Positivity results for the Hecke algebras of noncrystallographic finite Coxeter groups

This paper is a report on a computer check of some important positivity properties of the Hecke algebra in type H4, including the nonnegativity of the structure constants in the Kazhdan–Lusztig basis. This answers a long-standing question of Lusztig's. The same algorithm, carried out by hand, also allows us to deal with the case of dihedral Coxeter groups.