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

TL;DR

This paper introduces a combinatorial framework for symmetric $ ext{sl}_2$-webs, establishing an equivalence with quantum $ ext{sl}_2$-modules using symmetric Howe duality, and offers new perspectives on Jones-Wenzl projectors and colored Jones polynomials.

Contribution

It constructs a symmetric $ ext{sl}_2$-web category equivalent to quantum $ ext{sl}_2$-modules via symmetric Howe duality, providing novel insights into Jones-Wenzl projectors.

Findings

Established a braided monoidal equivalence between symmetric $ ext{sl}_2$-webs and quantum $ ext{sl}_2$-modules.

Provided new interpretations of Jones-Wenzl projectors.

Connected symmetric Howe duality with the structure of colored Jones polynomials.

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.