We define and study the category of symmetric -webs. This category is a combinatorial description of the category of all finite dimensional quantum -modules. Explicitly, we show that (the additive closure of) the symmetric -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 describes the full subcategory of quantum -modules generated by tensor products of the -dimensional vector representation of quantum .
By Karoubi completion, we get the whole category of finite dimensional -modules. Thus, it is a striking question if one can give a diagrammatic description of as well.
We provide a new diagrammatic description of the entire category of finite dimensional quantum -modules.
To this end, we introduced our new description of the representation theory of quantum , the category of symmetric -webs akin to the category of “usual” -webs studied by many people (recall that these “usual” -webs give a diagrammatic presentation of the full subcategory of all finite dimensional -modules whose objects are finite tensor products of the fundamental -representations.
Our main tool is the usage of symmetric -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.