# Stratifying Endomorphism Algebras

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*Edward Cline; Brian Parshall; Leonard Scott*

Suppose that \(R\) is a finite dimensional algebra
and \(T\) is a right \(R\)-module. Let \(A = \mathrm{
End}_R(T)\) be the endomorphism algebra of \(T\).
This memoir presents a systematic study of the relationships between
the representation theories of \(R\) and \(A\),
especially those involving actual or potential structures on
\(A\) which ”stratify” its homological algebra. The
original motivation comes from the theory of Schur algebras and the
symmetric group, Lie theory, and the representation theory of finite
dimensional algebras and finite groups.

The book synthesizes common features of many of the above areas, and
presents a number of new directions. Included are an
abstract ”Specht/Weyl module” correspondence, a new theory
of stratified algebras, and a deformation theory for them. The
approach reconceptualizes most of the modular representation theory
of symmetric groups involving Specht modules and places that theory in
a broader context. Finally, the authors formulate some
conjectures involving the theory of stratified algebras and finite
Coexeter groups, aiming toward understanding the modular representation
theory of finite groups of Lie type in

#### Table of Contents

# Table of Contents

## Stratifying Endomorphism Algebras

- Contents v6 free
- Introduction 110 free
- Chapter 1. Preliminaries 615 free
- Chapter 2. Stratified algebras 3746
- Chapter 3. Stratifying endomorphism algebras 4554
- 3.1. Constructing stratified algebras 4554
- 3.2. Filtrations and tilting modules 5059
- 3.3. A converse result: stratified algebras as endomorphism algebras 5362
- 3.4. Tilting modules and the Ringel dual of a highest weight category 5564
- 3.5. Stratification and recollement 5968
- 3.6. An abstract theory of twisted Young modules 6271
- 3.7. An abstract theory of Specht module socles 6574
- 3.8. Ext[sup(1)]–vanishing and abstract Specht/Weyl equivalences 6877

- Chapter 4. Stratifications and orders in semisimple algebras 7079
- 4.1. Integral endomorphism algebras 7079
- 4.2. Z–stratified algebras 7685
- 4.3. Integral stratification and recollement 7887
- 4.4. An abstract theory of twisted integral Young modules 8089
- 4.5. Integral theory and Specht module socles 8998
- 4.6. Integral Ext[sup(1)]–vanishing and Specht/Weyl equivalences 92101
- 4.7. Integral Ext[sup(1)]–vanishing and integral stratification 93102

- Chapter 5. Examples 95104
- Chapter 6. Some conjectures for finite Coxeter groups and further remarks 108117
- References 117126