Group actions on semimatroids

TL;DR

This paper explores group actions on semimatroids and geometric semilattices, generalizing enumerative results and introducing new combinatorial interpretations, including nonrealizable arithmetic matroids and connections to toric arrangements.

Contribution

It develops a framework for group actions on semimatroids, extending cryptomorphisms to infinite cases and generalizing enumerative results for arithmetic matroids and toric arrangements.

Findings

Introduces orbit-counting functions and Tutte polynomials for group actions on semimatroids.

Generalizes enumerative results to nonrealizable arithmetic matroids.

Establishes connections between group actions, semimatroids, and toric arrangements.

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.