Planar flows and quadratic relations over semirings
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.
Cite
@article{arxiv.1102.2578,
title = {Planar flows and quadratic relations over semirings},
author = {Vladimir I. Danilov and Alexander V. Karzanov and Gleb A. Koshevoy},
journal= {arXiv preprint arXiv:1102.2578},
year = {2012}
}
Comments
35 pages. This is the revised version accepted for publication in J. Algebraic Comb. (The final publication is available at springerlink.com.) Also in the Appendix we add the assertion that the function of minors of any matrix over a field is generated (using Lindstr\"om method) by flows in a weighted planar acyclic directed graph