Some new examples in 1-cohomology
The author falsifies Conjecture 2 of \textit{R. M. Guralnick} and \textit{C. Hoffman} [Groups and geometries. Proc. Conf., Siena, Italy, September 1-7, 1996. Basel: Birkhäuser. Trends in Mathematics. 81-89 (1998; Zbl 0894.20039)]. That is, he shows that the dimension of \(H^1(\text{PSL}(6,q),V)\) may be strictly larger than two for some absolutely irreducible representation \(V\) (in characteristic \(p\)), where \(q\) is any sufficiently large power of a sufficiently large prime \(p\). The method relies on computer computations (by his student McDowell) of the top coefficient of a Kazhdan-Lusztig polynomial. It needs the generic cohomology of \textit{E. Cline, B. Parshall, L. Scott}, and \textit{W. van der Kallen} [Invent. Math. 39, 143-163 (1977; Zbl 0336.20036)] and the asymptotic truth of the Lusztig conjecture as proved by \textit{H. H. Andersen, J. C. Jantzen}, and \textit{W. Soergel} [Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220. Paris: Société Mathématique de France (1994; Zbl 0802.17009)]. The author discusses the background and also gives cross characteristic variations.
scientific article; zbMATH DE number 1901830
scientific article