Decompositions of q-Matroids Using Cyclic Flats
Combinatorics
2023-02-23 v2 Information Theory
math.IT
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.
Cite
@article{arxiv.2302.02260,
title = {Decompositions of q-Matroids Using Cyclic Flats},
author = {Heide Gluesing-Luerssen and Benjamin Jany},
journal= {arXiv preprint arXiv:2302.02260},
year = {2023}
}