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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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