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arXiv · 2412.02614

Cacti, Toggles, and Reverse Plane Partitions

Devin Brown·Balazs Elek·Iva Halacheva
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Abstract

The cactus group acts combinatorially on crystals via partial Schützenberger 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.

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Devin Brown, Balazs Elek, Iva Halacheva. 2024-12-03. Cacti, Toggles, and Reverse Plane Partitions. https://arxiv.org/abs/2412.02614

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