Our main result here is that the specialization at $t=1/q$ of the $Q_{km,kn}$operators studied in [4] may be given a very simple plethystic form. Thisdiscovery yields elementary and direct derivations of several identitiesrelating these operators at $t=1/q$ to the Rational Compositional Shuffleconjecture of [3]. In particular we show that if $m,n $ and $k$ are positiveintegers and $(m,n)$ is a coprime pair then $$ q^{(km-1)(kn-1)+k-1\over 2}Q_{km,kn}(-1)^{kn}\Big|_{t=1/q} \,=\, \textstyle{[k]_q\over [km]_q} e_{km}\big[X[km]_q\big] $$ where as customarily, for any integer $s \geq 0$ andindeterminate $u$ we set $[s]_u=1+u+\cdots +u^{s-1}$. We also show that thesymmetric polynomial on the right hand side is always Schur positive. Moreover,using the Rational Compositional Shuffle conjecture, we derive a preciseformula expressing this polynomial in terms of Parking functions in the$km\times kn$ lattice rectangle.
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机译：我们在这里的主要结果是，可以对在[4]中研究的$ Q_ {km，kn} $运算符在$ t = 1 / q $处的专业化给出非常简单的赋形形式。这一发现产生了将这些算子在$ t = 1 / q $处与[3]的有理成分混洗猜想相关的几个恒等式的基本和直接推导。特别地，我们表明如果$ m，n $和$ k $是正整数，而$（m，n）$是互质对，则$$ q ^ {（km-1）（kn-1）+ k-1 \超过2} Q_ {km，kn}（-1）^ {kn} \ Big | _ {t = 1 / q} \，= \，\ textstyle {[k] _q \ over [km] _q} e_ {km } \ big [X [km] _q \ big] $$，通常，对于任何整数$ s \ geq 0 $并不确定$ u $，我们设置$ [s] _u = 1 + u + \ cdots + u ^ {s- 1} $。我们还表明，右侧的不对称多项式始终为Schur正。此外，使用有理成分混洗猜想，我们推导出了一个精确公式，该公式用$ km \ times kn $格子矩形中的Parking函数表示该多项式。
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