Integrable deformations of cluster maps of type $D_{2N}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912661797601280 |
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| author | Kim, Wookyung |
| author_facet | Kim, Wookyung |
| contents | In this paper, we extend one of the main results from our joint work with Hone and Mase, in which we studied a deformed type $D_{4}$ map, to the general case of the type $D_{2N}$ for $N\geq3$. This can be achieved through a ``local expansion" operation, introduced in our joint work with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type $D_{4}$ map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type $D_{6}$ map and thus enables systematic generalization to higher ranks $D_{2N}$. We also study the degree growth of deformed type $D_{2N}$ map via the tropical method and conjecture that, for each $N$, the deformed map is an integrable, as indicated by the algebraic entropy test, the criterion for detecting integrability in the discrete dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrable deformations of cluster maps of type $D_{2N}$ Kim, Wookyung Exactly Solvable and Integrable Systems Mathematical Physics Combinatorics In this paper, we extend one of the main results from our joint work with Hone and Mase, in which we studied a deformed type $D_{4}$ map, to the general case of the type $D_{2N}$ for $N\geq3$. This can be achieved through a ``local expansion" operation, introduced in our joint work with Grabowski and Hone. This operation involves inserting a specific subquiver into the quiver arising from the Laurentification of the deformed type $D_{4}$ map. This insertion yields a new quiver, obtained through the Laurentification of the deformed type $D_{6}$ map and thus enables systematic generalization to higher ranks $D_{2N}$. We also study the degree growth of deformed type $D_{2N}$ map via the tropical method and conjecture that, for each $N$, the deformed map is an integrable, as indicated by the algebraic entropy test, the criterion for detecting integrability in the discrete dynamical systems. |
| title | Integrable deformations of cluster maps of type $D_{2N}$ |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2506.06182 |