Cacti, Toggles, and Reverse Plane Partitions
Devin Brown, Balazs Elek, Iva Halacheva

TL;DR
This paper extends the understanding of cactus group actions on crystals from type A to type D, using combinatorial toggles on reverse plane partitions, and confirms conjectures about generators of the full action.
Contribution
It proves an analogous cactus group action result for type D crystals using combinatorial toggles, addressing prior conjectures about generators.
Findings
Cactus group acts on type D crystals via combinatorial toggles.
Length one and two subdiagram elements generate the full cactus action.
Addresses conjectures of Dranowski, Kamnitzer, and Morton-Ferguson.
Abstract
The cactus group acts combinatorially on crystals via partial Sch\"utzenberger involutions. This action has been studied extensively in type and described via Bender-Knuth involutions. We prove an analogous result for the family of crystals in type . Our main tools are combinatorial toggles acting on reverse plane partitions of height . As a corollary, we show that the length one and two subdiagram elements generate the full cactus action, addressing conjectures of Dranowski, the second author, Kamnitzer, and Morton-Ferguson.
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Taxonomy
TopicsBotanical Research and Applications
