Journal of Nonlinear Mathematical Physics

Volume 20, Issue Supplement 1, November 2013, Pages 34 - 47

Combinatorics of Matrix Factorizations and Integrable Systems

Authors
Anton Dzhamay
School of Mathematical Sciences, The University of Northern Colorado, Campus Box 122, 501 20th Street, Greeley, CO 80639, USA, adzham@unco.edu
Received 2 September 2012, Accepted 13 June 2013, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2013.862433How to use a DOI?
Keywords
discrete integrable systems, matrix refactorization, discrete Painlevé equations
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.

Copyright
© 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/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
20 - Supplement 1
Pages
34 - 47
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2013.862433How to use a DOI?
Copyright
© 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  -