Decompositions of q-Matroids Using Cyclic Flats

Heide Gluesing-Luerssen; Benjamin Jany

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arXiv · 2302.02260

Decompositions of q-Matroids Using Cyclic Flats

Heide Gluesing-Luerssen·Benjamin Jany

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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.

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BibTeX RIS

Heide Gluesing-Luerssen, Benjamin Jany. 2023-02-04. Decompositions of q-Matroids Using Cyclic Flats. https://arxiv.org/abs/2302.02260

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