Toric Promotion with Reflections and Refractions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917375462342656 |
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| author | Adams, Ashleigh Defant, Colin Striker, Jessica |
| author_facet | Adams, Ashleigh Defant, Colin Striker, Jessica |
| contents | Inspired by recent work on refraction billiards in dynamics, we introduce a notion of refraction for combinatorial billiards. This allows us to define a generalization of toric promotion that we call toric promotion with reflections and refractions, which is a dynamical system defined via a graph $G$ whose edges are partitioned into a set of reflection edges and a set of refraction edges. This system is a discretization of a billiards system in which a beam of light can pass through, reflect off of, or refract through each toric hyperplane in a toric arrangement. Generalizing the main theorem known about toric promotion, we give a simple formula for the orbit structure of toric promotion with reflections and refractions when $G$ is a forest. We also completely describe the orbit sizes when $G$ is a cycle with an even number of refraction edges; this result is new even for ordinary toric promotion (i.e., when there are no refraction edges). When $G$ is a cycle of even size with no reflection edges, we obtain an interesting instance of the cyclic sieving phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03649 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric Promotion with Reflections and Refractions Adams, Ashleigh Defant, Colin Striker, Jessica Combinatorics Dynamical Systems 05A05, 05E18, 37E99 Inspired by recent work on refraction billiards in dynamics, we introduce a notion of refraction for combinatorial billiards. This allows us to define a generalization of toric promotion that we call toric promotion with reflections and refractions, which is a dynamical system defined via a graph $G$ whose edges are partitioned into a set of reflection edges and a set of refraction edges. This system is a discretization of a billiards system in which a beam of light can pass through, reflect off of, or refract through each toric hyperplane in a toric arrangement. Generalizing the main theorem known about toric promotion, we give a simple formula for the orbit structure of toric promotion with reflections and refractions when $G$ is a forest. We also completely describe the orbit sizes when $G$ is a cycle with an even number of refraction edges; this result is new even for ordinary toric promotion (i.e., when there are no refraction edges). When $G$ is a cycle of even size with no reflection edges, we obtain an interesting instance of the cyclic sieving phenomenon. |
| title | Toric Promotion with Reflections and Refractions |
| topic | Combinatorics Dynamical Systems 05A05, 05E18, 37E99 |
| url | https://arxiv.org/abs/2404.03649 |