Webs and quantum skew Howe duality

By Sabin Cautis, Joel Kamnitzer, S. Morrison. 2012

249 citations recorded by the source corpus.

We give a diagrammatic presentation in terms of generators and relations of the representation category of $$U_q({\mathfrak {sl}}_n) $$Uq(sln). More precisely, we produce all the relations among $$\mathrm{{SL}}_n$$SLn-webs, thus describing the full subcategory $$\otimes $$⊗-generated by fundamental representations $${\textstyle \bigwedge ^{k}_{}} {\mathbb C}^n$$⋀kCn (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 $$U_q\left( {\mathfrak {sl}}_n\right) $$Uqsln. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/58cff9c76cd1085fe2cf38eb10959fa6585019d4

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Citations in the corpus

Light ladders and clasp conjectures (2015, unknown): “sting patterns appeared in this computation, giving hope that something may be adapted to the general case. We can now derive the ﬁnal recursion relations for n= 4. They are: (3.35) κ(b,c,d),(0101) = [3]κ(b,c,d−1),(0100) − κ(b,c+1,d−2),(0110) κ(b,c,d−1),(1101) − κ2 (b,c,d−1),(0100) κ(b,c,d−1),(0111)”
The finiteness conjecture for skein modules (2019, background): “The first such description was given in [61] for G = SL3; the construction was generalized in [30,84] to G = SLN .”
Functoriality of colored link homologies (2017, background): “They can be seen — in a straightforward way — as a categorification of the Murakami–Ohtsuki–Yamada state-sum model [41] for the Reshetikhin–Turaev link invariants of type A, or, equivalently, the graphical calculus for the corresponding representation category, see Cautis–Kamnitzer–Morrison [11].”
Deformations of colored sl(N) link homologies via foams (2015, background): “Using this splitting of the deformed cobordism category, Bar-Natan and Morrison compute a decomposition for the chain complexes arising in the definition of the deformed link invariant, whose objects are planar 1-manifolds that arise as resolutions of the link diagram.”
Clasp technology to knot homology via the affine Grassmannian (2012, methodology): “′ k1i, E (ik) k 1i and 1iF (ik) k used above to deﬁne crossings, cups and caps are equal to the Reshetikhin-Turaev maps [RT] up to sign and multiplication by some power of q. Proof. This follows from [CKM] where we construct a functor from Uq(sln) (thought of as a category where the objects are weig”
Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality (2013, unknown): “The quantum (but non-categorified) version can be found in [52], see also [22], and a partial categorification (with only one of the two commuting actions) can be found in [69], [51].”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Clark-Morrison-Walker sl3 foams.”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “Our new conventions are chosen to match with both the categorified quantum group and the standard choice of basis for Cq , and can be obtained from those in [11] by turning their diagrams sideways and rotating about a horizontal axis.”