On bases of tropical Pl\"ucker functions

TL;DR

This paper studies tropical analogs of Pl"ucker relations on functions over certain integer boxes, constructing bases, characterizing subclasses, and exploring properties like Laurentness in the tropical setting.

Contribution

It constructs a basis for tropical Pl"ucker functions on truncated integer boxes and characterizes submodular and concave subclasses via restrictions to this basis.

Findings

Constructed a basis for tropical Pl"ucker functions on specified integer boxes.

Characterized submodular, skew-submodular, and discrete concave functions within this framework.

Discussed a tropical analogue of the Laurentness property.

Abstract

We consider functions $f:B\to\Rset$ that obey tropical analogs of classical Pl\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\{x\in \Zset^n\colon 0\le x\le a, m\le x_1+...+x_n\le m'\}$ (a rectangular integer box ``truncated from below and above''). We construct a basis for the set $\Tscr$ of tropical Pl\"ucker functions on $B$, a subset $\Bscr\subseteq B$ such that the restriction map $\Tscr\to\Rset^\Bscr$ is bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so-called skew-submodular, and discrete concave functions in $\Tscr$, discuss a tropical analogue of the Laurentness property, and present other results.