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${𝔤 𝔩}_{n}$ -webs, categorification and Khovanov–Rozansky homologies

D.T.Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, Switzerland

E-mail Address: daniel.tubbenhauer@math.uzh.ch

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https://doi.org/10.1142/S0218216520500741Cited by:2 (Source: Crossref)

Abstract

In this paper, we define an explicit basis for the ${𝔤 𝔩}_{n}$ -web algebra $H_{n} (k)$ (the ${𝔤 𝔩}_{n}$ generalization of Khovanov’s arc algebra) using categorified $q$ -skew Howe duality.

Our construction is a ${𝔤 𝔩}_{n}$ -web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and $H_{n} (k)$ , and it gives an explicit graded cellular basis of the $2$ -hom space between two ${𝔤 𝔩}_{n}$ -webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky ${𝔤 𝔩}_{n}$ -link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$ .

Keywords:

AMSC: 57M25, 57M27, 17B10, 20C08

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