Hecke algebras of type A: Auslander--Reiten quivers and branching rules

  • The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\] In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.

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Author:Simon Schmider
Advisor:Susanne Danz
Document Type:Doctoral Thesis
Language of publication:English
Publication Date:2016/06/01
Year of Publication:2016
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2016/05/25
Date of the Publication (Server):2016/06/01
Number of page:XII, 224
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):16-XX ASSOCIATIVE RINGS AND ALGEBRAS (For the commutative case, see 13-XX) / 16Gxx Representation theory of rings and algebras / 16G70 Auslander-Reiten sequences (almost split sequences) and Auslander- Reiten quivers
Licence (German):Standard gemäß KLUEDO-Leitlinien vom 30.07.2015