Cyclic sieving, promotion, and representation theory
Cyclic sieving, promotion, and representation theory
复制标题
循环筛选、提升和表示理论
DOI：
10.1016/j.jcta.2009.03.017
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
B. Rhoades
中科院分区：
文献类型：
--
作者：
B. Rhoades
我们证明了White[D.White，Personal Communications，2007]的猜想，以及Abuzzahab，Korson，Li和Meyer[O.Abuzzahab，M.Korson，M.Li，S.Meyer，Cycle and Dihedral Screing for Plane Partitions，U.S.Minneota REU Report，2005]，以及Reiner和White[V.Reiner，Personal Communications，2007；D.White，Personal Communications，2007]关于Reiner，Stanton和White[V.Reiner，D.Stanton，D.White，The Cycle Screing现象，J.Combin，2007]的一些相关猜想的集合。理论系列。A 108(2004)]，因为它适用于长方形场景中的Jeu-de-taquin推广。为此，我们利用Kazhdan-Lusztig理论和C[x11，…]的对偶正则基的刻划，xnn]由于Skandera[M.Skandera，on Dual Canonical and Kazhdan-Lusztig base and 3412,4231-避免置换，2006年提交出版]。然后，我们将我们的结果推广到分析了由提升和疏散产生的矩形画面上的二面体作用的不动点，这表明二面体群可能存在筛选现象。最后，我们给出了这一理论在超八面体群的长元素和不相交划分中涉及约化词的循环筛选现象的应用。
We prove a collection of conjectures of White [D. White, personal communication, 2007], as well as some related conjectures of Abuzzahab, Korson, Li, and Meyer [O. Abuzzahab, M. Korson, M. Li, S. Meyer, Cyclic and dihedral sieving for plane partitions, U. Minnesota REU Report, 2005] and of Reiner and White [V. Reiner, personal communication, 2007; D. White, personal communication, 2007], regarding the cyclic sieving phenomenon of Reiner, Stanton and White [V. Reiner, D. Stanton, D. White, The cyclic sieving phenomenon, J. Combin. Theory Ser. A 108 (2004)] as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan–Lusztig theory and a characterization of the dual canonical basis of C[x11,…,xnn] due to Skandera [M. Skandera, On the dual canonical and Kazhdan–Lusztig bases and 3412, 4231-avoiding permutations, 2006, submitted for publication]. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.