Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality Title: Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality Authors: David Rose and Daniel Tubbenhauer Status: Int. Math. Res. Not. (IMRN), 2016-17 (2016), 5249-5290. Last update: Mon, 5 Jan 2015 16:33:17 GMT ArXiv link: http://arxiv.org/abs/1501.00915 ArXiv version differs from the published version LaTex Beamer presentation: Slides1, Slides2, Slides3, Slides4, Slides5, Slides6, Slides7, Slides8 Poster: Poster 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.   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. NEWS I am still a fool. My paper got accepted. The arXiv version of this paper was updated. My paper got accepted. The arXiv version of this paper was updated. "There are two ways to do mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent." - based on a quotation from Raoul Bott. Upcoming event where you can meet me: Visit Faro Click
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