Cacti, Toggles, and Reverse Plane Partitions

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 $A$ and described via Bender-Knuth involutions. We prove an analogous result for the family of crystals $B(n\varpi_1)$ in type $D$. Our main tools are combinatorial toggles acting on reverse plane partitions of height $n$. 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.