Combinatorics of Matrix Factorizations and Integrable Systems

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