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We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite-dimensional algebra, whose representation theory is analogous to blocks of Bernstein—Gelfand—Gelfand category\mathcal{O}. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden—Licata—Proudfoot—Webster. Applying our construction to non-linear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.
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Equivariant Cohomology and Conditional Oriented Matroids
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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- PAR ID:
- 10523837
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 11
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 9292 to 9322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnad025
