From weakly separated collections to matroid subdivisions

We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex $\Delta_{k,n}$, and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex $e_J$ induces an $\ell$-split matroid subdivision of $\Delta_{k,n}$, where $\ell$ is the number of cyclic intervals in the $k$-element subset $J$. We prove that a given collection of $k$-element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the blade $((1,2,\ldots, n))$ on the corresponding vertices of $\Delta_{k,n}$ induces a matroid (in fact, a positroid) subdivision. In this way we obtain a compatibility criterion for (planar) multi-splits of a hypersimplex, generalizing the rule known for 2-splits. We study in an extended example the case $(k,n) = (3,7)$ the set of arrangements of $(k-1)(n-k-1)$ weakly separated vertices of $\Delta_{k,n}$.Comment: 29 pages, 10 figures. v3: added proof of Corollary 3

Valuated matroid polytopes and linking system composition.

PhD Theses.Valuated matroids are a generalisation of matroids; matroids themselves being an abstraction of the notion of independence. Valuated matroids have many equivalent de nitions including via independent sets and circuits, and in this thesis we show that a valuated matroid has an equivalent de nition in terms of a rank function which we construct by analogy with the matroid rank function by looking at matroid and valuated matroid polytopes. We separately construct a hyperoperation which is an extension of a previously studied operation of composing valuated matroids, this being the composition of valuated linking systems. The composition of valuated linking systems can be seen as a generalisation of matrix multiplication to tropical linear spaces. In particular, the hyperoperation we introduce has been in uenced by viewing matrices as representing linear spaces, which we can do by looking at their row space, and consequently by how these relate to Pl ucker coordinates. Working tropically, since tropical linear spaces are equivalent to valuated matroids, which are also known as tropical Pl ucker vectors, we create the hyperoperation by using the parallels with matrices representing linear spaces over a eld. We describe the hyperproduct completely for small rank, where this operation forms a hypergroup. In higher rank we investigate what known matroid subdivisions it contains, as well as also showing that it does not form a fan, and nor is it convex in general. We also conjecture this hyperoperation forms a hypergroup for higher rank, and present some investigation towards this

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

Topology of Arrangements and Representation Stability

The workshop “Topology of arrangements and representation stability” brought together two directions of research: the topology and geometry of hyperplane, toric and elliptic arrangements, and the homological and representation stability of configuration spaces and related families of spaces and discrete groups. The participants were mathematicians working at the interface between several very active areas of research in topology, geometry, algebra, representation theory, and combinatorics. The workshop provided a thorough overview of current developments, highlighted significant progress in the field, and fostered an increasing amount of interaction between specialists in areas of research