Functors and projective summands of permutation modules (Q1320148)

Let \(G\) be a finite group, \(H\) be a subgroup of \(G\), \(k\) be a field. If \(X\) is a subset of \(G\), \([X] = \sum_{x \in X} x\). Let \(Y = kG [H]\) and \(E(Y)\) be the Hecke algebra of \(Y\). Let \(T\) be a transversal for the set of \((H,H)\)-cosets of \(G\). Suppose that \(V\) is the direct \(kG\)-summand of \(Y\) with multiplicity \(d\). Let \(S\) be the corresponding \(d\)-dimensional irreducible representation of \(E(Y)\). Let \(a_t\) be such that \(a_t ([H]) = [Ht H]\). The author proposes the following criterion for projectivity of \(V\). Theorem. \(V\) is projective if and only if \([H] \Delta [H] \neq 0\), where \(\Delta = \sum_{t\in T} s_{ij} (a_t) t\). The result is applied to the Steinberg component of a finite group of Lie type and Young modules of symmetric groups.