2024 Workshop on (Mostly) Matroids

Almost all q-matroids are not representable

20 Aug 2024, 13:55

25m

S236 (IBS Science Culture Center)

Presentation (25 min)

Speaker

Sebastian Degen (Bielefeld University)

Description

The $q$-analoge of a combinatorial object arises by replacing finite sets with finite dimensional vector spaces. In particular we can view $q$-matroids as $q$-analogues of matroids. One motivation to study $q$-matroids stems from coding theory, as the representable $q$-matroids arise from rank-metric codes. In the matroidal setting Peter Nelson proved in 2018 that asymptotically almost all matroids are non-representable,
therefore one can ask if the same holds true in the $q$-analogue. In this talk we investigate this question and provide a positive answer to it. For this purpose we give a lower bound on the number of all fixed dimensional $q$-matroids, using the theory of constant dimension codes and an upper bound on the number of all representable $q$-matroids, using the concept of zero patterns.