A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t = 1/q
A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t = 1/q
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一种新的体积对称函数算子和 t = 1/q 时的有理组合洗牌猜想
DOI：
10.1016/j.jcta.2016.07.001
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
G. Xin
中科院分区：
文献类型：
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作者：
A. Garsia;E. Leven;N. Wallach;G. Xin
这里我们的主要结果是，在t= 1/q的Q k m，k n算子研究Bergeron等人的专业化。[2]可以被赋予非常简单的体积增大形式。这一发现产生的几个身份有关这些运营商在t= 1/q的理性组合洗牌猜想Bergeron等人的初等和直接推导。[3]的文件。特别地，我们证明了如果m，n和k是正整数，并且（m，n）是互质对，则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]其中，对于任何整数s≥ 0和不确定的u，我们设置[s] u= 1+ u+ n + u s− 1。我们还表明，对称多项式的右手边总是舒尔积极的。利用有理组合洗牌猜想，我们得到了该多项式在km × kn格点矩形上用Parking函数表示的精确公式.
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≥ 0 and indeterminate u we set [s] u= 1+ u+⋯+ 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.