Explicit forms in lower degrees of rank 2 cluster scattering diagrams
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916084859273216 |
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| author | Akagi, Ryota |
| author_facet | Akagi, Ryota |
| contents | In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the explicit forms of wall elements including the Badlands, which is the complement of $G$-fan. In this paper, we write one up to 7 degrees. Also, by using this method, we derive some walls independent of their degrees. Second, we find a certain admissible form of them. In the proof of these facts, we introduce a matrix action on a structure group, which we call a similarity transformation, and we argue the relation between this action and ordered products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15470 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit forms in lower degrees of rank 2 cluster scattering diagrams Akagi, Ryota Combinatorics Algebraic Geometry In this paper, we study wall elements of rank 2 cluster scattering diagrams based on dilogarithm elements. We derive two major results. First, we give a method to calculate wall elements in lower degrees. By this method, we may see the explicit forms of wall elements including the Badlands, which is the complement of $G$-fan. In this paper, we write one up to 7 degrees. Also, by using this method, we derive some walls independent of their degrees. Second, we find a certain admissible form of them. In the proof of these facts, we introduce a matrix action on a structure group, which we call a similarity transformation, and we argue the relation between this action and ordered products. |
| title | Explicit forms in lower degrees of rank 2 cluster scattering diagrams |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2309.15470 |