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Symmetric Webs, Jones-Wenzl recursions, and $$q$$-Howe duality. (English) Zbl 1404.17025
Summary: We define and study the category of symmetric $$\mathfrak{sl}_2$$-webs. This 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 category of symmetric $$\mathfrak{sl}_{2}$$-webs 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 colored Jones polynomials.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 18B99 Special categories 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
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