Stranding $\mathfrak{sl}_n$ webs

Abstract

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.