Cohomology and Ext for blocks whose Brauer trees are lines or stars (Q6600690)

Given a finite group \(G\) and a field \(k\) of characteristic \(r\) dividing \(|G|\), the groups \(\mathrm{Ext}^{n}_{G}(V, W)\) for \(V,W \in \mathrm{Irr}_{k}G\), and in particular the cohomology groups \(H^{n}(G,V)\), give important structural information about \(G\) and its representation theory over \(k\).\N\NIn the paper under review, the authors compute the dimensions of \(\mathrm{Ext}^{n}_{G}(V,W)\) for all irreducible modules \(V, W\) lying in \(r\)-blocks of cyclic defect whose Brauer tree is either a star or a line, which in the line case extends a result of \textit{O. Dudas} [Rev. Unión Mat. Argent. 65, No. 1, 155--164 (2023; Zbl 07769970)]. They also make use of this information to aid in completing the same computation for all cross characteristic \(r\)-blocks of cyclic defect in rank one groups of Lie type.