## Final Report Summary - D07.SYMGPS.OX (Vertices of simple modules for the symmetric and related finite groups)

This project aimed at studying modular representations of symmetric groups and related finite groups, as well as of finite-dimensional algebras. Such representations were a fundamental concept since they realised groups and algebras via linear actions on vector spaces. For modular representations, the underlying coefficients were in a field of prime characteristic. The objectives of the project were to investigate vertices and sources as well as homological invariants of distinguished classes of modules for symmetric groups, and related finite groups, and to develop fast algorithms for the computation of vertices and sources. Vertices and sources were group theoretic invariants of indecomposable modules. The homological invariant that was necessary to understand was the Auslander-Reiten quiver, which could be viewed as part of a presentation of the category of all representations.

Regarding the accomplishment of research objectives as presented in the original proposal the fellow, in joint work with Erdmann, gave a complete description of the vertices and sources of two-modular Specht modules labelled by partitions of the form (n-2,2). Furthermore, they analysed the position of Specht modules in the Auslander-Reiten quiver. The main results were that all Specht modules in blocks of weight two had quasi-length one.

Furthermore, in recent work with Bryant, Erdmann and Muller, Danz proved groundbreaking results on the Lie module of the symmetric group. These modules were of interest not only in algebra and combinatorics, but also in algebraic topology. Until recently, the structure for p-power degrees was a complete mystery. Supported by computer calculations, they determined the projective-free part of this module for certain small degrees, for which, nevertheless, the module itself was huge. Surprisingly, these had elementary abelian vertex and endo-permutation sources and led to a general conjecture.

In addition, in joint work with J. Muller, the fellow proved the Feit conjecture, on sources of simple modules for a large class of groups related to symmetric groups, and they also obtained new results on vertices of simple modules for such classes of groups.

In terms of the new lines of research that were opened by the project, various directions emerged and were followed up during the project period. In joint work with Ellers and Murray, the fellow studied centraliser algebras of general group algebras, which was largely computational. Moreover, in joint work with Boltje and Kulshammer, the fellow worked in the depth on group algebra extensions. The depth was important in the study of operator algebras; however in terms of modular representation theory, this was pioneering work. The obtained results included upper and lower bounds.

Finally, Boltje and Danz proved previous results on the double Burnside ring for finite groups. This ring encoded permutation actions on a higher level and became of central interest in connection with the Broue conjecture, as well as for the theory of fusion systems.

Regarding the accomplishment of research objectives as presented in the original proposal the fellow, in joint work with Erdmann, gave a complete description of the vertices and sources of two-modular Specht modules labelled by partitions of the form (n-2,2). Furthermore, they analysed the position of Specht modules in the Auslander-Reiten quiver. The main results were that all Specht modules in blocks of weight two had quasi-length one.

Furthermore, in recent work with Bryant, Erdmann and Muller, Danz proved groundbreaking results on the Lie module of the symmetric group. These modules were of interest not only in algebra and combinatorics, but also in algebraic topology. Until recently, the structure for p-power degrees was a complete mystery. Supported by computer calculations, they determined the projective-free part of this module for certain small degrees, for which, nevertheless, the module itself was huge. Surprisingly, these had elementary abelian vertex and endo-permutation sources and led to a general conjecture.

In addition, in joint work with J. Muller, the fellow proved the Feit conjecture, on sources of simple modules for a large class of groups related to symmetric groups, and they also obtained new results on vertices of simple modules for such classes of groups.

In terms of the new lines of research that were opened by the project, various directions emerged and were followed up during the project period. In joint work with Ellers and Murray, the fellow studied centraliser algebras of general group algebras, which was largely computational. Moreover, in joint work with Boltje and Kulshammer, the fellow worked in the depth on group algebra extensions. The depth was important in the study of operator algebras; however in terms of modular representation theory, this was pioneering work. The obtained results included upper and lower bounds.

Finally, Boltje and Danz proved previous results on the double Burnside ring for finite groups. This ring encoded permutation actions on a higher level and became of central interest in connection with the Broue conjecture, as well as for the theory of fusion systems.