Decompositions of \(q\)-matroids using cyclic flats (Q6654115)

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.