Additive higher representation theory

Thesis

This thesis is devoted to the study of higher representation theory as introduced in [Rou4]. As this theory is in its early days, it is essential to seek out modules that can rightfully be named building blocks and allow one to express as much of the structure of arbitrary modules as possible in their terms. We contribute towards this undertaking in the case of additive higher representation theory. Inspiration is drawn from Soergel bimodules which categorify the Hecke algebra.

We introduce functorially cyclic modules as well as (strongly) universal cell modules. Examples include the minimal categorifications of [Rou4]. Properties of such modules are discussed and universal properties in terms of representable 2-functors are established. This leads to constructions and classifications in terms of split Frobenius objects, using a new variant of the Barr-Beck theorem for additive categories. Furthermore, we encounter a new class of modules so called coinvariant modules which arise from automorphism group actions. We also construct canonical cofiltrations and demonstrate why the Jordan-Holder theory of [Rou4] does not readily generalise. Throughout, we comment on the succession [MaMi1]-[MaMi5] that tackles the same questions, however arrives at different conclusions.

As applications, we first show that the 2-category of singular Soergel bimodules of [Wi2] arises naturally within the additive higher representation theory of Soergel bimodules. Second, we establish (weak) equivalences between certain associated universal cell modules together with a categorification of cell module homomorphisms of the Hecke algebra. Third, we show that singular Soergel bimodules constructed with a faithful representation categorify the Schur algebroid, generalising the main result of [Li]. Fourth given a group and a subgroup, we recover the additive monoidal category of representations of the subgroup from the corresponding category for the group without invoking Tannakian formalism.

Authors: Klein, F

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Soergel Bimodule; Higher Representation Theory; Jordan-Hölder Theorem; Category Theory; Barr-Beck Theorem
• Subjects: Representations of groups