UNIQUENESS OF LIFTS OF SPECHT MODULES IN THE SYMMETRIC GROUP

Abstract. Recently Fayers has proven a combinatorial condition on partitions that classifies exactly when the Specht module associated to that partition remains irreducible modulo some odd prime p. In this note we show that a uniqueness condition holds as well. More specifically, if p is an odd prime and λ and σ are partitions of n such that Sλ ∼ = Sσ, and both are irreducible as modules over Fp, then λ = σ. 1