A diagrammatic categorification of the q-Schur algebra

Abstract

In this paper we categorify the q-Schur algebra S(n,d) as a quotient of Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our 2-category contains Soergel's monoidal category of bimodules of type A, which categorifies the Hecke algebra H(d), as a full sub-2-category if d does not exceed n. For the latter result we use Elias and Khovanov's diagrammatic presentation of Soergel's monoidal category of type A.

60 pages, lots of figures. v3: Substantial changes. To appear in Quantum Topology

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• 2-Kac-Moody algebras
• sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k)
• Equivariant colored sl(N)-homology for links
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• Annular Evaluation and Link Homology
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Trivalent vertices and bordered knot Floer homology in the standard basis
• Trace and categorical sl(n) representations
• A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$