Tropicalization of Principal Minors

Abstract: Tropicalization is a way to understand the asymptotic behavior of algebraic (or semi-algebraic) sets through polyhedral geometry. In this talk, I will describe the tropicalization of the principal minors of real symmetric and Hermitian matrices. This gives a combinatorial way of understanding their asymptotic behavior and discovering new inequalities on these minors. For positive semidefinite matrices, the resulting tropicalization will have a nice combinatorial structure called M-concavity and be closely related to the tropical Grassmannian and tropical flag variety. For general Hermitian matrices, this story extends to valuated delta matroids.

This is based on joint works with Abeer Al Ahmadieh, Nathan Cheung, Tracy Chin, Gaku Liu, Felipe Rincón, and Josephine Yu.

algebraic geometrycombinatorics

Audience: researchers in the topic

Combinatorics and Geometry BLT Seminar