Consider a finite Coxeter group \$W\$ with corresponding braid group \$B_W\$. The dual braid monoid of \$W\$ is a homogeneous monoid with group of fractions \$B_W\$. It was defined in general by D. Bessis, who gave a presentation for it based on the type \$A\$ case due to Birman, Ko, and Lee. This monoid has nice combinatorial properties, for instance the noncrossing partition lattice \$NC(W)\$ is naturally embedded in it.
In this talk we will study the \$\mathbf{k}\$-algebras \$\mathcal{A}(W)\$ of these monoids, \$\mathbf{k}\$ a field. These turn out to belong to the class of Koszul algebras. Furthermore, positive elements of the cluster complex attached to \$W\$ naturally index a family of elements of the algebra \$\mathcal{A}^{\dagger}(W)\$ , the Koszul dual of \$\mathcal{A}(W)\$). We will explain why this family forms (conjecturally) a basis of \$\mathcal{A}^{\dagger}(W)\$, and give a complete answer in type \$A\$.