2023/03/12 by Tetsu Masuda, Masuda, Tetsu, Naoto Okubo +3 · 1 citation
Chemistry · Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Molecular spectroscopy and chirality #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2303.06704
openalex publication_date 2023/03/12 · openalex created_date 2023/03/16 · openalex updated_date 2026/07/28
A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to derive a tropical, i.e. subtraction-free birational, representation of Weyl groups from cluster algebras. Our results provide an extensive class of Weyl group actions, including previously known examples with algebro-geometric background, and hence are relevant to the q-Painleve equations and their higher-order extensions. Key ingredients of the argument are the combinatorial aspects of the reflection associated with a cycle subgraph in the quiver. We also study symplectic structures of the discrete dynamical systems thus obtained. The normal form of a skew-symmetric integer matrix allows us to choose Darboux coordinates while preserving the birationality.