Skew-growth function for dual Artin monoid

Artin group is the fundamental group of the compliment of the discriminant loci of classical Lie types, and is defined by the Artin braid relations. The monoid inside the Artin group generated by simple generators, called Artin monoid, is a lattice and is used to give a discripsion of the universal covering of the compliment of the discriminant. Recently, people found another lattice structure in the Artin group by using generators corresponding to all reflections, and call it the dual Artin monoid. We show that the skew growth function of the dual Artin monoid has exactly the rank number of zero loci on the interval $(0,1]$. The same statement for the original Artin monoid still remains to be a conjecture. Joint work with T. Ishibe.