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Stratifying Endomorphism Algebras
Edward Cline, University of Oklahoma, Norman, OK, and Brian Parshall and Leonard Scott, University of Virginia, Charlottesville, VA
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Memoirs of the American Mathematical Society
1996; 119 pp; softcover
Volume: 124
ISBN-10: 0-8218-0488-X
ISBN-13: 978-0-8218-0488-9
List Price: US$43
Individual Members: US$25.80
Institutional Members: US$34.40
Order Code: MEMO/124/591
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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 all characteristics.

Readership

Graduate students and research mathematicians interested in representation theory of algebraic and finite groups, finite-dimensional algebras, and Lie algebras.

Table of Contents

  • Preliminaries
  • Stratified algebras
  • Stratifying endomorphism algebras
  • Stratifications and orders in semisimple algebras
  • Examples
  • Some conjectures for finite Coxeter groups and further remarks
  • References
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