Sylow branching coefficients for symmetric groups

Abstract

Let p⩾5 be a prime and let n be a natural number. In this article, we describe the irreducible constituents of the induced characters ϕ↑frakturSn for arbitrary linear characters ϕ of a Sylow p‐subgroup Pn of the symmetric group Sn, generalising results of Giannelli and Law (J. Algebra 506 (2018) 409–428). By doing this, we introduce Sylow branching coefficients for symmetric groups.

Sylow branching coefficients and a conjecture of Malle and Navarro

“…We observe that 𝜆 has exactly three addable 3-hooks. In particular, we have that 𝑉 (3,1) [3] = 𝜒 (6,1) − 𝜒 (3,2,2) + 𝜒 (3,1 4 ) .…”

“…Example 3.11. Following on from Example 3.8, we can compute 𝑉 (3,1) [3,3], a virtual character of 𝔖 10 : 𝑉 (3,1) [3, 3] = (−1) 0 ⋅ 𝑉 (6,1) [3] + (−1) 1 ⋅ 𝑉 (3,2,2) [3] + (−1) 2 ⋅ 𝑉 (3,1 4 ) [3] = 𝜒 (9,1) + 𝜒 (6,4) − 2𝜒 (6,2,2) + 2𝜒 (6,1 4 ) + 𝜒 (4,4,2) + 2𝜒 (3,2 3 ,1) − 𝜒 (3,2,2,1 3 ) − 𝜒 (3,3,2,1,1) + 𝜒 (3,1 7 ) .…”

“…For a partition $$\lambda \in \mathcal {P}(n)$$, we denote the corresponding irreducible character of $${\mathfrak {S}}_n$$ by $$\chi ^\lambda$$. Given $$P_n$$ a Sylow $$p$$‐subgroup of $${\mathfrak {S}}_n$$ and $$\varphi \in \operatorname{Irr}(P_n)$$, we use the notation introduced in [6] by letting $$Z^\lambda _\varphi$$ denote the natural number defined by $$\begin{equation*} Z^\lambda _\varphi :=[(\chi ^\lambda )_{P_n},\varphi ]. \end{equation*}$$These multiplicities are called Sylow branching coefficients for symmetric groups.…”

Linear characters of Sylow subgroups of symmetric groups

“…We fix a parametrisation of the linear characters of P n for all n as follows (see [L19, §2.3.2] or [GL20] for more detail). First we consider the case n = p k .…”

“…The irreducible constituents of ½ P  Sn were completely described in [GL18]. Later in [GL20], the focus was extended to the study of any monomial character φ  Sn , where φ is an arbitrary linear character of P ∈ Syl p (S n ). The present article complements the study begun in [GL20] by extending to symmetric groups a result proved by Navarro for p-solvable groups [N03,Theorem C].…”