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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.21989 |
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| _version_ | 1866917041936531456 |
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| author | Cowan, Lucas Adams Eilfort, Ronja Seekamp, Kerry Tymoczko, Julianna |
| author_facet | Cowan, Lucas Adams Eilfort, Ronja Seekamp, Kerry Tymoczko, Julianna |
| contents | Web graphs form a family of planar directed graphs with boundary that can be used to model quantum $\mathfrak{sl}_n$-invariant vectors. Standard Young tableaux on an $n \times k$ rectangle naturally index a basis for $\mathfrak{sl}_n$ web graphs. We prove that evacuation of the tableau $T$ corresponds to reflection of the associated web graph $w_T$ up to equivalence under a specific set of edge-flip relations. This extends a result of Patrias and Pechenik for the cases $n=2,3$ and mirrors analogous results about rotation of web graphs corresponding to promotion of tableau by Peterson-Pylyavskyy-Rhoades for $n=3$ and Gaetz-Pechenik-Pfannerer-Striker-Swanson for $n=4$. We use an intermediate object called a multicolored noncrossing matching, which is closely related to the notion of strandings recently introduced by Russell and the fourth author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21989 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Evacuation of rectangular standard Young tableaux corresponds to reflection of $\mathfrak{sl}_n$ webs Cowan, Lucas Adams Eilfort, Ronja Seekamp, Kerry Tymoczko, Julianna Combinatorics Representation Theory 05C10, 05C25, 05A18, 20G42 Web graphs form a family of planar directed graphs with boundary that can be used to model quantum $\mathfrak{sl}_n$-invariant vectors. Standard Young tableaux on an $n \times k$ rectangle naturally index a basis for $\mathfrak{sl}_n$ web graphs. We prove that evacuation of the tableau $T$ corresponds to reflection of the associated web graph $w_T$ up to equivalence under a specific set of edge-flip relations. This extends a result of Patrias and Pechenik for the cases $n=2,3$ and mirrors analogous results about rotation of web graphs corresponding to promotion of tableau by Peterson-Pylyavskyy-Rhoades for $n=3$ and Gaetz-Pechenik-Pfannerer-Striker-Swanson for $n=4$. We use an intermediate object called a multicolored noncrossing matching, which is closely related to the notion of strandings recently introduced by Russell and the fourth author. |
| title | Evacuation of rectangular standard Young tableaux corresponds to reflection of $\mathfrak{sl}_n$ webs |
| topic | Combinatorics Representation Theory 05C10, 05C25, 05A18, 20G42 |
| url | https://arxiv.org/abs/2510.21989 |