Star Reducible Coxeter Groups

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 . We also classify the star reducible Coxeter groups and show that they form nine infinite families with two exceptional cases.