On the principal blocks of \({\mathfrak S}_9\) and \({\mathfrak S}_{10}\) over a field of characteristic \(3\)
Following the work of \textit{S. Martin} and \textit{L. Russell} [J. Algebra 185, No. 2, 440-480 (1996; Zbl 0894.20016)], the author constructs the Ext-quiver of the principal block of the group algebra \(F{\mathfrak S}_9\) over an algebraically closed field \(F\) of characteristic 3. He finds in particular that there are no self-extensions of simple modules, the \(\text{Ext}^1\)-space between two simple modules is at most one-dimensional, and the Ext-quiver is bipartite. He further computes the complete Loewy structures of the principal indecomposable modules finding that they all have Loewy length 7. Using the results for \(F{\mathfrak S}_9\), he constructs the Ext-quiver of the principal block of \(F{\mathfrak S}_{10}\). The same extension results hold and he also determines the Loewy structures of the principal indecomposable modules. Again they all have Loewy length 7. In addition to their intrinsic value, these results are of interest because these blocks have non-Abelian defect groups. Whereas, previous investigations have primarily focused on blocks with Abelian defect group.
scientific article; zbMATH DE number 1526264
scientific article