Stranding $\mathfrak{sl}_n$ webs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913175291559936 |
|---|---|
| author | Russell, Heather M. Tymoczko, Julianna |
| author_facet | Russell, Heather M. Tymoczko, Julianna |
| contents | Webs are planar directed graphs that encode invariant vectors for tensor products of fundamental $U_q(\mathfrak{sl}_n)$-representations. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, web calculus is governed by effective reduction rules and well-understood reduced-web bases, but these features break down in higher rank. In this paper we develop a global combinatorial framework for untagged $\mathfrak{sl}_n$ webs, called strandings, which organizes local labeling data into systems of colored directed paths on the web graph.
Our main result is an explicit state-sum formula for the invariant vector of an untagged web as a weighted sum over its valid strandings. The usual labeling-based contruction requires a choice of decomposition into elementary web pieces and a vertex-by-vertex coefficient calculation. Our global reformulation avoids both. We prove that the vectors produced by strandings are $U_q(\mathfrak{sl}_n)$-invariant, compare the untagged theory with the tagged-web framework of Cautis, Kamnitzer, and Morrison, and use this comparison to obtain a complete set of relations for untagged web graphs. We also give applications of strandings to nonvanishing results, basis constructions from tableaux, and connections with Springer-theoretic combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stranding $\mathfrak{sl}_n$ webs Russell, Heather M. Tymoczko, Julianna Representation Theory Combinatorics 05E10, 17B37, 57K16 Webs are planar directed graphs that encode invariant vectors for tensor products of fundamental $U_q(\mathfrak{sl}_n)$-representations. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, web calculus is governed by effective reduction rules and well-understood reduced-web bases, but these features break down in higher rank. In this paper we develop a global combinatorial framework for untagged $\mathfrak{sl}_n$ webs, called strandings, which organizes local labeling data into systems of colored directed paths on the web graph. Our main result is an explicit state-sum formula for the invariant vector of an untagged web as a weighted sum over its valid strandings. The usual labeling-based contruction requires a choice of decomposition into elementary web pieces and a vertex-by-vertex coefficient calculation. Our global reformulation avoids both. We prove that the vectors produced by strandings are $U_q(\mathfrak{sl}_n)$-invariant, compare the untagged theory with the tagged-web framework of Cautis, Kamnitzer, and Morrison, and use this comparison to obtain a complete set of relations for untagged web graphs. We also give applications of strandings to nonvanishing results, basis constructions from tableaux, and connections with Springer-theoretic combinatorics. |
| title | Stranding $\mathfrak{sl}_n$ webs |
| topic | Representation Theory Combinatorics 05E10, 17B37, 57K16 |
| url | https://arxiv.org/abs/2510.12035 |