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arXiv · 2208.04855

Equivariant cohomology and conditional oriented matroids

Galen Dorpalen-Barry·Nicholas Proudfoot·Jidong Wang
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Abstract

We give a cohomological interpretation of the Heaviside filtration on the Varchenko--Gelfand ring of a pair $(\mathcal{A},\mathcal{K})$, where $\mathcal{A}$ is a real hyperplane arrangement and $\mathcal{K}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where $\mathcal{K}$ is the ambient vector space. We also define the Gelfand--Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand--Rybnikov ring of an oriented matroid and the aforementioned Varchenko--Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.

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Galen Dorpalen-Barry, Nicholas Proudfoot, Jidong Wang. 2022-08-09. Equivariant cohomology and conditional oriented matroids. https://arxiv.org/abs/2208.04855

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