## Final Report Summary - GLORY (Global-Local Methods in Representation Theory)

Symmetry phenomena are well known to play a remarkable role in mathematics and its applications to chemistry, physics, and many other branches of natural sciences. It is, therefore, important to develop mathematical models to analyze symmetries in a unified way. Group theory provides such models by way of permutation representations, that is, groups acting on sets. After linearizing, these actions gain more structure, which can be encoded in matrices, facilitating both abstract investigations and concrete calculations. A natural question to ask then is, how far global properties of such linear representations of some complicated group are determined locally by better behaved subgroups. All of these are objectives of representation theory. This project, which has been carried out by Dr Susanne Danz at the University of Kaiserslautern, contributes to this central problem of understanding the interplay of global and local representation theory, by defining two main objectives and establishing tight connections between them. The first one concerns block theory and Auslander--Reiten theory of symmetric groups and finite groups of Lie type; the second objective focuses on the double Burnside ring of a finite group. Overall, the project aimed, in particular, at gaining new evidence for the long-standing conjectures of Broué and Feit, and at developing novel methods to subsequently attack these in greater generality. This required a combination of very classical methods from block theory and homological algebra, powerful up-to-date methods from computer algebra, and the promising novel concept of bisets and double Burnside rings, which links representation theory with group theory as well as with algebraic topology.

Through the work on this CIG project, Dr Danz has strengthened her international collaborations. She has worked in collaborations with researchers from Germany, Europe, Singapore, and the US.

With R. Boltje (UC Santa Cruz) Danz established ghost algebras of double Burnside algebras and algebras related to the category of biset functors. The crux in the work of Boltje and Danz is a construction of double Burnside algebras from twisted category algebras via idempotent condensation. In recent years, twisted category algebras have attracted much interest through work of B. Steinberg, M. Linckelmann, and others. They, in particular, open up very new perspectives on many well-known algebras. In a series of two papers, Boltje and Danz showed that certain twisted category algebras are quasi-hereditary. As a consequence, they also obtained new results on representations of prominent classes of algebras such as Brauer algebras.

Building on the results of Boltje and Danz, on earlier work of Boltje and Xu, and on yet unpublished work of Boltje and Perepelitsky, Danz started a collaboration with Boltje and J. Müller (University of Jena). They have started to develop computer programs, based on the computer algebra system GAP, to explicitly compute with p-permutation equivalences. The notion of a p-permutation equivalence is due to Boltje and Xu, and is the object of interest in a variation of the abovementioned Broué Conjecture. The next goal is now to exploit the newly developed GAP programs to gain a better understanding of the structure of p-permutation equivalences, and to verify Boltje—Xu’s version of Broué’s Conjecture for distinguished (classes of) finite groups and blocks.

With K. Erdmann (University of Oxford), Danz has shown that twisted category algebras also arise naturally as certain Hopf algebras, called bismash products. This led to a very precise description of the algebra structure of bismash products and generalizations.

In a partly computational project with R. Bryant (University of Manchester), K. Erdmann and J. Müller, Danz has studied local invariants such as vertices and sources of the Lie module of the symmetric group, over fields of positive characteristic.

In joint work with E. Giannelli (University of Kaiserslautern) Danz completed the classification of the vertices of simple modules of symmetric groups that are labelled by hook partitions.

In very recent work with K.J. Lim (Singapore) Danz studied important classes of modules of symmetric groups. Lim and Danz, in particular, solved the problem of parametrizing simple Specht modules in terms of signed Young modules, which had been open for 10 years.

With her PhD student S. Schmider, Danz has investigated the structure of Auslander--Reiten quivers of symmetric groups and Hecke algebras of type A.

Danz’s PhD student P. Luka studies local invariants of finite groups of Lie type over fields of non-defining characteristic, with a particular focus on Feit’s Conjecture for finite classical groups over fields of non-defining characteristic.

The results accomplished so far have been published in seven original research articles, which have appeared in peer-reviewed journals or are currently under review. The results obtained during the past three years will pave the way for ample future research directions, so that the collaborations initiated in connection with this CIG will continue also after the end of the project.

