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<p>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 <i>toric promotion with reflections and refractions</i>, which is a dynamical system defined via a graph <i>G</i> whose edges are partitioned into a set of <i>reflection edges</i> and a set of <i>refraction edges</i>. 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 <i>G</i> is a forest. We also completely describe the orbit sizes when <i>G</i> 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 <i>G</i> is a cycle of even size with no reflection edges, we obtain an interesting instance of the cyclic sieving phenomenon.</p>

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Toric promotion with reflections and refractions

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Ashleigh Adams,
  • Colin Defant,
  • Jessica Striker

摘要

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.