The polytope of all $q$-rank functions

TL;DR

This paper characterizes the set of all $q$-rank functions as a polytope, revealing structural properties and identifying vertices corresponding to $q$-matroids, with a focus on convex combinations and special classes.

Contribution

It identifies the polytope of $q$-rank functions, shows it contains no interior lattice points, and explores properties of convex combinations, especially for paving and uniform $q$-matroids.

Findings

The polytope of $q$-rank functions has no interior lattice points.

Vertices of the polytope correspond to $q$-matroids.

Convex combinations preserve certain properties in $q$-matroids.

Abstract

A $q$-rank function is a real-valued function defined on the subspace lattice that is non-negative, upper bounded by the dimension function, non-drecreasing, and satisfies the submodularity law. Each such function corresponds to the rank function of a $q$-polymatroid. In this paper, we identify these functions with points in a polytope. We show that this polytope contains no interior lattice points, implying that the points corresponding to $q$-matroids are among its vertices. We investigate several properties of convex combinations of two lattice points in this polytope, particularly in terms of independence, flats, and cyclic flats. Special attention is given to the convex combinations of paving and uniform $q$-matroids.