Flag arrangements and triangulations of products of simplices

We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and effective criterion to ensure the vanishing of many Schubert structure constants in the flag manifold, and a new perspective on Billey and Vakil's method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo

Of matroid polytopes, chow rings and character polynomials

Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients

Tropical Discriminants

Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the projective toric variety given by an integer matrix A, is shown to coincide with the Minkowski sum of the row space of A and of the tropicalization of the kernel of A. This leads to an explicit positive formula for the extreme monomials of any A-discriminant, without any smoothness assumption.Comment: Major revisions, including several improvements and the correction of Section 5. To appear: Journal of the American Mathematical Societ

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Combinatorics of Grassmannian Decompositions

This thesis studies several combinatorially defined families of subsets of the Grassmannian. We introduce and study a family of subsets called “basis shape loci” associated to transversal matroids. Additionally, we study the Deodhar and positroid decompositions of the Grassmannian. A basis shape locus takes as input data a zero/nonzero pattern in a matrix, which is equivalent to a specific presentation of a transversal matroid. The locus is defined to be the set of points in the Grassmannian which are the row spaces of matrices with the prescribed zero/nonzero pattern. We show that this locus depends only on the transversal matroid, not on the specific presentation. When a transversal matroid is a positroid, the closure of its basis shape locus is exactly the positroid variety labelled by the matroid. We give a sufficient, and conjecturally necessary, condition for when a transversal matroid is a positroid. Components in the Deodhar decomposition are indexed by Go-diagrams, certain fillings of Ferrers shapes with white stones, black stones, and pluses. Le-diagrams are a common combinatorial object indexing positroids; all Le-diagrams are Go-diagrams. We give a system of local flips on fillings of Ferrers shapes which may be used to turn arbitrary diagrams into Go-diagrams. When a Go-diagram is a Le-diagram, these flips are exactly the previously studied Le-moves. Using these local flips, we conjecture a combinatorial condition describing when one Deodhar component is contained in the closure of another within a Schubert cell. We define a variety containing and conjecturally equal to the closure of a Deodhar component and prove that this combinatorial criterion implies a containment of these varieties. We further show that there is no reasonable description of Go-diagrams in terms of forbidden subdiagrams by providing an injection from the set of valid Go- diagrams into the set of minimal forbidden subdiagrams. In lieu of such a description, we give an algorithmic characterization of Go-diagrams. Finally, we use the above results to prove several corollaries about Wilson loop cells, which arise in the study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Notably, it was previously known that the matroid represented by a generic point in a Wilson loop cell is a positroid. We show that the closure of the Wilson loop cell agrees with the positroid variety labelled by this positroid