relations among symmetrized cycle-sums, and (iii) a set of principled geometric interpretations that illuminate the PMAP and its symmetrized variant. These results yield both constructive criteria for realizability of prescribed cycle-sums and a clearer picture of the underlying algebraic geometry governing principal minors.
Abstract
We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector $v\in \mathbb{C}^{n}$ there is an $n\times n$ matrix that has all $i\times i$ principal minors equal to $v_{i}$. We use a special isomorphism (a non-linear change of coordinates to cycle-sums) that simplifies computation and reveals hidden structure. We use the symmetries that preserve symmetrized principal minors and cycle-sums to treat 3 cases: symmetric, skew-symmetric and general square matrices. We describe the matrices that have such symmetrized principal minors as well as the ideal of relations among symmetrized principal minors / cycle-sums. We also connect the resulting algebraic varieties of symmetrized principal minors to tangential and secant varieties, and Eulerian polynomials.