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

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).

38 pages, only small changes

References in corpus (5)

• 2-Kac-Moody algebras
• Graded decomposition numbers for cyclotomic Hecke algebras
• An invariant of tangle cobordisms via subquotients of arc rings
• sl(N)-Web categories
• Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)

Cited by in corpus (6)

• Deformations of colored sl(N) link homologies via foams
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• sl(N)-Web categories
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Trace and categorical sl(n) representations
• An Exceptional Splitting of Khovanov's Arc Algebras in Characteristic 2