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 (nontrivial) 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.
| Original language | English |
|---|---|
| Pages (from-to) | 2940-2970 |
| Number of pages | 31 |
| Journal | SIAM Journal on Discrete Mathematics |
| Volume | 38 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2024 |
Bibliographical note
Publisher Copyright:© 2024 Society for Industrial and Applied Mathematics.
Funding
\ast Received by the editors April 3, 2023; accepted for publication (in revised form) August 20, 2024; published electronically December 3, 2024. https://doi.org/10.1137/23M156358X Funding: The first author was partially supported by grant 422479 from the Simons Foundation. \dagger Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027 USA (heide.gl@ uky.edu). \ddagger Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, the Netherlands ([email protected]).
| Funders |
|---|
| Simons Foundation |
Keywords
- cyclic flats
- decomposition
- direct sum
- irreducibility
- q-matroid
ASJC Scopus subject areas
- General Mathematics