On the degrees of representations of groups not divisible by 2k

Abstract

Let [Formula: see text] be a finite group. For a given pair of non-negative integers [Formula: see text] and [Formula: see text], we aim to give an explicit formula to count the number of elements of [Formula: see text], the set of all irreducible representations of [Formula: see text] whose degree is not divisible by [Formula: see text]. In this paper, we give formulas for [Formula: see text], when [Formula: see text] is symmetric group, generalized symmetric group, and alternating group. Further, we also have a complete characterization of the elements in [Formula: see text], [Formula: see text], and the self-conjugate partitions of [Formula: see text] and [Formula: see text] with degree not divisible by [Formula: see text].