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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 $J_E$ 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 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 [Banica], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.
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Idempotents, traces, and dimensions in Hecke categories
The paper provides closed formulas for recursible local intersection forms and recursive partial trace formulas that reduce categorical dimensions in asymptotic Hecke categories to diagrammatic computations.
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