Decompositions of \(q\)-matroids using cyclic flats
type: Item
instance of: scholarly article
title: Decompositions of \(q\)-matroids using cyclic flats
author: Heide Gluesing-Luerssen
author: Benjamin Jany
published in: SIAM Journal on Discrete Mathematics
publication date: 2024-12-18T00:00:00Z
review text: In this paper, one investigates the cyclic flats of a direct sum of \(q\)-matroids. To be more precise, one shows that the cyclic flats of a direct sum of \(q\)-matroids are the direct sums of the cyclic flats of each \(q\)-matroid. One uses this to give a simplified description of the rank function of a direct sum of \(q\)-matroids. In addition, one characterizes irreducibility of a \(q\)-matroid in terms of its cyclic flats. One also shows that any \(q\)-matroid is a direct sum of irreducible \(q\)-matroids in an essentially unique way.\N\NThe article has \(8\) sections, including the introduction where one explains the background and purpose of the paper. In Section 2 one gives some basic notation, and in Section 3 one focuses in particular on the important concept of the cyclic core of a subspace of the ground space of a \(q\)-matroid. In Section 4 one presents the lattice of cyclic flats of a \(q\)-matroid and gives new results describing independence and rank functions in terms of cyclic flats. In Section 5 one recalls known facts about direct sums of \(q\)-matroids, gives new results relating the rank function to cyclic flats, and shows how duality commutes with taking direct sums. In Section 6 one describes the cyclic flats of a direct sum of two \(q\)-matroids, uses this to show the associativity of direct sums, and a result on vector rank metric codes that are ``direct-sum-like''. In Section 7 one gives and proves many results, including important ones on irreducibility and decomposition of \(q\)-matroids. In Section 8 one uses another result from Section 7 (Theorem 7.9) to give a complete classification of all \(q\)-matroids with \(3\)-dimensional ground space. One also describes \(q\)-matroids with \(4\)-dimensional ground spaces.
reviewed by: Trygve Johnsen
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05C70
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 7959402
zbMATH Keywords: q-matroid
zbMATH Keywords: cyclic flats
zbMATH Keywords: direct sum
zbMATH Keywords: irreducibility
zbMATH Keywords: decomposition
MaRDI profile type: Publication
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DOI: 10.1137/23M156358X
description: scientific article; zbMATH DE number 7959402