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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.06704 |
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| _version_ | 1866912558047297536 |
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| author | Masuda, Tetsu Okubo, Naoto Tsuda, Teruhisa |
| author_facet | Masuda, Tetsu Okubo, Naoto Tsuda, Teruhisa |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06704 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras Masuda, Tetsu Okubo, Naoto Tsuda, Teruhisa Exactly Solvable and Integrable Systems Representation Theory 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. |
| title | Birational Weyl group actions and q-Painleve equations via mutation combinatorics in cluster algebras |
| topic | Exactly Solvable and Integrable Systems Representation Theory |
| url | https://arxiv.org/abs/2303.06704 |