Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

Abstract

We define and study the category of symmetric ${\mathfrak{sl}}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum ${\mathfrak{sl}}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric ${\mathfrak{sl}}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.

32 pages, lots of figures, comments welcome

References in corpus (2)

• Knot invariants and higher representation theory
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality

Cited by in corpus (7)

• Super $q$-Howe duality and web categories
• Cellular structures using ${\text{U}}_q$-tilting modules
• SL2 tilting modules in the mixed case
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• On a symplectic quantum Howe duality
• On rank one 2-representations of web categories
• Webs of Type P