Total positivity for matroid Schubert varieties

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2025年9月1日

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Xuhua He,
Connor Simpson,
Kaitao Xie

摘要

We define the totally nonnegative matroid Schubert variety $\mathcal Y_V$ $Y_{V}$ of a linear subspace $V \subset \mathbb R^n$ $V \subset R^{n}$ . We show that $\mathcal Y_V$ $Y_{V}$ is a regular CW complex homeomorphic to a closed ball, with strata indexed by pairs of acyclic flats of the oriented matroid of V. This closely resembles the regularity theorem for totally nonnegative generalized flag varieties. As a corollary, we obtain a regular CW structure on the real matroid Schubert variety of V.