On heights of characters of finite groups.(English)Zbl 1481.20029
Summary: 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 \(t \geq 3\) 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 \(p\).
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