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Webs and quantum skew Howe duality. (English) Zbl 1387.17027
Summary: We give a diagrammatic presentation in terms of generators and relations of the representation category of $$U_q(\mathfrak{sl}_n)$$. More precisely, we produce all the relations among $$\mathrm{SL}_n$$-webs, thus describing the full subcategory $$\otimes$$-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 G. Kuperberg] in [Commun. Math. Phys. 180, No. 1, 109–151 (1996; Zbl 0870.17005)] and affirms conjectures of Dongseok Kim in [Graphical calculus on representations of quantum lie algebras, Ph. D. thesis, University of California, Davis (2003), arxiv:math/0310143] and Scott Morrison in [A diagrammatic category for the representation theory of $$U_q(\mathfrak{sl}_n)$$. PhD thesis, University of California, Berkeley (2007), arxiv:0704.1503]. Our main tool is an application of quantum skew Howe duality. This is the published version of arxiv:1210.6437.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras
Zbl 0870.17005
Full Text:
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