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

摘要

Our main result here is that the specialization at t = 1/q of the Q(km,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/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) is a coprime pair then