A new approach to the $SL_n$ spider

TL;DR

This paper introduces a new diagrammatic framework called a spider that extends the existing $SL_n$ spider, using combinatorial data from the root system of $SL_n$ to define generators and relations.

Contribution

It presents a novel, combinatorially defined spider that encompasses the $SL_n$ spider, providing a new tool for understanding the representation category of $U_q(sl_n)$.

Findings

Defines a new $SL_n$-containing spider with generators and relations

Uses combinatorial data from the root system of $SL_n$

Simplifies the diagrammatic encoding 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 $SL_n$.