Combinatorics of Matrix Factorizations and Integrable Systems

TL;DR

This paper explores the combinatorial and geometric relations between eigenvectors of certain rational matrix functions, revealing a cube diagram structure that encodes eigenvector relations and connects to integrable systems.

Contribution

It introduces a novel cube diagram representation for eigenvector relations of matrix functions, linking combinatorics, geometry, and integrable systems in a new way.

Findings

Cube diagram encodes eigenvector relations

Faces of the cube correspond to coordinate systems

Generating functions on faces relate to Lagrangians of integrable systems

Abstract

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial-geometric way using a diagram of a cube. In this representation, vertices of the cube represent eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.