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.
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.
| Translated title of the contribution | Investigations about James' Conjecture about Iwahori-Hecke-Algebras of Type A: PhD thesis, RWTH Aachen |
|---|---|
| Original language | German |
| Place of Publication | Aachen |
| Publisher | RWTH Aachen |
| Publication status | Published - 2003 |