${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies

Abstract

In this paper we define an explicit basis for the ${\mathfrak{gl}}_n$-web algebra $H_n(\vec{k})$ (the ${\mathfrak{gl}}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a ${\mathfrak{gl}}_n$-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 $H_n(\vec{k})$, and it gives an explicit graded cellular basis of the $2$-hom space between two ${\mathfrak{gl}}_n$-webs. We use this to give a (in principle) computable version of colored Khovanov-Rozansky ${\mathfrak{gl}}_n$-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only $F$.

New version with additional results and an improved presentation, following a referee report. 59 pages, lots of figures, comments welcome. To appear in J. Knot Theory Ramifications

References in corpus (18)

• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Graded decomposition numbers for cyclotomic Hecke algebras
• Knot invariants and higher representation theory
• Derived equivalences for symmetric groups and sl\_2-categorification
• Category O and sl(k) link invariants
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• sl(N)-Web categories
• Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)
• Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of Type A
• Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)
• Matrix factorizations and colored MOY graphs
• Categorified sl(N) invariants of colored rational tangles
• Indecomposable 1-morphisms of \dot{U}^+_3 and the canonical basis of U_q^+(sl_3)
• The sl(N)-web algebras and dual canonical bases
• A large family of indecomposable projective modules for the Khovanov-Kuperberg algebra of $s{l}_3$-webs
• A characterisation of indecomposable web-modules over Khovanov-Kuperberg Algebras
• ${\mathfrak{sl}}_3$-web bases, intermediate crystal bases and categorification

Cited by in corpus (5)

• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Cellularity and subdivision of KLR and weighted KLRW algebras
• Drinfel'd doubles of the $n$-rank Taft algebras and a generalization of the Jones polynomial
• Real Springer fibers and odd arc algebras
• A formula to evaluate type A webs and link polynomials