Results on the algebraic matroid of the determinantal variety

Tsakiris, Manolis

doi:10.1090/tran/9055

Results on the algebraic matroid of the determinantal variety
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by Manolis C. Tsakiris;

Trans. Amer. Math. Soc. 377 (2024), 731-751

DOI: https://doi.org/10.1090/tran/9055

Published electronically: October 25, 2023

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Abstract:

We make progress towards characterizing the algebraic matroid of the determinantal variety defined by the minors of fixed size of a matrix of variables. Our main result is a novel family of base sets of the matroid, which characterizes the matroid in special cases. Our approach relies on the combinatorial notion of relaxed supports of linkage matching fields that we introduce, our interpretation of the problem of completing a matrix of bounded rank from a subset of its entries as a linear section problem on the Grassmannian, and a connection that we draw with a class of local coordinates on the Grassmannian described by Sturmfels $\&$ Zelevinsky in 1993.

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