Linear and Non-Linear Algebra

Speaker: Luke Oeding

Title: 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 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. 

This is joint work with Huajun Huang, http://arxiv.org/pdf/1510.02515v1.pdf