A minimaj-preserving crystal on ordered multiset partitions
A minimaj-preserving crystal on ordered multiset partitions
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DOI：
10.1016/j.aam.2017.11.006
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发表时间：
2017-07
期刊：
影响因子：
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通讯作者：
G. Benkart;L. Colmenarejo;P. Harris;R. Orellana;G. Panova;A. Schilling;Martha Yip
中科院分区：
文献类型：
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作者：
G. Benkart;L. Colmenarejo;P. Harris;R. Orellana;G. Panova;A. Schilling;Martha Yip
我们提供了一个晶体结构上的有序多集分区，最近出现在追求的德尔塔猜想。这个猜想是由Haglund，Remmel和Wilson提出的，作为Shuffle猜想的推广。在Delta猜想的组合分析中出现了各种关于有序多集划分的统计，其中之一是minimaj统计，它是单词主要索引统计的变体。我们的晶体具有的性质，minimaj统计是恒定的连接组件的晶体。特别是，这产生了另一个证明舒尔积极性的分次Frobenius系列的推广R n，k由于Haglund，Rhoades和Shimozono的coinvariant代数R n。晶体结构也使我们能够证明的minimaj统计与有序多集分区的主要指标统计的等分布性。
We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization R n, k due to Haglund, Rhoades and Shimozono of the coinvariant algebra R n. The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.