The Cloud and Flock Polynomials of q-Matroids

TL;DR

This paper introduces the cloud and flock polynomials for q-matroids, demonstrating they determine the Whitney function and the entire configuration, and explores their behavior under duality and direct sums.

Contribution

It establishes that the cloud-flock polynomials and the configuration fully determine the q-matroid's Whitney function and structure, extending matroid theory to q-matroids.

Findings

Cloud and flock polynomials determine the Whitney function.

The configuration and cloud-flock lattice are preserved under duality and direct sums.

The Whitney function alone does not behave well under direct sums.

Abstract

We show that the Whitney function of a q-matroid can be determined from the cloud and flock polynomials associated to the cyclic flats. These polynomials capture information about the corank (resp., nullity) of certain spaces whose cyclic core (resp., closure) is the given cyclic flat. Going one step further, we prove that the Whitney function, and in fact the cloud-flock lattice, are determined by the configuration of the q-matroid, which is the abstract lattice of cyclic flats together with the corank-nullity data. Furthermore, we show that the configuration and cloud-flock lattice behave well under duality and direct sums, whereas the Whitney function does not contain enough information to behave well under taking direct sums. As an aside we show that every configuration of a matroid arises as a configuration of a q-matroid, whereas the converse is not true.