Geodesic
Conserved quantities of Q-systems from dimer integrable systems
Panupong Vichitkunakorn1
1Prince of Songkla University
The electronic journal of combinatorics, Tome 25 (2018) no. 1
Cet article a éte moissonné depuis la source The Electronic Journal of Combinatorics website

Voir la notice de l'article

We study a discrete dynamic on weighted bipartite graphs on a torus, analogous to dimer integrable systems in Goncharov-Kenyon 2013. The dynamic on the graph is an urban renewal together with shrinking all 2-valent vertices, while it is a cluster transformation on the weight. The graph is not necessary obtained from an integral polygon. We define the Hamiltonians of a weighted graph as partition functions of all weighted perfect matchings with a common homology class, then show that they are invariant under a move on the weighted graph. This move coincides with a cluster mutation, analog to Y-seed mutation in dimer integrable systems. We construct graphs for Q-systems of type A and B and show that the Hamiltonians are conserved quantities of the systems. This reproves the results of Di Francesco-Kedem 2010 and Galashin-Pylyavskyy 2016 for the Q-systems of type A, and gives new results for that of type B. Similar to the results in Di Francesco-Kedem 2010, the conserved quantities for Q-systems of type B can also be written as partition functions of hard particles on a certain graph. For type A, we show that the conserved quantities Poisson commute under a nondegenerate Poisson bracket.
DOI : 10.37236/6994
Classification : 82B20, 13F60, 05C22, 37K10
Mots-clés : discrete dynamical systems, cluster algebras, conserved quantities, dimers, perfect matchings

Panupong Vichitkunakorn  1

1 Prince of Songkla University 
@article{10_37236_6994,
     author = {Panupong Vichitkunakorn},
     title = {Conserved quantities of {Q-systems} from dimer integrable systems},
     journal = {The electronic journal of combinatorics},
     year = {2018},
     volume = {25},
     number = {1},
     doi = {10.37236/6994},
     zbl = {1387.82017},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/6994/}
}
TY  - JOUR
AU  - Panupong Vichitkunakorn
TI  - Conserved quantities of Q-systems from dimer integrable systems
JO  - The electronic journal of combinatorics
PY  - 2018
VL  - 25
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.37236/6994/
DO  - 10.37236/6994
ID  - 10_37236_6994
ER  - 
%0 Journal Article
%A Panupong Vichitkunakorn
%T Conserved quantities of Q-systems from dimer integrable systems
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6994/
%R 10.37236/6994
%F 10_37236_6994
Panupong Vichitkunakorn. Conserved quantities of Q-systems from dimer integrable systems. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/6994

Cité par Sources :