Interpolation Polynomials, Bar Monomials, and Their Positivity

We prove a conjecture of Knop-Sahi on the positivity of interpolation polynomials, which is an inhomogeneous generalization of Macdonald's conjecture for Jack polynomials. We also formulate and prove the nonsymmetric version of this conjecture, and in fact, we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade.