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Formulas for primitive idempotents in Frobenius algebras and an application to decomposition maps

Authors: Max Neunhöffer and Sarah Scherotzke
Journal: Represent. Theory 12 (2008), 170-185
MSC (2000): Primary 16G30; Secondary 16G99, 20C08, 20F55
Published electronically: March 19, 2008
MathSciNet review: 2390671
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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 Frobenius-Schur relations for semisimple algebras.

The second part of this paper considers $\mathcal {O}$-free $\mathcal {O}$-algebras of finite $\mathcal {O}$-rank over a discrete valuation ring $\mathcal {O}$ and their decomposition maps under modular reduction modulo the maximal ideal of $\mathcal {O}$, 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-Hecke algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly.

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Additional Information

Max Neunhöffer
Affiliation: School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland, United Kingdom

Sarah Scherotzke
Affiliation: Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Keywords: Frobenius algebra, symmetric algebra, idempotent, explicit formula, Frobenius-Schur relations, projective indecomposable module, simple module, Grothendieck group, decomposition map, Coxeter group, Iwahori-Hecke algebra, James’ conjecture
Received by editor(s): May 8, 2007
Received by editor(s) in revised form: February 9, 2008
Published electronically: March 19, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.