A note on Alperin's conjecture (Q1327060)

Let \(F\) be an algebraically closed field of characteristic \(p \neq 0\), \(G\) be a finite group, \(\Delta\) be the simplicial complex of non-trivial \(p\)- subgroups of \(G\) and \(B\) be a block of the group algebra \(FG\). The Lefschetz conjugation module of \(B\) is defined by \[ L_{\text{conj}}(B) := \sum_ C(-1)^{| C|}\text{Ind}^ G_{G_ C}(B_ C), \] considered as an element in the Green ring of \(FG\). In this definition \(C\) ranges over a set of representatives for the orbits of \(G\) on \(\Delta\), \(G_ C\) denotes the stabilizer of a simplex \(C\) in \(\Delta\), and \(B_ C\) is the sum of the Brauer correspondents of \(B\) in \(G_ C\), considered as a \(G_ C\)-module under conjugation. The author shows that \[ \sum_ S \text{Hom}_ F(S,P_ G(S)) = \sum_ Q \text{Ind}^ G_{N_ G(Q)}(P_{N_ G(Q)}(L_{\text{conj}}(\overline{B(Q)}))); \] here \(S\) ranges over a set of representatives for the isomorphism classes of simple \(B\)-modules, \(P_ G(S)\) denotes the projective cover of \(S\), \(Q\) ranges over a set of representatives for the conjugacy classes of \(p\)-subgroups of \(G\), \(B(Q)\) is the sum of the Brauer correspondents of \(B\) in \(N_ G(Q)\) and \(\overline{B(Q)}\) is the image of \(B(Q)\) in \(F[N_ Q(Q)/Q]\). This is of interest in connection with the reformulation [J. Lond. Math. Soc., II. Ser. 39, No. 1, 48-60 (1989; Zbl 0672.20005)] of Alperin's conjecture by the author and \textit{R. Knörr}.