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On the Subregular J-Rings of Coxeter Systems

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Abstract

We recall Lusztig’s construction of the asymptotic Hecke algebra J of a Coxeter system (W,S) via the Kazhdan–Lusztig basis of the corresponding Hecke algebra. The algebra J has a direct summand JE for each two-sided Kazhdan–Lusztig cell of W, and we study the summand JC corresponding to a particular cell C called the subregular cell. We develop a combinatorial method involving truncated Clebsch–Gordan rules to compute JC without using the Kazhdan–Lusztig basis. As applications, we deduce some connections between JC and the Coxeter diagram of W, and we show that for certain Coxeter systems JC contains subalgebras that are free fusion rings in the sense of Banica and Vergnioux (Noncommut. Geom. 3(3), 327–359, 2009), thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.

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Acknowledgements

It is my great pleasure to thank Victor Ostrik for numerous helpful suggestions. I am very grateful to Alexandru Chirvasitu and Amaury Freslon for helpful discussions about free fusion rings and compact quantum groups. I would also like to acknowledge the mathematical software SageMath [11], which was used extensively in our computations.

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    Tianyuan Xu

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  1. Tianyuan Xu
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Presented by: Peter Littelmann

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Xu, T. On the Subregular J-Rings of Coxeter Systems. Algebr Represent Theor 22, 1479–1512 (2019). https://doi.org/10.1007/s10468-018-9829-x

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