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arXiv · 1308.0566

The sl(N)-web algebras and dual canonical bases

Marco Mackaay
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Abstract

In this paper, which is a follow-up to my paper with Yonezawa "sl(N)-web categories", I define and study sl(N)-web algebras for any N greater than one. For N=2 these algebras are isomorphic to Khovanov's arc algebras and for N=3 they are Morita equivalent to the sl(3)-web algebras which I defined and studied with Pan and Tubbenhauer. The main result of this paper is that the sl(N)-web algebras are Morita equivalent to blocks of certain level-N cyclotomic KLR algebras, for which I use the categorified quantum skew Howe duality from the aforementioned paper with Yonezawa. Using this Morita equivalence and Brundan and Kleshchev's work on cyclotomic KLR algebras, I show that there exists an isomorphism between a certain space of sl(N)-webs and the split Grothendieck group of the corresponding sl(N)-web algebra, which maps the dual canonical basis elements to the Grothendieck classes of the indecomposable projective modules (with a certain normalization of their grading).

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Marco Mackaay. 2013-11-06. The sl(N)-web algebras and dual canonical bases. https://arxiv.org/abs/1308.0566

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