Planar flows and quadratic relations over semirings

TL;DR

This paper generalizes quadratic relations for flow-generated functions over semirings, unifying known relations like Pl"ucker and tropical relations through a combinatorial framework.

Contribution

It provides a combinatorial description of universal quadratic relations for flow-generated functions over arbitrary semirings, extending classical matrix relations.

Findings

Unified framework for quadratic relations over semirings

Connections to Pl"ucker and tropical relations

Applications discussed in combinatorics and algebra

Abstract

Adapting Lindstr\"om's well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of "universal" quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of known quadratic relations for minors of matrices (e.g., Pl\"ucker relations) and the tropical counterparts of such relations. Also some applications and related topics are discussed.