The free product of $q$-matroids
Gianira N. Alfarano, Eimear Byrne, Andrew Fulcher
TL;DR
This work introduces and develops the free product operation for pairs of $q$-matroids, providing three cryptomorphic descriptions (independent spaces, rank function, cyclic flats) and establishing its maximality in a weak-order class along with a unique factorisation into irreducibles. It shows how cyclic flats govern the product’s structure and offers representability results, including a block-structure description and a detailed analysis for uniform factors. In the rank-one uniform case, representability is linked to geometric objects called clubs on the projective line, connecting finite geometry to $q$-matroid theory. The paper also delineates open questions about representability, evasivity, and combinatorial counts of $q$-matroids, highlighting both theoretical insights and concrete geometric constraints with potential applications to rank-metric codes and related areas.
Abstract
We introduce the notion of the free product of $q$-matroids, which is the $q$-analogue of the free product of matroids. We study the properties of this noncommutative binary operation, making an extensive use of the theory of cyclic flats. We show that the free product of two $q$-matroids $M_1$ and $M_2$ is maximal with respect to the weak order on $q$-matroids having $M_1$ as a restriction and $M_2$ as the complementary contraction. We characterise $q$-matroids that are irreducible with respect to the free product and we prove that the factorization of a $q$-matroid into a free product of irreducibles is unique up to isomorphism. We discuss the representability of the free product, with a particular focus on rank one uniform $q$-matroids and show that such a product is represented by clubs on the projective line.