sln-webs, categorification and Khovanov-Rozansky homologies

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  • Daniel Tubbenhauer, Denmark
In this paper we define an explicit basis for the \mathfrak{sl}_n-web algebra H_n(\vec{k}), the \mathfrak{sl}_n generalization of Khovanov's arc algebra H_{2}(m), using categorified q-skew Howe duality.

Our construction, which can be seen as a \mathfrak{sl}_n-web version of Hu and Mathas graded cellular basis for the cyclotomic KL-R algebras of type A, has two major applications. The first is that it is a graded cellular basis. The second is that it can be explicitly computed for any \mathfrak{sl}_n-web w=v^*u which gives a basis of the corresponding 2-hom space between u and v. We use this fact to give a (in principle) computable version of Khovanov-Rozansky \mathfrak{sl}_n-link homology. The complex we define for this purpose can be realized in the KL-R setting and needs only F's and no E's.

Moreover, we discuss some application of our construction on the uncategorified level related to dual canonical bases of the \mathfrak{sl}_n-web space W_n(\vec{k}) and the MOY-calculus. Latter gives rise to a method to compute colored Reshetikhin-Turaev \mathfrak{sl}_n-link polynomials.
Original languageEnglish
PublisherarXiv.org
Number of pages95
Publication statusPublished - 2014

    Research areas

  • Categorification, Categorial representation theory, Combinatorial representation theory, Quantum groups, Link homologies

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