Sign Conjugacy Classes of the Symmetric Groups

Abstract

A conjugacy class C of a finite group G is a sign conjugacy class if every irreducible character of G takes value 0, 1 or -1 on C. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.

Sign conjugacy classes of the alternating groups

“…we can apply Lemma 2.1 to results from Section 2 of [2] and, as α 1 > α 2 = a ≥ 5, obtain that if β is self conjugate then α is either (2a, a, a − 1, 1) or (a + 1, a, a − 1, 1), as in all other cases…”

“…• For α as in Theorem 3.13 of [2] we have that α 3 > α 4 ≥ α h−1 = 2 as h ≥ 5 and then α 1 > α 2 > α 3 + α 4 ≥ 5. In particular α 1 ≥ 7 and then β ′ 2 = 4 < α 1 − 2 = β 2 so that β is not self conjugate.…”

“…For example (3) is a sign partition which is not an A n -sign partition, while (2,2) is an A n -sign partition which is not a sign partition. In general however A n -sign partitions are almost always sign partitions and almost all even sign partitions are A n sign partitions, as will be seen from Theorem 1.6 and Theorem 1.3 of [2].…”

“…We will now prove Lemmas 1.7 and 1.8. To do this we will use results from [2] and [4]. Since most of the partitions considered there are of the form (a, b, 1 c ) we will first classify in the next lemma which such partitions are self conjugate.…”

“…We will divide the proof of the lemma in the following cases: 1) α = (α 2 + 2a − 1, α 2 , a, a − 1, 1) with α 2 > 2a and a ≥ 4, 2) all other cases. Case 2) will be divided in subcases corresponding to the different cases of the proof of Theorem 1.6 of [2]. 1) For α = (α 2 + 2a − 1, α 2 , a, a − 1, 1) with α 2 > 2a and a ≥ 4 let β := (α 2 + 2a, 3, 1 α 2 +2a−4 ).…”