Topic
Cryptomorphism
About: Cryptomorphism is a research topic. Over the lifetime, 3 publications have been published within this topic receiving 69 citations.
Papers
TL;DR: Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements of hyperplanes as discussed by the authors, and they have been used for representability, moduli problems and invariants of matroids.
Abstract: This article is an introduction to matroid theory aimed at algebraic geometers Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements Not all matroids come from linear subspaces; those that do are said to be representable Still, one may apply linear algebraic constructions to non-representable matroids There are a number of different definitions of matroids, a phenomenon known as cryptomorphism In this survey, we begin by reviewing the classical definitions of matroids, develop operations in matroid theory, summarize some results in representability, and construct polynomial invariants of matroids Afterwards, we focus on matroid polytopes, introduced by Gelfand–Goresky–MacPherson–Serganova, which give a cryptomorphic definition of matroids We explain certain locally closed subsets of the Grassmannian, thin Schubert cells, which are labeled by matroids, and which have applications to representability, moduli problems, and invariants of matroids following Fink–Speyer We explain how matroids can be thought of as cohomology classes in a particular toric variety, the permutohedral variety, by means of Bergman fans, and apply this description to give an exposition of the proof of log-concavity of the characteristic polynomial of representable matroids due to the author with Huh
69 citations
Posted Content•
TL;DR: In this article, the authors study group actions on (possibly infinite) semimatroids and geometric semilattices and show that these actions give rise to a matroid over the ring of integers.
Abstract: We initiate the study of group actions on (possibly infinite) semimatroids and geometric semilattices. To every such action is naturally associated an orbit-counting function, a two-variable "Tutte" polynomial and a poset which, in the realizable case, coincides with the poset of connected components of intersections of the associated toric arrangement. In this structural framework we recover and strongly generalize many enumerative results about arithmetic matroids, arithmetic Tutte polynomials and toric arrangements by finding new combinatorial interpretations beyond the realizable case. In particular, we thus find a class of natural examples of nonrealizable arithmetic matroids. Moreover, under additional conditions these actions give rise to a matroid over the ring of integers. As a stepping stone toward our results we also prove an extension of the cryptomorphism between semimatroids and geometric semilattices to the infinite case.
16 citations
TL;DR: This work provides a suitable setting to reformulate and extend conjecture of Stanley about $h$-vectors of matroids which is expected to be tractable with techniques that are out of reach for matroIDS alone.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2016 | 1 |
| 2015 | 1 |