Some Simple Projective Brauer Quotients of Simple Modules for the Symmetric Groups in Characteristic Two

Abstract

where S is the symmetric group on n symbols and k is a field of characteristic n p ) 0. In this paper we answer the question, ''When is the Brauer quotient of a simple F S -module V with respect to a subgroup H of S both simple and 2 n n Ž . projective as an N H rH-module?,'' in some special cases. Remarkably, in each S n Ž . case there is only one such subgroup H up to conjugacy . ᮊ

On some trivial source Specht modules

“…[22, Lemma 2.5] Let H ≤ S n and r be an integer such that 0 < r ≤ n. Let H act on n. If H fixes n − r + 1, . .…”

Some modules of symmetric groups

“…(ii) [59,Proposition 2.1] We have N (P ) ∼ = U (P ) ⊕ V (P ) as F[N G (P )/P ]-modules if we have N ∼ = U ⊕ V for some FG-modules U and V .…”

“…[59, Lemma 2.5] Let H ≤ S n and r be an integer such that 0 < r ≤ n. Let H act on[1, n]. If H fixes n − r + 1, .…”