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

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• 作者：Adriano Garsia^a ; ¹ ; ^{garsia@math.ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Emily Sergel Leven^a ; ² ; ^{esergel@ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Nolan Wallach^a ; ³ ; ^{nwallach@ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Guoce Xin^b ; ⁴ ; ^{guoce.xin@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Parking function ; Shuffle conjecture ; Plethysm
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：57-100
• 全文大小：653 K

文摘

Our main result here is that the specialization at t=1/q$t=1/q$ of the Q_km,kn$Q_{km,kn}$ 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$t=1/q$ 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)$(m,n)$ is a coprime pair then where as customarily, for any integer s≥0$s\ge 0$ and indeterminate u   we set [s]_u=1+u+⋯+u^s−1${\left[s\right]}_u=1+u+\cdots +u^{s-1}$. 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 km×kn$km\times kn$ lattice rectangle.