The Non-projective Part of the Lie Module for the Symmetric Group

The Lie module of the group algebra FSn of the symmetric group is known to be not projective if and only if the characteristic p of F divides n. We show that in this case its nonprojective summands belong to the principal block of FSn. Let V be a vector space of dimension m over F , and let L(V ) be the n-th homogeneous part of the free Lie algebra on V ; this is a polynomial representation of GLm(F ) of degree n, or equivalently, a module of the Schur algebra S(m, n). Our result implies that, when m ≥ n, every summand of L(V ) which is not a tilting module belongs to the principal block of S(m, n), by which we mean the block containing the n-th symmetric power of V .