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Abstract
We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector there is an
matrix that has all
principal minors equal to
. 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 three 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.
Acknowledgements
Oeding thanks Bernd Sturmfels for introducing us to this question, and for his continued excellence in mentorship. Oeding is also grateful for the partial support provided by the South Korean National Institute for Mathematical Sciences (NIMS) where some of this work was carried out. The authors are also grateful to the developers of Macaulay2, where the initial examples in this paper were all computed. The authors are also thankful for the careful review of two referees whose remarks improved the exposition of this work, and helped simplify the conditions on Theorem 1.2(1).
Notes
No potential conflict of interest was reported by the authors.