Categorical geometric skew Howe duality

By Sabin Cautis, Joel Kamnitzer, Anthony M. Licata. 2009

51 citations recorded by the source corpus.

AbstractWe categorify the R-matrix isomorphism between tensor products of minuscule representations of $U_{q}({\mathfrak{sl}}_{n})$ by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of $U_{q}({\mathfrak{sl}}_{2})$ which are related to representations of $U_{q}({\mathfrak{sl}}_{n})$ by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e543bde2f955533e1234a824f97e7d2ef42f275c

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In Geometry and Topology

1. David E. V. Rose
2. Hoel Queffelec
3. Sabin Cautis
4. Aaron D. Lauda
5. Paul Wedrich
6. Joel Kamnitzer
7. Anthony M. Licata
8. Ben Webster

In Mathematical Physics

1. David E. V. Rose
2. Hoel Queffelec
3. Sabin Cautis
4. Aaron D. Lauda
5. Paul Wedrich
6. Joel Kamnitzer
7. Anthony M. Licata
8. S. Morrison

Citations in the corpus

• COHERENT SHEAVES AND CATEGORICAL sl2 ACTIONS (2009, methodology): “In [ CKL1 ] we gave an example of a geometric categorical sl2 action which is essentially a natural compactication of the one here.”
• Implicit structure in 2-representations of quantum groups (2011, background): “The geometric categorical sl2 action on ⊕N k=0 DCoh(T G(k,N)) defined in [6] and [5] induces a 2-representation U̇(sl2) on K where: • the objects of K are the derived categories of coherent sheaves DCoh(T G(k,N)), • the 1-morphisms of K are the Fourier-Mukai kernels in DCoh(T G(k + r,N)× T G(k,N)), ”
• Heisenberg categorification and Hilbert schemes (2010, background): “These computations are analogous to those from [4] and [3] in the setting of categorical sln-actions.”
• Braiding via geometric Lie algebra actions (2010, background): “e (n = 2 case). A similar description of Ti is possible in general. In the rest of this section we prove Theorem 3.1. 3.3. Proof of sl2 conditions (i) - (vii). In [CKL2] (based on the computations in [CKL1]) we proved that the sl2 relations (i) - (vii) hold for cotangent bundles to Grassmannians (i.”
• Clasp technology to knot homology via the affine Grassmannian (2012, methodology): “ies to categorify the sub-module Λ∞ q (C2∞)⊗m. These two approaches are closely related and give the same homological knot invariants. The main tool we use is skew Howe duality, building on work from [CKL1]. The key technical construction is the categorificationof the Jones-Wenzl projectors and their”
• Deformations of colored sl(N) link homologies via foams (2015, background): “Recall that work of Cautis-Kamnitzer-Licata [CKL] and Cautis-Kamnitzer-Morrison [CKM] shows that the commuting (skew Howe dual) actions of quantum glm and slN on the vector spaces∧k q (C m q ⊗C N q ) induce a functor ϕm : Uq(glm) → Rep(Uq(slN )) which sends a glm weight a = (a1, . . . , am) to the t”
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, methodology): “Cautis, Kamnitzer, and Licata related the braiding of fundamental Uq(sln) representations to the Weyl group action using a version of Howe duality for exterior algebras they called skew Howe duality [15].”
• The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “In [8], the authors show that the slm quantum Weyl group elements correspond under skew Howe duality to the braiding on Uq(sln) representations given by the R-matrix.”
• Derived equivalences for cotangent bundles of Grassmannians via categorical 𝔰𝔩2 actions (2009, background): “sheaf. Similar geometric settings for equivalences constructed from strong categorical sl 2 actions, including convolutions of affine Schubert varieties and Nakajima quiver varieties, are considered in [CKL1] and [CKL3], respectively. 1.4. Acknowledgements. First and foremost, we would like to thank J”
• Webs and quantum skew Howe duality (2012, result): “ztig’s quantum group element T2U_ q(gl 2) is taken to the R-matrix braidings : V k qC n V l qC n! V l q C n V k q C n. This last fact was previously proven by the rst and second authors and Licata in [CKL10], following the approach from [TL02] who established the analogous result for symmetric Howe ”