The largest character degrees of the symmetric and alternating groups. (Q2790913)

Given a finite group \(G\), it is interesting to ask how large the largest degree of an irreducible character of \(G\) is, in terms of \(|G|\). There are general results and conjectures about this, and one of the most difficult cases is that of the alternating groups. This paper makes an important contribution to this case by showing that the sum of the squares of the degrees of the irreducible characters of non-maximal degree exceeds the square of the maximal degree. This result may not appear surprising, but is quite difficult to prove. The authors prove it by starting with the symmetric group, and comparing character degrees using the hook-length formula. They define a graph on the set of partitions of \(n\) in which two partitions are joined by an edge if one of them is obtained from the other by moving a box from the first row of the Young diagram to the first column. They then compare the degree of the character labelled by a given partition with the degrees of the characters labelled by its neighbours, and use this (together with a computer calculation to deal with finitely many small cases) to prove their main result for the symmetric groups.NEWLINENEWLINE Passing from the symmetric to the alternating groups requires some additional analysis involving the branching rule.NEWLINENEWLINE The paper is well written, self-contained and sufficiently elementary to have broad appeal. The only disappointment is the misprint in Proposition 4.2, whose conclusion is a sum with no summand; but the alert reader can fill in the details as an exercise.