Stranding $\mathfrak{sl}_n$ webs

TL;DR

This work provides a precise, computation-friendly framework for $\mathfrak{sl}_n$ webs by leveraging Fontaine's web graphs and the global combinatorial device of strandings. The authors introduce a state-sum construction $f(G)$ that yields $U_q(\mathfrak{sl}_n)$-invariant vectors from a Fontaine web $G$, prove invariance, and connect Fontaine and CKM web frameworks via explicit maps and signs. They establish a complete set of Fontaine relations, relate strandings to binary labelings, and demonstrate how to build basis webs from rectangular Young tableaux, including coherent structures and connections to promotion/evacuation. The approach yields nonvanishing criteria, a canonical base stranding, and a bridge to geometric objects like Springer fibers, culminating in a surjectivity result for $f$ and a kernel description via CKM relations. Collectively, these results enable explicit, scalable computations with $\mathfrak{sl}_n$ webs for $n\ge 4$, unify multiple web formalisms, and link diagrammatics to representation theory and geometry.

Abstract

Webs are a kind of planar, directed, edge-labeled graph that encode invariant vectors for quantum representations of $\mathfrak{sl}_n$. The theory of webs developed organically for $\mathfrak{sl}_2$, where they are also known as noncrossing matchings and the Temperley-Lieb algebra, before being formalized by Kuperberg for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ as the morphisms in a diagrammatic categorification of quantum representations called the spider category. Various models extend webs to $n \geq 4$. 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 ($U_q(\mathfrak{sl}_n)$-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.

Figures (32)

• Figure 1: A Fontaine web graph for $\mathfrak{sl}_4$ with three valid strandings
• Figure 2: The six flows on two strandings of an $\mathfrak{sl}_4$ web graph, corresponding to $x_1 \otimes x_2 \otimes x_1 \otimes x_3 \otimes x_2 \otimes x_3 \otimes x_4 \otimes x_4$ and $q^{-1}x_1 \otimes x_2 \otimes x_3 \otimes x_4 \otimes x_1 \otimes x_2 \otimes x_3 \otimes x_4$. On the left, blue, red, and green strands are $(1,2)$, $(2,3)$, and $(3,4)$ flows, respectively; and on the right, $(1,3), (2,4)$, and $(1,4)$ flows are shown in violet, orange, and teal, respectively
• Figure 3: A Fontaine web graph for $\mathfrak{sl}_4$
• Figure 4: Flipping edges in a Fontaine web graph
• Figure 5: A CKM web graph for $\mathfrak{sl}_4$

Theorems & Definitions (135)

• Remark 1
• Definition 2: Fontaine webs
• Example 3
• Remark 4
• Remark 5
• Definition 6
• Example 7
• Lemma 8
• proof
• Definition 9