2025/10/14 by Russell, Heather M., Tymoczko, Julianna
#05E10 #17B37 #57K16 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2510.12035
Webs are a kind of planar, directed, edge-labeled graph that encode invariant vectors for quantum representations of \mathfraksln. The theory of webs developed organically for \mathfraksl2, where they are also known as noncrossing matchings and the Temperley-Lieb algebra, before being formalized by Kuperberg for \mathfraksl2 and \mathfraksl3 as the morphisms in a diagrammatic categorification of quantum representations called the spider category. Various models extend webs to n ≥ 4. Only Cautis-Kamnitzer-Morrison prove a full set of relations for their webs, though Fontaine's webs are better adapted to computations, more graph-theoretically natural, and directly generalize webs for n=2 and n=3. This paper formalizes the theory of Fontaine's webs, proving the existence of a deep and powerful global structure on these webs called strandings. We do three key things: 1) give a state-sum formula to construct (Uq(\mathfraksln)-invariant) web vectors from the orientation of strandings on Fontaine's webs; 2) list and prove a complete set of relations, connecting strandings to the local data of binary labelings that are well-established in the literature; and 3) provide applications and examples of how strandings facilitate computations.