Prime Power Degree Representations of the Double Covers of the Symmetric and Alternating Groups

Abstract

In 1998, Zalesskii proposed to classify all instances of irreducible characters of quasi-simple groups which are of prime power degree. In joint work, Malle and Zalesskii then dealt with all quasi-simple groups with the exception of the alternating groups and their double covers [5]. In an earlier article [1] we have classified all the irreducible characters of Sn of prime power degree and have deduced from this also the corresponding classification for the alternating groups. In the present article we complete Zalesskii's programme by dealing with the final case left open in [5], the double covers of the alternating groups. We derive this result from a corresponding result on the double covers of the symmetric groups. If one is only interested in spin characters, one easily sees that only 2-powers can occur as prime power degrees. However from a combinatorial point of view it is natural to ask more generally: when is the number of shifted standard tableaux of a given shape a prime power? Since our method is independent of the prime, we will answer this question, showing that apart from a few accidental cases for small n only the ‘obvious’ partitions satisfy the prime power condition. Thus in turn for the spin characters, apart from exceptions for small n, ‘obvious’ spin characters of 2-power degree are indeed the only ones (this confirms the conjecture stated in [5]).