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978-3-8439-1674-5, Reihe Mathematik
On restricted rational Cherednik algebras
400 Seiten, Dissertation Technische Universität Kaiserslautern (2014), Softcover, A5
The aim of this thesis is to shed some light on the representation theory of restricted rational Cherednik algebras - primarily for exceptional complex reflection groups for which almost no results exist so far. The central topics are specific problems posed by Gordon concerning the structure of the Verma modules and the simple modules of restricted rational Cherednik algebras, and Martino's conjecture which relates Calogero-Moser families defined by the block structure of these algebras with Rouquier families defined by the block structure of Hecke algebras.
We give - for the first time - explicit results about the block structure, the dimensions of the simple modules, the decomposition matrices of the Verma modules, and the structure of the simple modules as graded modules for the underlying reflection group - and thus the answers to Gordon's questions - for generic parameters for around half of the 34 exceptional complex reflection groups. While we can confirm in this way the generic part of Martino's conjecture for all these groups, it turns out as a surprise that the conjecture is wrong for the group G25. This is the first and only counter-example to this conjecture so far, and there is no abstract theoretical explanation for the failure yet.
Due to the exceptional nature of exceptional complex reflection groups the approach taken in this thesis differs from the existing literature in so far as it is computational and experimental. This thesis is the first approach to computations in rational Cherednik algebras and their modules, and even though we make use of the powerful computer algebra system Magma, we still have to introduce several new computational methods along with their implementation to deal with these problems. This includes for example a new Las Vegas algorithm for computing the head of a local module and for computing the decomposition matrix of a constituent-closed family of local modules. These computational aspects culminated in the development of a Cherednik Algebra Magma Package (CHAMP for short) which provides a user-friendly and flexible way to perform basic computation in (restricted) rational Cherednik algebras and Verma modules in arbitrary characteristic (CHAMP is freely available at http://thielul.github.io/CHAMP/).
The hope is that the results presented here - along with CHAMP itself - will lead to new theoretical insight, which is hard to gain when the situation for the exceptional groups is not understood.