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 / q of the Q k m , k n 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 q ( k m - 1 ) ( k n - 1 ) + k - 1 2 Q k m , k n ( - 1 ) k n | t = 1 / q = k q k m q e k m X k m q where as customarily, for any integer s ź 0 and indeterminate u we set s u = 1 + u + ź + u s - 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 k m × k n lattice rectangle.