Independent Spaces of q -Polymatroids
Heide Gluesing-Luerssen, Benjamin Jany · Algebraic Combinatorics · 2022
This paper is devoted to the study of independent spaces of q -polymatroids. With the aid of an auxiliary q -matroid it is shown that the collection of independent spaces satisfies the same properties as for q -matroids. However, in contrast to q -matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q -polymatroid, and this fact can be exploited to derive a cryptomorphism of q -polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be elaborated on.