Trivial source modules in blocks with cyclic defect groups. (Q966452)

Let \(A\) be a block of a finite group \(G\) with cyclic defect group \(P\neq 1\), let \(B\) be the Brauer correspondent of \(A\) in \(N_G(P)\), and let \(B_1\) be the Brauer correspondent of \(B\) in \(N_G(P_1)\) where \(P_1\) denotes the unique minimal subgroup of \(P\). The authors prove that the following assertions are equivalent: (1) \(A\) and \(B\) are Puig equivalent; (2) all simple modules in \(A\) have trivial sources; (3) \(\chi(u)>0\) for every non-exceptional character \(\chi\) in \(A\) and every \(u\in P\); (4) the Brauer tree of \(A\) is a star with the exceptional vertex in the center, and there exists a non-exceptional character \(\chi\) in \(A\) such that \(\chi(u)>0\) for every \(u\in P\). -- The authors also show that \(B\) and \(B_1\) are Puig equivalent if and only if there exists a non-exceptional character \(\chi\) in \(A\) such that either \(\chi(u)>0\) for all \(u\in P\setminus\{1\}\) or \(\chi(u)<0\) for all \(u\in P\setminus\{1\}\). Reviewer's remark: The results in this paper are related to a conjecture by \textit{S. Danz} and \textit{B. Külshammer} [J. Algebra 322, No. 11, 3919-3949 (2009; Zbl 1228.20011)] and to a recent result by \textit{L. Puig} and \textit{Y. Zhou} [Adv. Math. 226, No. 4, 3088-3104 (2011; Zbl 1213.20011)].