PhD Defense Jeroen van der Meer
Title: Higher-algebraic Picard invariants in modular representation theory
This thesis consists of three main parts, prefaced by a general introduction.
The first part is based on a paper joint with Richard Wong. We exhibit an ∞-categorical decomposition of the stable module category of a general finite group, and we show that, in the case of certain particularly simple finite p-groups, this decomposition can be interpreted as an instance of Galois descent. We then use this perspective to produce proof-of-concept calculations of the group of endotrivial modules for these p-groups.
In the second part, we move on to computations for more complicated groups. Of particular interest will be the case of the extraspecial groups, which have traditionally played a fundamental role in the theory of endotrivial modules. We analyse the Picard spectral sequence for the extraspecial groups and show that the E2-page inherits a great deal of structure from a certain Tits building of isotropic subspaces with respect to a quadratic form.
In the third and final part, we move on to study the Dade group of endopermutation modules. We investigate how it can be realised as the Picard group of a certain ∞-category of genuine equivariant spectra. On our way, we produce a general framework for studying modules whose endomorphisms are trivial up to a specified subcategory of the representation category. This produces invariants that interpolate between the group of endotrivial modules and the Dade group, as well as other more
exotic invariants that are of independent interest.
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Advisor: Jesper Grodal, University of Copenhagen
Chair: Jesper Michael Møller, University of Copenhagen
Tobias Barthel, Max Planck Institute for Mathematics
Nadia Mazza, Lancaster University