Untersuchungen zu James' Vermutung über Iwahori-Hecke-Algebren vom Typ A
type: CreativeWork
description: This thesis deals with a conjecture by Gordon James from 1990 and a variant by Meinolf Geck about decomposition maps of the generic Iwahori-Hecke algebra H of the symmetric group on n points arising from specialization to characteristic l>0. If the parameter of the algebra is specialized to an element q of GF(l) with multiplicative order e, the conjecture states, that, for el>n, the decomposition map does not depend on l and q, but only on e. The main result of the present work is the following reformulation of the James-Geck conjecture: For el>n every primitive idempotent in the extension of scalars AH is primitive as idempotent in the algebra A'H, where A and A' are rings of characteristic 0 (which are explicitly constructed for l and q), such that A is contained in A' and therefore AH in A'H. Thus an equivalent statement is given, which contains only rings and algebras of characteristic 0. To prove equivalence, generalizations of the well-known methods of lifting of idempotents and of Brauer reciprocity are developed. In addition results and observations are presented, that might lead to an approach to a proof of the above mentioned reformulation. A method is presented, how one can derive explicit formulae for primitive idempotents in non-semi-simple symmetric algebras using matrix representations on projective indecomposable modules. Such matrix representations seem to arise naturally from the Kazhdan-Lusztig cell modules. Another possible attack for a proof stems from another result of the present thesis, an explicit construction of a Wedderburn decomposition for H using the Kazhdan-Lusztig basis.
abstract: This thesis deals with a conjecture by Gordon James from 1990 and a variant by Meinolf Geck about decomposition maps of the generic Iwahori-Hecke algebra H of the symmetric group on n points arising from specialization to characteristic l>0. If the parameter of the algebra is specialized to an element q of GF(l) with multiplicative order e, the conjecture states, that, for el>n, the decomposition map does not depend on l and q, but only on e. The main result of the present work is the following reformulation of the James-Geck conjecture: For el>n every primitive idempotent in the extension of scalars AH is primitive as idempotent in the algebra A'H, where A and A' are rings of characteristic 0 (which are explicitly constructed for l and q), such that A is contained in A' and therefore AH in A'H. Thus an equivalent statement is given, which contains only rings and algebras of characteristic 0. To prove equivalence, generalizations of the well-known methods of lifting of idempotents and of Brauer reciprocity are developed. In addition results and observations are presented, that might lead to an approach to a proof of the above mentioned reformulation. A method is presented, how one can derive explicit formulae for primitive idempotents in non-semi-simple symmetric algebras using matrix representations on projective indecomposable modules. Such matrix representations seem to arise naturally from the Kazhdan-Lusztig cell modules. Another possible attack for a proof stems from another result of the present thesis, an explicit construction of a Wedderburn decomposition for H using the Kazhdan-Lusztig basis.
author: Neunhöffer, Max
language: ger
keywords: info:eu-repo/classification/ddc/510, Mathematik, Iwahori Hecke, Algebra, James Conjecture, Kazhdan Lusztig, Cell Module Type A
date published: 2003
date modified: 2026-07-30T19:42:31Z
provider: RWTH Aachen University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Publikationsserver der RWTH Aachen University