Pfaffian systems of A-hypergeometric equations I: Bases of twisted cohomology groups

We consider bases of Pfaffian systems for A-hypergeometric systems. These are given by Gröbner deformations, they also provide bases for twisted cohomology groups. For a hypergeometric system associated with a class of order polytopes, these bases have a combinatorial description. The size of the bases associated with a subclass of the order polytopes has a growth rate of polynomial order.