This is an exposition of some recent developments related to the object in the title, particularly the computation of the Gromov-Witten invariants of the flag manifold [5] and the quadratic algebra approach [6]. The notes are largely based on the papers [5] and [6], authored jointly with S.~Gelfand, A.~N.~Kirillov, and A.~Postnikov. This is by no means an exhaustive survey of the subject, but rather a casual introduction to its combinatorial aspects.