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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.02614 |
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| _version_ | 1866917855780405248 |
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| author | Brown, Devin Elek, Balazs Halacheva, Iva |
| author_facet | Brown, Devin Elek, Balazs Halacheva, Iva |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cacti, Toggles, and Reverse Plane Partitions Brown, Devin Elek, Balazs Halacheva, Iva Combinatorics Representation Theory 05E10, 17B10 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. |
| title | Cacti, Toggles, and Reverse Plane Partitions |
| topic | Combinatorics Representation Theory 05E10, 17B10 |
| url | https://arxiv.org/abs/2412.02614 |