The gluing problem for some block fusion systems. (Q961023)

Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(P\). Then \(B\) determines a fusion system \(\mathcal F\) on \(P\). Following work of \textit{B. Külshammer} and \textit{L. Puig} [Invent. Math. 102, No. 1, 17-71 (1990; Zbl 0739.20003)], one can attach an element \(\alpha_Q\in H^2(\Aut_{\mathcal F}(Q),k^\times)\) to every \(\mathcal F\)-centric subgroup \(Q\) of \(P\). The ``gluing problem'' asks whether these elements can be ``glued'' uniquely to an element \(\alpha\in H^2(\mathcal F^c,k^\times)\) where \(\mathcal F^c\) denotes the full subcategory of \(\mathcal F\) whose objects are the \(\mathcal F\)-centric subgroups of \(P\). In the paper under review, the author considers the cases where \(B\) is either tame (so that \(p=2\) and \(P\) is dihedral, semidihedral or quaternion) or the principal \(p\)-block of \(\text{PSL}_3(p)\) where \(p\neq 2\) (so that \(P\) is extraspecial of order \(p^3\) and exponent \(p\)). In these cases he proves the existence of \(\alpha\) but observes that \(\alpha\) is not unique when \(p\equiv 1\pmod 3\) and \(B\) is the principal \(p\)-block of \(\text{PSL}_3(p)\).