From October 2011 until March 2015 Dr Danz held a position as a Junior Professor at the Department of Mathematics at the University of Kaiserslautern. She has built up a small research group, has supervised a master student, and is currently still supervising two PhD students. In May 2014 a committee of internal and external referees has evaluated Dr Danz’s achievements in research, teaching, and administration. This evaluation procedure provided Dr Danz with a qualification equivalent to the German Habilitation. In October 2014 Dr Danz was offered a permanent professorship (W2) at the Catholic University of Eichstätt-Ingolstadt, which she has now been holding since April 2015. This CIG project was specifically tailored to be carried out at the University of Kaiserslautern. Thus, after having moved to the University of Eichstätt-Ingolstadt, Dr Danz, unfortunately, had to terminate the CIG project GLoRy one year prior to the initial end date.

Through the work on this CIG project, Dr Danz has strengthened her international collaborations. She has worked in collaborations with researchers from Germany, Europe, Singapore, and the US.

With R. Boltje (UC Santa Cruz) Danz established ghost algebras of double Burnside algebras and algebras related to the category of biset functors. The crux in the work of Boltje and Danz is a construction of double Burnside algebras from twisted category algebras via idempotent condensation. In recent years, twisted category algebras have attracted much interest through work of B. Steinberg, M. Linckelmann, and others. They, in particular, open up very new perspectives on many well-known algebras. In a series of two papers, Boltje and Danz showed that certain twisted category algebras are quasi-hereditary. As a consequence, they also obtained new results on representations of prominent classes of algebras such as Brauer algebras.

Building on the results of Boltje and Danz, on earlier work of Boltje and Xu, and on yet unpublished work of Boltje and Perepelitsky, Danz started a collaboration with Boltje and J. Müller (University of Jena). They have started to develop computer programs, based on the computer algebra system GAP, to explicitly compute with p-permutation equivalences. The notion of a p-permutation equivalence is due to Boltje and Xu, and is the object of interest in a variation of the abovementioned Broué Conjecture. The next goal is now to exploit the newly developed GAP programs to gain a better understanding of the structure of p-permutation equivalences, and to verify Boltje—Xu’s version of Broué’s Conjecture for distinguished (classes of) finite groups and blocks.

With K. Erdmann (University of Oxford), Danz has shown that twisted category algebras also arise naturally as certain Hopf algebras, called bismash products. This led to a very precise description of the algebra structure of bismash products and generalizations.

In a partly computational project with R. Bryant (University of Manchester), K. Erdmann and J. Müller, Danz has studied local invariants such as vertices and sources of the Lie module of the symmetric group, over fields of positive characteristic.

In joint work with E. Giannelli (University of Kaiserslautern) Danz completed the classification of the vertices of simple modules of symmetric groups that are labelled by hook partitions.

In very recent work with K.J. Lim (Singapore) Danz studied important classes of modules of symmetric groups. Lim and Danz, in particular, solved the problem of parametrizing simple Specht modules in terms of signed Young modules, which had been open for 10 years.

With her PhD student S. Schmider, Danz has investigated the structure of Auslander--Reiten quivers of symmetric groups and Hecke algebras of type A.

Danz’s PhD student P. Luka studies local invariants of finite groups of Lie type over fields of non-defining characteristic, with a particular focus on Feit’s Conjecture for finite classical groups over fields of non-defining characteristic.

The results accomplished so far have been published in seven original research articles, which have appeared in peer-reviewed journals or are currently under review. The results obtained during the past three years will pave the way for ample future research directions, so that the collaborations initiated in connection with this CIG will continue also after the end of the project.

From October 2011 until March 2015 Dr Danz held a position as a Junior Professor at the Department of Mathematics at the University of Kaiserslautern. She has built up a small research group, has supervised a master student, and is currently still supervising two PhD students. In May 2014 a committee of internal and external referees has evaluated Dr Danz’s achievements in research, teaching, and administration. This evaluation procedure provided Dr Danz with a qualification equivalent to the German Habilitation. In October 2014 Dr Danz was offered a permanent professorship (W2) at the Catholic University of Eichstätt-Ingolstadt, which she has now been holding since April 2015. This CIG project was specifically tailored to be carried out at the University of Kaiserslautern. Thus, after having moved to the University of Eichstätt-Ingolstadt, Dr Danz, unfortunately, had to terminate the CIG project GLoRy one year prior to the initial end date.