Restricting irreducible characters to Sylow 𝑝-subgroups

Abstract

We restrict irreducible characters of finite groups of degree divisible by p p to their Sylow p p -subgroups and study the number of linear constituents.

On permutation characters and Sylow p-subgroups of Sn

“…The operator Ω q was first defined in [5,Section 3] and it is recalled below for the convenience of the reader. Given compositions µ and ν, we denote by µ the unique partition obtained by reordering the parts of µ, and by µ • ν the composition of |µ| + |ν| obtained by concatenating µ and ν.…”

A note on restriction of characters of alternating groups to Sylow subgroups

“…The above statement has been verified for various classes of groups in [3]. In particular, when G is the symmetric group S n it is shown that the restriction to a Sylow p-subgroup of any irreducible character of degree divisible by p has at least p distinct linear constituents.…”

“…In the second case we can write again δ(γ n ) = φ(γ n−1 )ψ(σ) = 0, by induction. If g = ω n then the result follows from [3,Lemma 3.11].…”

“…For the proof of Proposition 3.12 we will only need the following specific instance of that more general definition. The extended concept of q-sections of partitions was the key idea to prove Theorem A of [3] for symmetric groups. Let n be a natural number.…”

Restrictions of characters in p-solvable groups

“…In particular, all irreducible characters θ ∈ Irr(P ) appear as constituents of χ P with multiplicity at least θ(1) in this case. In [7,8,9], Giannelli and Navarro investigated a more general situation where χ(1) is divisible by p and χ P has at least one linear constituent λ ∈ Irr(P ). They conjectured (and proved in many cases) that χ P has at least p distinct linear constituents.…”