Abstract
We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from Korff, C., Stroppel, C.: The \(\hat {\mathfrak {sl}(n)_{k}}\)-WZNW fusion ring: a combinato-rial construction and a realisation as quotient of quantum cohomology. Adv. Math. 225(1), 200–268, (2010) and give a similar description of the \(\mathfrak {sp}_{2n}\)-fusion ring in terms of non-commutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings. Finally we also compute the fusion rings for type G 2.
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HHA was supported by the Danish National Research Foundation center of Excellence, Center for Quantum Geometry of Moduli Spaces (QGM); and CS by a visiting professorship at Chicago university. We thank Troels Bak Andersen and Stephen Griffeth for comments on a preliminary version of the paper.
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Andersen, H.H., Stroppel, C. Fusion Rings for Quantum Groups. Algebr Represent Theor 17, 1869–1888 (2014). https://doi.org/10.1007/s10468-014-9479-6
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DOI: https://doi.org/10.1007/s10468-014-9479-6