# Le radical quasi-hereditaire des q-algebres de Schur

Abstract : The subject of this work is the theory of quasi-hereditary algebras, which have been introduced by Cline, Parshall and Scott in 1988. An important example is given by the classical Schur algebra, whose origin lies in the theory of polynomial representations of $\mbox(GL)_n$; a more general class of examples is given by the $q$-Schur algebras introduced by Dipper and James in 1989. The aim of this work is to study the quasi-hereditary radical of such an algebra, both from a theoretical and a computational point of view. That radical has been introduced in a recent article by Geck, with an appendix by Donkin.\\ The first chapter of the thesis is a survey about basic definitions and results concerning quasi-hereditary algebras; the general theory is illustrated by the examples given by the $q$-Schur algebras. In the second chapter, we develop explicit methods for studying in detail the representations of the Schur algebras $S(2,r)$. In particular, we have programs in (\sf GAP) for computing the Weyl modules and the quasi-hereditary radical of these algebras. One of the principal results of this thesis establishes a link, via the quasi-hereditary radical, between James' conjecture'' concerning modular representations of $q$-Schur algebras and the theory of Kazhdan--Lusztig cells. This is the subject of Chapter~3. A result of this type can already be found in the above-mentioned article of Geck, but the proof uses a certain identity which is wrong. The fact that this identity is wrong was discovered by using our explicit computations in the example $S(2,r)$.
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Theses
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https://tel.archives-ouvertes.fr/tel-00004685
Contributor : Ammar Seddiq Mahmood <>
Submitted on : Monday, February 16, 2004 - 2:55:08 PM
Last modification on : Tuesday, November 19, 2019 - 2:37:28 AM
Long-term archiving on: : Wednesday, September 12, 2012 - 1:20:38 PM

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• HAL Id : tel-00004685, version 1

### Citation

Ammar Seddiq Mahmood. Le radical quasi-hereditaire des q-algebres de Schur. Mathématiques [math]. Université Claude Bernard - Lyon I, 2003. Français. ⟨tel-00004685⟩

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