Hall–Littlewood polynomials at roots of 1 and modular representations of the symmetric group

Abstract

We first give a brief introduction to Hall–Littlewood functions; we follow closely the notation used in Macdonald [3].Let λ = (λ1,…,λm) be a be a partition of n; that is λ1 + … + λm = n, λ1 ≥ λ2 ≥ … ≥ λm >0. We shall sometimes write l(λ) for m and refer to l(λ) as the length of λ and we shall write |λ| for ∑λi. Let x1, x2, … be an infinite set of indeterminates and t an indeterminate independent of the xi (i = 1,2, …). Let Pλ(x;t) = Pλ(x1, x2, …t) and Qλ(x,t) = Qλ(xl, x2, …t) be the Hall–Littlewood P- and Q-functions defined as in Macdonald.