On a conjecture of Alvis

Abstract

We exhibit for each integer n 15 an ordinary irreducible character of the symmetric group S n , which restricts irreducibly to A n , with the property that its degree is divisible by every prime less than or equal to n, thereby proving a conjecture of D.L. Alvis.

Degree graphs of simple orthogonal and symplectic groups

“…The lemma says that the irreducible characters of G F are in bijection with the set of pairs (χ s , μ s ), where χ s is the semisimple character corresponding to (s) and μ s is a unipotent character of C G * (s) F * . The degree of the character corresponding to (χ s , μ s ) is χ s (1)μ s (1).…”

“…In the notation of [3], this conjugacy class corresponds to the pair of partitions λ = (1 ), μ = ∅ with η (1) = ( − 1, 1) and ξ (1) = ∅. By [3, §8], the centralizer in G * is of order (q −1 + 1)(q + 1) and the degree χ c (1) is as claimed in all cases.…”

“…Now let 3, so that G has the character degrees listed in Lemma 2.2. Since χ α is a unipotent character, the simple group L also has an irreducible character of degree χ α (1).…”

“…Since 4, G has the character degrees listed in Lemma 2.3. As χ α is unipotent, L also has a unipotent character of degree χ α (1). Also, [G : L] = 2 implies that irreducible constituents of (χ 1 ) L and (χ c ) L have degrees divisible by χ 1 (1)/2 and χ c (1)/2, respectively.…”

“…These graphs are described briefly in [12]. The graphs for the alternating groups can be found using the Atlas character tables and the results of [1]. The graphs for the simple groups of exceptional Lie type and for the simple linear and unitary groups were determined by the author in [19,20], respectively.…”