From weakly separated collections to matroid subdivisions
N Early�- arXiv preprint arXiv:1910.11522, 2019 - arxiv.org
arXiv preprint arXiv:1910.11522, 2019•arxiv.org
We study arrangements of slightly skewed tropical hyperplanes, called blades by A.
Ocneanu, on the vertices of a hypersimplex $\Delta_ {k, n} $, and we investigate the
resulting induced polytopal subdivisions. We show that placing a blade on a vertex $ e_J $
induces an $\ell $-split matroid subdivision of $\Delta_ {k, n} $, where $\ell $ is the number of
cyclic intervals in the $ k $-element subset $ J $. We prove that a given collection of $ k $-
element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on�…
Ocneanu, on the vertices of a hypersimplex $\Delta_ {k, n} $, and we investigate the
resulting induced polytopal subdivisions. We show that placing a blade on a vertex $ e_J $
induces an $\ell $-split matroid subdivision of $\Delta_ {k, n} $, where $\ell $ is the number of
cyclic intervals in the $ k $-element subset $ J $. We prove that a given collection of $ k $-
element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on�…
We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex , and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex induces an -split matroid subdivision of , where is the number of cyclic intervals in the -element subset . We prove that a given collection of -element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the blade on the corresponding vertices of induces a matroid (in fact, a positroid) subdivision. In this way we obtain a compatibility criterion for (planar) multi-splits of a hypersimplex, generalizing the rule known for 2-splits. We study in an extended example the case the set of arrangements of weakly separated vertices of .
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