On heights of characters of finite groups
On heights of characters of finite groups
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关于有限群的特征的高度
DOI：
10.1016/j.jalgebra.2020.02.035
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发表时间：
2020
影响因子：
0.9
通讯作者：
Zhang Jiping
中科院分区：
文献类型：
--
作者：
Feng Zhicheng;Liu Yanjun;Zhang Jiping
本文首先提出了有限群的p-块中特征标的最大高度下界的一个猜想。然后证明了我们的猜想对散在单群的覆盖群的所有块，对G/Z（G）同构于A6，A7的拟单群G或具有例外覆盖群的李型单群的所有块，对对称群的所有块，以及对有限一般线性群或酉群的所有块都成立.为了证明对称群的情形，它涉及Olsson关于t-核划分存在性的一个公开问题，其中t≥ 3是整数。作为副产品，我们对对称群和一般线性或酉群的特征标高度的研究也为Isaacs-Moretó-Navarro-Tiep猜想提供了证据，该猜想声称任意有限群G的Sylow p-子群P的不同不可约特征标度的数目至多比G的不可约特征标度是p的倍数的数目多1。
In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z (G) isomorphic to A 6, A 7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group, as well as for all blocks of finite general linear or unitary groups. For the proof of the case with symmetric groups, it relates to an open question of Olsson on the existence of t-core partitions, where t≥ 3 is an integer. As a byproduct, our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the Isaacs-Moretó-Navarro-Tiep Conjecture, claiming that the number of distinct irreducible character degrees of a Sylow p-subgroup P of an arbitrary finite group G is at most one more than the number of irreducible character degrees of G that are multiples of p.