Matroids arising from algebraic shifting

TL;DR

This paper characterizes certain classes of shifted hypergraphs and graphs whose inverse exterior shifting images are matroid bases, revealing connections to known matroids like the simplicial and rigidity matroids.

Contribution

It provides a complete characterization of shifted hypergraphs and graphs that correspond to matroid bases, extending to symmetric shifting and higher uniformities.

Findings

Characterization of shifted simple graphs and 3-uniform hypergraphs as matroid bases

Identification of hypergraphs with hyperedges forming initial lex-segments

Extension of results to symmetric shifting and higher uniformities

Abstract

We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment. There are several examples of known matroids arising in this way: the simplicial matroid, the hyperconnectivity matroid and the area-rigidity matroid. For $k\ge 4$, we provide a similar characterization for shifted $k$-uniform hypergraphs satisfying an additional combinatorial condition. For symmetric shifting, we prove an analogous characterization for shifted simple graphs, where the classical generic rigidity matroid is an example of a matroid arising in this way.