Symmetrization of Principal Minors and Cycle-Sums

TL;DR

This paper solves the Symmetrized Principal Minor Assignment Problem by characterizing matrices with given symmetrized principal minors using cycle-sums and algebraic geometry, revealing structural insights across matrix types.

Contribution

It introduces a novel approach using cycle-sums and algebraic geometry to determine matrices with prescribed symmetrized principal minors, covering symmetric, skew-symmetric, and general matrices.

Findings

Characterization of matrices with specified symmetrized principal minors.

Description of algebraic relations among symmetrized principal minors and cycle-sums.

Connection of algebraic varieties to tangential, secant varieties, and Eulerian polynomials.

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.