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.

A few extra words

A classical result of Rumer, Teller and Weyl, modernly interpreted, states that the so-called Temperley-Lieb category \(\mathcal{TL}\) describes the full subcategory of quantum \(\mathfrak{sl}_2\)-modules generated by tensor products of the 2-dimensional vector representation V of quantum \(\mathfrak{sl}_2\).
By Karoubi completion, we get the whole category of finite dimensional \(\mathfrak{sl}_2\)-modules. Thus, it is a striking question if one can give a diagrammatic description of \(\mathbf{KAR}(\mathcal{TL})\) as well.
We provide a new diagrammatic description of the entire category of finite dimensional quantum \(\mathfrak{sl}_2\)-modules.
To this end, we introduced our new description of the representation theory of quantum \(\mathfrak{sl}_2\), the category of symmetric \(\mathfrak{sl}_2\)-webs akin to the category of “usual” \(\mathfrak{sl}_n\)-webs studied by many people (recall that these “usual” \(\mathfrak{sl}_n\)-webs give a diagrammatic presentation of the full subcategory of all finite dimensional \(\mathfrak{sl}_n\)-modules whose objects are finite tensor products of the fundamental \(\mathfrak{sl}_n\)-representations.
Our main tool is the usage of symmetric q-Howe duality.
In particular, the Jones-Wenzl projectors are included in our picture, but without any recursive formula. Namely, they are directly given as below.
Another corollary of our construction is a “MOY-calculus” for colored Jones polynomials.