The $(n,1)$-Reduced DKP Hierarchy, the String Equation and $W$ Constraints

The total descendent potential of a simple singularity satisfies the Kac–Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding $W$-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type $D$ in a different way, viz.as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the $W$ constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov–Schulman operators.