Abstract
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 \({\textstyle \bigwedge ^{k}_{}} {\mathbb C}^n\) (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg 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 \(U_q\left( {\mathfrak {sl}}_n\right) \). 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.
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Notes
The fullness of \(\Gamma _n\) was proven in Proposition 3.5.8 of [15] using Schur-Weyl duality instead of skew Howe duality, but the argument is essentially the same.
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Acknowledgments
The authors benefited from discussions with Arkady Berenstein, Bruce Fontaine, Stavros Garoufalidis, Greg Kuperberg, Valerio Toledano Laredo and Sebastian Zwicknagl. We would like to thank Dongho Moon who pointed out that the relation of Equation (2.8) was missing in the first 3 versions posted on the arXiv! S.C. was supported by NSF grant DMS-1101439 and the Alfred P. Sloan foundation, S.M. was supported by the Australian Research Council grant DE120100232 and by DOD-DARPA HR0011-12-1-0009, J.K. was supported by NSERC. We would also like to thank VIA Rail for providing the venue where much of this research was carried out.
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Cautis, S., Kamnitzer, J. & Morrison, S. Webs and quantum skew Howe duality. Math. Ann. 360, 351–390 (2014). https://doi.org/10.1007/s00208-013-0984-4
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DOI: https://doi.org/10.1007/s00208-013-0984-4