An official website of the United States government
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.
more »
« less
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.
more »
« less
- 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
More Like this
-
-
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $$\mathcal{M}$$-separated collections for any oriented matroid $$\mathcal{M}$$. We show that maximal by size $$\mathcal{M}$$-separated collections are in bijection with fine zonotopal tilings (if $$\mathcal{M}$$ is a realizable oriented matroid), or with one-element liftings of $$\mathcal{M}$$ in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid $$\mathcal{M}$$ is pure if $$\mathcal{M}$$-separated collections form a pure simplicial complex, i.e., any maximal by inclusion $$\mathcal{M}$$-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank $$3$$, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank $$3$$ is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an $$n$$-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank $$3$$, graphical, uniform).more » « less
-
Abstract We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $$G_{Q,\gamma }$$ constructed from a symmetric quiver $$Q$$ with enough loops and a dimension vector $$\gamma $$. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of $$S_{\gamma }$$-invariants of the Postnikov–Shapiro slim subgraph space attached to $$G_{Q,\gamma }$$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of $$S_{\gamma }$$-orbits under the permutation action on the set of break divisors on $$G$$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.more » « less
-
Abstract We show that the algebra $$D_\hbar (SL_{n}/U)$$ of differential operators on the base affine space of $$SL_{n}$$ is the quantized Coulomb branch of a certain 3d $$\mathcal{N} = 4$$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer–Hanany–Kirwan about the universal hyperkähler implosion of $$SL_{n}$$. We also formulate and prove a generalization identifying the Hamiltonian reduction $$T^{*} SL_{n}\ /\!/ _{\psi } U$$ as a Coulomb branch for an arbitrary unipotent character $$\psi $$. As an application of our results, we provide a new interpretation of the Gelfand–Graev symmetric group action on $$D_\hbar (SL_{n}/U)$$.more » « less
-
Thefoundationof a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids arepastures, a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroidswithout large uniform minors. i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if is positively oriented then is representable over every field with at least 3 elements. Two matroids are said to belong to the samerepresentation classif they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1093/imrn/rnad025
