A new approach to the $SL_n$ spider

TL;DR

The paper introduces a new diagrammatic framework—the cobweb spider—that contains the existing $SL_n$ spider and is built from root-system combinatorics of $rak{sl}_n$. It defines a state-sum map from $SL_n$ webs to cobwebs and proves this map is well-defined and injective, leveraging the known equivalence between the $SL_n$ spider and $ ext{Rep}(U_q(rak{sl}_n))$. The cobweb calculus is shown to be nontrivial and compatible with key web relations, with explicit checks for tag operations, $I=H$-type relations, and bubble-splitting moves like bursting a digon and a square. The construction suggests a simpler, more natural presentation of the quantum-group spider via root combinatorics and points toward connections with graph planar algebras and Ocneanu cells, offering a potential path to purely diagrammatic understanding of quantum group representations.

Abstract

The $SL_n$ spider gives a diagrammatic way to encode the representation category of the quantum group $U_q(sl_n)$. The aim of this paper is to define a new spider that contains the $SL_n$ spider. The new spider is defined by generators and relations, according to fairly simple rules that start with combinatorial data coming from the root system of $SL_n$.

Figures (6)

• Figure 1: The four kinds of vertex in a web.
• Figure 2: Convention for removing a strand labeled $n$.
• Figure 3: Web relations. Some labels are omitted, but can be deduced.
• Figure 4: Virtual crossings and bivalent vertices in a cobweb.
• Figure 5: Cobweb relations. Unlabeled strands can have any consistent orientations and labels.

Theorems & Definitions (10)

• Remark
• Proposition 1
• proof
• Proposition 2
• proof
• Remark
• Theorem 1
• proof
• Theorem 2
• proof