
Symmetric webs, JonesWenzl recursions and Howe duality
Abstract 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 JonesWenzl projectors and the colored Jones polynomials.A few extra words A classical result of Rumer, Teller and Weyl, modernly interpreted, states that the socalled TemperleyLieb 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 JonesWenzl 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 “MOYcalculus” for colored Jones polynomials. 
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Last update: 20.01.2018 or later ·
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