Webs and quantum skew Howe duality

TL;DR

This paper provides a complete diagrammatic presentation of the representation category of quantum sl_n, using webs and quantum skew Howe duality, confirming several longstanding conjectures and questions in the field.

Contribution

It offers the first full set of relations for SL_n-webs, describing the subcategory generated by fundamental representations, and applies quantum skew Howe duality to achieve this.

Findings

All relations among SL_n-webs are explicitly described.

The subcategory generated by fundamental representations is fully characterized.

The results affirm conjectures by Kim and Morrison.

Abstract

We give a diagrammatic presentation in terms of generators mod relations of the representation category of $U_q(\mathfrak{sl}_n)$. More precisely, we produce all the relations among $\rm{SL}_n$-webs, thus describing the full subcategory tensor-generated by fundamental representations $\bigwedge^k \mathbb{C}^n$ (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim [arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an application of quantum skew Howe duality.