The direct sum of q-matroids and their decomposition into irreducible components

Abstract: $q$-Matroids, the $q$-analogue of matroids, have been intensively studied in recent years in coding theory due to their close connection with rank metric codes. In fact, it was shown in 2018 by Jurrius and Pellikaan that a rank metric code induces a $q$-matroid that captures many of the code's invariants. Many results from classical matroid theory were found to have natural $q$-analogues. One of the major differences between classical matroids and $q$-matroids, however, arose with the introduction of the direct sum of $q$-matroids. In this talk, I will discuss several combinatorial and algebraic properties of the direct sum of $q$-matroids and show how they differ from the properties established for classical matroids. I will first show that similarly to matroids, $q$-matroids can be decomposed uniquely (up to equivalence) into the direct sum of irreducible components. However, we will see that this result cannot be achieved via a natural analogue of the matroid notion of connected components. I will then discuss the representability of the direct sum of $q$-matroids. A $q$-matroid is said to be representable if it is induced by a rank metric code. I will show that unlike classical matroids, the direct sum of $q$-matroids does not necessarily preserve representability.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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