Matroids, Ordered Structures, Arrangements in Combinatorics 25 - 26 July 2022
Manchester, UK

A framework is a structure consisting of stiff bars connected at joints with full rotational freedom. When is the framework unique up to isometric transformations of the ambient space? This is the global rigidity question which is well studied in the Euclidean case. I will survey how, in the generic case, global rigidity depends only on the underlying graph (edges for bars and vertices for joints) and not the joint positions or the bar lengths. Moreover, in dimension 2, generic global rigidity can be understood combinatorially as a property of an appropriate sparsity matroid defined on the graph. Then I will report on joint work with Sean Dewar and John Hewetson where we have generalised the theory to apply to a wide class of normed spaces and, again, in the plane case, can give a combinatorial understanding of global rigidity in terms of an appropriate matroid.

Matroids were introduced to generalize the independence structure that arises from columns in a matroid as well as the independence structure arising from forests of a graph. The circuits of a matroid are the minimal dependent sets. A collection of circuits that exhibits all the non-flats of a matroid is a tropical basis. We are interested in minimal tropical basis for nested matroids. We will begin with an introduction to matroids, and tropical basis. This is joint work with Yelana Mandelshtam.

In the early 2000's Chapoton introduced three counting polynomials associated with finite Coxeter groups. Despite the fact that these polynomials count different objects---the F-triangle enumerates faces of the cluster complex, the H-triangle enumerates nonnesting partitions and the M-triangle enumerates noncrossing partitions---Chapoton conjectured that these polynomials are related by (invertible) variable transformations. Armstrong generalized these definitions to the Fuß--Catalan setting, and the conjectured relations between the polynomials were proven by Athanasiadis, Krattenthaler, Thiel and Tzanaki.
In this talk we give an elementary combinatorial proof of the transformation between the F- and the H-triangle which relies on an interpretation of these polynomials in terms of the Tamari lattice. In this setting, both polynomials enumerate certain lattice paths with respect to appropriate statistics. In addition, we exhibit an instance of combinatorial reciprocity that relates Dyck paths and certain Schröder paths.
This talk is based on joint work with Cesar Ceballos and Eleni Tzanaki.

For every crystallographic root system Φ, the dominant regions and bounded dominant regions of the corresponding m -Catalan arrangement are enumerated by the m -Catalan number Cat (Φ, m ) and the positive m -Catalan number Cat + (Φ, m ) respectively. Athanasiadis has shown that dominant regions biject to integer points of a certain simplex depending on Φ and m . This bijection leads to a refined counting of the regions giving rise to Narayana numbers. Analogous results hold for bounded regions. After revising the above bijectons, we show that by making use of integer points in the simplex, one can explain a surprising reciprocal relation on the Catalan numbers as an instance of Ehrhart reciprocity. We extend and explain this reciprocity to Narayana numbers. Finally, we discuss a refined face counting of the regions based on Chapoton’s F = H -triangles. More precisely, we give a combinatorial interpretation of the F -triangle in terms of flats of the dominant regions and of the H -triangle in terms of m -Dyck paths. This talk is based on past joint work with C. Athanasiadis and current work with H. Mühle.

Given a Weyl group W, one can define its Coxeter arrangement and its Shi arrangement, the later being a sub-arrangement of the former. Loosely speaking, the regions of the Coxeter arrangement, called alcoves, correspond to the elements of W and thus the Shi regions correspond to sets of these elements. In 1987 J.-Y. Shi proved that the regions of the Shi Arrangement, seen as sets of elements of W, contain a unique minimal element. In 1999, Athanasiadis and Linusson, gave a bijection between the regions of the Shi arrangement and parking functions. In this talk, we show how this bijection can be used to combinatorially compute the minimal element of a Shi region in type A, then, using a result from Armstrong, Reiner and Rhoades '15 expanding [AL '99], we give a construction for types ABCD and finally a less combinatorial but type free construction for all Weyl groups.

The v-Tamari lattice is a partial order on the set of lattice paths lying weakly above a given lattice path v. It was introduced by Préville-Ratelle and Viennot in 2017, and generalises the m-Tamari lattice which arose in connection to representation theory of higher diagonal harmonics modules. In this talk, we will show how the v-Tamari lattice can be geometrically realised in terms of the edge graph of a polytopal subdivision, which is induced by a collection of tropical hyperplane arrangements. This settles an open problem by Francois Bergeron about geometric realisations of m-Tamari lattices. This is based on joint work with Arnau Padrol and Camilo Sarmiento.

We study the tropicalization of the set of matrices with positive entries and bounded rank, i.e. the positive part of determinant varieties. Given a (d x n)-matrix of fixed rank r, we can interpret the columns of the tropicalization of this matrix as n points in d-dimensional space, lying on a common r-dimensional tropical linear space. We consider such tropical point configurations, and introduce a combinatorial criterion to characterize configurations which can be obtained from the tropicalization of matrices with positive entries.

We consider arrangements of tropical hyperplanes, where the apices of the hyperplanes can be `taken to infinity' in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is defined by its `type' data, analogous to the covectors of an oriented matroid. By work of Develin-Sturmfels and Fink-Rincon it is known that these `tropical complexes' are dual to (regular) subdivisions of root polytopes, which in turn are in bijection with mixed subdivisions of certain generalized permutohedra. Extending previous work with Joswig-Sanyal we show how a natural monomial labeling of these complexes describe polynomial relations (syzygies) among various `type ideals'. One such type ideal is shown to be Alexander dual to the initial ideals of a class of toric ideals, leading to novel ways of studying their algebraic properties. For instance our methods of studying the dimension of a tropical complex provides new bounds on certain homological invariants of toric edge ideals of bipartite graphs. Such ideals have been extensively studied in the commutative algebra community and we hope that this perspective leads to new avenues of research. This is joint work with Ayah Almousa and Ben Smith.

In some cases, tropicalizations of moduli spaces reveal beautiful combinatorial structure and provide interesting connections, e.g. tropical Grassmannians, or spaces of tropical stable curves or maps. We investigate a generalization by studying spaces of lines in a plane, or, more generally, spaces of codimension-1-linear subspaces of a fixed linear space. They can be viewed as building block for flag varieties. The reason to restrict to this case is that the equations for such a space are linear, such its tropicalization can be described in terms of the underlying matroid. We provide examples of tropicalizations of such spaces to showcase the combinatorial structures.

This is joint work with Philipp Jell, Felipe Rincon and Benjamin Schroeter.

The intersection theory of tropical linear spaces, under the name of the matroid Chow ring, has recently been in the spotlight for its involvement in matroid Hodge theory. Berget, Eur, Spink and Tseng have developed tools to easily pass between tropical intersection (Chow) and polytope algebra (K-theory) computations in this setting. I'll present this as well as ongoing joint work of mine with Eur, Larson and Spink which extends this to delta-matroids, their Coxeter type BC counterpart. Consequences include enumerative results on type B generalised permutohedra and generalisation of some results of Postnikov.