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On the subregular $$J$$-rings of Coxeter systems. (English) Zbl 1439.20041
Summary: 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 $$J_E$$ for each two-sided Kazhdan-Lusztig cell of $$W$$, and we study the summand $$J_C$$ corresponding to a particular cell $$C$$ called the subregular cell. We develop a combinatorial method involving truncated Clebsch-Gordan rules to compute $$J_C$$ without using the Kazhdan-Lusztig basis. As applications, we deduce some connections between $$J_C$$ and the Coxeter diagram of $$W$$, and we show that for certain Coxeter systems $$J_C$$ contains subalgebras that are free fusion rings in the sense of T. Banica and R. Vergnioux [J. Noncommut. Geom. 3, No. 3, 327–359 (2009; Zbl 1203.46048)], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16T20 Ring-theoretic aspects of quantum groups 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 20C08 Hecke algebras and their representations
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