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Star reducible Coxeter groups. (English) Zbl 1149.20034
Summary: We define “star reducible” Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan-Lusztig bases of these groups have a nice projection property to the Temperley-Lieb type quotient, and furthermore that the images of the basis elements \(C_w'\) (for fully commutative \(w\)) in the quotient have structure constants in \(\mathbb{Z}^{\geq 0}[v,v^{-1}]\). We also classify the star reducible Coxeter groups and show that they form nine infinite families with two exceptional cases.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20C08 Hecke algebras and their representations
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