Our main result here is that the specialization at
t=1/q of the
Qkm,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
e30d03eead71033e8dd933c78819d88" title="Click to view the MathML source">(m,n) is a coprime pair then
where as customarily, for any integer
s≥0 and indeterminate
u we set
[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×kn lattice rectangle.