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Abstract
In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect
to a certain choice of basis. The proofs use a generalisation of the well known
FrobeniusSchur relations for semisimple algebras.
The second part of this paper considers Ofree Oalgebras of finite Orank
over a discrete valuation ring O and their decomposition maps under modular reduction modulo the maximal ideal of O, thereby studying the modular
representation theory of such algebras.
Using the formulas from the first part we derive general criteria for such
a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective
indecomposable modules.
Finally we show how this approach could eventually be used to attack a
conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori
HeckeAlgebras, provided the necessary matrix representations on projective
indecomposable modules could be constructed explicitly.
to a certain choice of basis. The proofs use a generalisation of the well known
FrobeniusSchur relations for semisimple algebras.
The second part of this paper considers Ofree Oalgebras of finite Orank
over a discrete valuation ring O and their decomposition maps under modular reduction modulo the maximal ideal of O, thereby studying the modular
representation theory of such algebras.
Using the formulas from the first part we derive general criteria for such
a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective
indecomposable modules.
Finally we show how this approach could eventually be used to attack a
conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori
HeckeAlgebras, provided the necessary matrix representations on projective
indecomposable modules could be constructed explicitly.
Original language  English 

Pages (fromto)  170185 
Number of pages  16 
Journal  Represent. Theory 
Volume  12 
DOIs  
Publication status  Published  19 Mar 2008 
Keywords
 Frobenius algebra, symmetric algebra, idempotent, explicit formula, FrobeniusSchur relations, projective indecomposable module, simple module, Grothendieck group, decomposition map, Coxeter group, IwahoriHecke algebra, James' conjecture
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 1 Finished

EP/C523229/1: Multidisciplinary Critical Mass in Computational Algebra and Applications
Linton, S. A. (PI), Gent, I. P. (CoI), Leonhardt, U. (CoI), Mackenzie, A. (CoI), Miguel, I. J. (CoI), Quick, M. (CoI) & Ruskuc, N. (CoI)
1/09/05 → 31/08/10
Project: Standard