A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t = 1/q

Our main result here is that the specialization at t=1/qt=1/q of the Qkm,knQkm,kn operators studied in Bergeron et al. [2] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at t=1/qt=1/q to the Rational Compositional Shuffle conjecture of Bergeron et al. [3]. In particular we show that if m, n and k   are positive integers and (m,n)(m,n) is a coprime pair thenq(km−1)(kn−1)+k−12Qkm,kn(−1)kn|t=1/q=[k]q[km]qekm[X[km]q] where as customarily, for any integer s≥0s≥0 and indeterminate u   we set [s]u=1+u+⋯+us−1[s]u=1+u+⋯+us−1. We also show that the symmetric polynomial on the right hand side is always Schur positive. Moreover, using the Rational Compositional Shuffle conjecture, we derive a precise formula expressing this polynomial in terms of Parking Functions in the km×knkm×kn lattice rectangle.