More Like this

1. Foundations of Matroids, Part 1: Matroids without Large Uniform Minors

   https://doi.org/10.1090/memo/1536  

   Baker, Matthew; Lorscheid, Oliver (January 2025, Memoirs of the American Mathematical Society)

   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 $M$without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if $M$is positively oriented then $M$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
2. Intersection Theory of Matroids: Variations on a Theme

   https://doi.org/10.1017/9781009490559.002  

   Ardila–Mantilla, Federico (June 2024, Cambridge University Press)

   Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μ_k(M) = deg_M(α^{r−k}β^k) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of M. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids. Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference. 

   more » « less
3. The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron

   https://doi.org/10.1093/imrn/rnad010  

   Łukowski, Tomasz; Parisi, Matteo; Williams, Lauren K (March 2023, International Mathematics Research Notices)

   Abstract The positive Grassmannian $$Gr^{\geq 0}_{k,n}$$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$$\mu $$ onto the hypersimplex [ 31] and the amplituhedron map$$\tilde{Z}$$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $$\mathcal{N}=4$$ super Yang-Mills. We define a map we call T-duality from cells of $$Gr^{\geq 0}_{k+1,n}$$ to cells of $$Gr^{\geq 0}_{k,n}$$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $$\Delta _{k+1,n}$$ to positroid dissections of the amplituhedron $$\mathcal{A}_{n,k,2}$$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $$\mathcal{A}_{n,k,2}$$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $$Gr_{k,k+2}$$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $$m$$, we define the momentum amplituhedron for any even $$m$$. 

   more » « less
4. Computation of Dressians by dimensional reduction

   https://doi.org/10.1515/advgeom-2022-0016  

   Brandt, Madeline; Speyer, David E. (July 2022, Advances in Geometry)

   Abstract We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions. 

   more » « less
5. Stellahedral geometry of matroids

   https://doi.org/10.1017/fmp.2023.24  

   Eur, Christopher; Huh, June; Larson, Matt (January 2023, Forum of Mathematics, Pi)

   Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

   more » « less