Final Report Summary - B10NONABBLCKSETH (Representation Theory of Blocks of Group Algebras with Non-abelian Defect Groups) The project was conducted in pure mathematics in the areas of representation theory of associative algebras and Lie theory. The aim of the project was to study representations of symmetric groups, Hecke algebras and other related associative algebras, over fields of non-zero characteristic. Many basic problems, such as finding the irreducible representations, i.e. simple modules, are not solved in general for these algebras. In order to get a deeper insight into these algebras, one needs to construct and understand invariants of modules over these algebras. Accomplishment of research objectives as presented in the original proposalWe focussed our research on the distinguished class of blocks of symmetric groups with non-abelian defect groups. We investigated decomposition numbers, Ext-quivers, indecomposable modules, Cartan matrices and properties of gradings on these algebras. We used methodology and techniques of Lie theory interconnected with methods of representation theory of associative algebras. Several results on connections between graded representation theory and the structure of the projective indecomposable modules have been obtained. In particular, we managed to establish such a connection for a class of blocks of symmetric groups with defect 2. An elementary combinatorial proof of the fact that the structure of the projective indecomposable modules is entirely determined by a certain special kind of gradings on these algebras is given. As a corollary of this result, an elementary combinatorial algorithm for construction of quivers of these blocks is established. This work has already been published in a peer reviewed journal (doi: 10.1142/S0219498812502209). Also, these results gave us some insight on how to apply a similar construction on a class of blocks of symmetric groups with non-abelin defect groups. This is our main approach when trying to prove Turner's conjecture about the structure of non-abelian blocks. This is still work in progress as Turner's conjecture turned out to be very difficult to prove. We made some progress in this direction when blocks of small defect are concerned, and especially when the RoCK blocks are concerned. At the moment, these results are being written up and will be sent for publication in the next two months. We plan to use our results for small defect blocks and to generalise them for wider classes of blocks. Our attempts to prove Turner's conjecture supplied us with a plethora of problems we intend to work on in the future. After the end of the project, we will continue our work on these problems, which are interesting enough to occupy our attention for the next few years. New objectives and new lines of researchDuring our project, we found several new lines of research, some of which we present now. Most notably, there are interesting connections between the algebra Dn from a paper which was co-authored by the scientist in charge (doi: 10.1016/j.jpaa.2010.06.032). To Dn, we can apply a construction that associates to each finite dimensional algebra A of global dimension 2, a triangulated category, which is a cluster category in the case A is a hereditary algebra. This new category will contain some stable categories of modules. This will help us to describe certain classes of modules possessing a special kind of filtration which will be useful in determining Richardson orbits of the action of a parabolic subgroup of an orthogonal group. This algebra will be a good testing ground for our new construction of quivers with additional data containing grading structure. We expect to have several cluster categories as a result of these constructions. This should give us some ideas on how to proceed with the construction of an appropriate geometric model that would include quivers with additional data describing graded structures of the above mentioned blocks of algebras.Another paper, about gradings on blocks with quaternion defect groups, is in the finishing stages of preparation and will be sent for publication in a matter of days. In this paper, we prove that there are no non-trivial gradings on blocks with quaternion defect groups. This is proved by using a result from our previous paper (doi:10.4064/cm122-2-1) that deals with maximal tori of groups of outer automorphisms of these blocks. This result gives us a nice criterion for determining when a given algebra possesses no non-trivial gradings (up to graded Morita equivalence and rescalling).
