Combinatorics of Matrix Factorizations and Integrable Systems
- https://doi.org/10.1080/14029251.2013.862433How to use a DOI?
- discrete integrable systems, matrix refactorization, discrete Painlevé equations
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.
- © 2013 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
Cite this article
TY - JOUR AU - Anton Dzhamay PY - 2021 DA - 2021/01 TI - Combinatorics of Matrix Factorizations and Integrable Systems JO - Journal of Nonlinear Mathematical Physics SP - 34 EP - 47 VL - 20 IS - Supplement 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2013.862433 DO - https://doi.org/10.1080/14029251.2013.862433 ID - Dzhamay2021 ER -