Decompositions of q-Matroids Using Cyclic Flats

TL;DR

This paper explores the structure of q-matroids through their cyclic flats, providing a decomposition into irreducible components and simplifying the understanding of their direct sums.

Contribution

It introduces a characterization of irreducibility of q-matroids via cyclic flats and proves unique decomposition into irreducible q-matroids.

Findings

Cyclic flats of a direct sum are sums of cyclic flats of summands.

Rank function of a direct sum is simplified using cyclic flats.

Every q-matroid decomposes uniquely into irreducible q-matroids.

Abstract

We study the direct sum of q-matroids by way of their cyclic flats. Using that the rank function of a q-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats of the direct sum of two q-matroids are exactly all the direct sums of the cyclic flats of the two summands. This simplifies the rank function of the direct sum significantly. A q-matroid is called irreducible if it cannot be written as a (non-trivial) direct sum. We provide a characterization of irreducibility in terms of the cyclic flats and show that every q-matroid can be decomposed into a direct sum of irreducible q-matroids, which are unique up to equivalence.