Title: The polytope of all $q$-rank functions
Abstract: A $q$-rank function is a real-valued function defined on the subspace lattice of $\mathbb{F}_q^n$ that is non-negative, upper bounded by the dimension function, non-decreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a $q$-polymatroid. Intuitively, we can view these objects as $q$-analogues of polymatroids. In this viewpoint the concept of $q$-analogue can be interpreted as generalizing from finite sets to finite-dimensional vector spaces over finite fields. Their original motivation comes from algebraic coding theory, as the representable $q$-polymaroids arise from so-called rank-metric codes.
In this talk, we identify these functions with points in a polytope. The lattice points of that polytope correspond to integer-valued $q$-polymatroids, also called $q$-matroids. We show that all lattice points are among the vertices of the polytope and investigate several properties of convex combinations of two such vertices. In particular, we do so with regard to independence, flats, and cyclic flats of the $q$-polymatroids formed by these convex combinations. Here special attention is drawn to convex combinations of paving and uniform $q$-matroids.
The advanced seminar begins at 4:00 pm s.t..