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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
Software:
GitHub; SageMath
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