Symmetrization of Principal Minors and Cycle-Sums

2015/10/08 by Huang, Huajun, Oeding, Luke · 1 citation
#05A05 #15A69 #15B05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.02515

Abstract

We solve the Symmetrized Principal Minor Assignment Problem, that is we show how to determine if for a given vector v∈ ℂⁿ there is an n× n matrix that has all i× 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.

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• Wick theorem for analytic functions of Gaussian fields

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