Symmetrization of Principal Minors and Cycle-Sums

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.

23 pages, updated content to match journal version in Linear and Multilinear Algebra, 2016

References in corpus (6)

• Eynard-Mehta theorem, Schur process, and their pfaffian analogs
• Hyperdeterminantal relations among symmetric principal minors
• Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices
• Secants of minuscule and cominuscule minimal orbits
• Hyperdeterminants of Polynomials
• Set-theoretic defining equations of the tangential variety of the Segre variety

Cited by in corpus (1)

• On matrices in finite free position