Webs and quantum skew Howe duality

We give a diagrammatic presentation in terms of generators and relations of the representation category of U q ( sl n ) . More precisely, we produce all the relations among SL n -webs, thus describing the full subcategory ⊗ -generated by fundamental representations ⋀ k C n (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg in Commun Math Phys 180(1):109–151, ( 1996 ) and affirms conjectures of Kim in Graphical calculus on representations of quantum lie algebras, Ph. D. thesis, University of California, Davis, ( 2003 ) and Morrison in A Diagrammatic Category for the Representation Theory of U q sl n . PhD thesis, University of California, Berkeley, ( 2007 ). Our main tool is an application of quantum skew Howe duality. This is the published version of arXiv:1210.6437 .