Hecke algebras and the socle of the projective indecomposable modules (Q1359029)

Let \(G\) be a finite group, \(H\) a subgroup of \(G\), \(k\) a field, of characteristic \(r\), \([H]:=\sum_{h\in H} h\in kG\) and \(Y=kG[H]\) the transitive permutation module on the \(H\)-cosets. Then the endomorphism ring \(E:=\text{End}_{kG}(Y)\) has a \(k\)-basis \(\{a_t\mid t\in T\}\), indexed by a transversal \(T\) of \(H\)-double cosets in \(G\). Let \(V\) be an indecomposable \(kG\)-direct summand of \(Y\) appearing with multiplicity \(d\), \(S\colon E\to\text{Mat}_d(k)\) with \(S(a)=(s_{ij}(a))\) the corresponding matrix-representation of \(E\) and \(\Delta_{ij}:=\sum_{t\in T} s_{ij}(a_t)t^{-1}\in kG\). The main result of the paper is the following theorem: \(V\) is a projective \(kG\)-module if and only if \([H]\Delta_{ij}[H]\neq 0\) for some pair \(i,j\), in which case the socle \(\text{Soc}(V)\) is isomorphic to \(kG[H]\Delta_{ij}[H]\). The author uses his result to recover results about the socle of the Steinberg module for finite groups of Lie type and Specht modules for symmetric groups.