The lattice of flats $\mathcal{L}_M$ of a matroid $M$ is combinatorially well-behaved and, when $M$ is realizable, admits a geometric model in the form of a “Schubert variety of hyperplane arrangement”. In contrast, the lattice of flats of a polymatroid exhibits many combinatorial pathologies and admits no similar geometric model.
We address this situation by defining the lattice $\mathcal{L}_P$ of “combinatorial flats” of a polymatroid $P$. Combinatorially, $\mathcal{L}_P$ exhibits good behavior analogous to that of $\mathcal{L}_M$: it is graded, determines $P$ when $P$ is simple, and is top-heavy. When $P$ is realizable over a field of characteristic 0, we show that $\mathcal{L}_P$ is modeled by “the Schubert variety of a subspace arrangement”.
Our work generalizes a number of results of Ardila–Boocher and Huh–Wang on Schubert varieties of hyperplane arrangements; however, the geometry of Schubert varieties of subspace arrangements is noticeably more complicated than that of Schubert varieties of hyperplane arrangements. Many natural questions remain open.
Revised:
Accepted:
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Keywords: polymatroid, subspace arrangement, lattice of flats, Schubert variety, vector group, equivariant compactification
Crowley, Colin 1; Simpson, Connor ; Wang, Botong 2
CC-BY 4.0
@article{ALCO_2025__8_5_1285_0,
author = {Crowley, Colin and Simpson, Connor and Wang, Botong},
title = {Combinatorial flats and {Schubert} varieties of subspace arrangements},
journal = {Algebraic Combinatorics},
pages = {1285--1312},
year = {2025},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {5},
doi = {10.5802/alco.447},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.447/}
}
TY - JOUR AU - Crowley, Colin AU - Simpson, Connor AU - Wang, Botong TI - Combinatorial flats and Schubert varieties of subspace arrangements JO - Algebraic Combinatorics PY - 2025 SP - 1285 EP - 1312 VL - 8 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.447/ DO - 10.5802/alco.447 LA - en ID - ALCO_2025__8_5_1285_0 ER -
%0 Journal Article %A Crowley, Colin %A Simpson, Connor %A Wang, Botong %T Combinatorial flats and Schubert varieties of subspace arrangements %J Algebraic Combinatorics %D 2025 %P 1285-1312 %V 8 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.447/ %R 10.5802/alco.447 %G en %F ALCO_2025__8_5_1285_0
Crowley, Colin; Simpson, Connor; Wang, Botong. Combinatorial flats and Schubert varieties of subspace arrangements. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1285-1312. doi: 10.5802/alco.447
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