STRATIFYING ENDOMORPHISM ALGEBRAS - INTRODUCTION

Suppose that R is a finite dimensional algebra and T is a right R-modu le. Let A=End(R)(T) be the endomorphism algebra of T. This paper prese nts a systematic study of the relationships between the representation theories of R and A, especially those involving actual or potential q uasi-hereditary structures on the tatter algebra. Our original motivat ion comes from the theory of Schur algebras, work of Soergel on the Be rnstein-Gelfand-Gelfand category Omicron, and recent results of Dlab-H eath-Marko realizing certain endomorphism algebras as quasi-hereditary algebras. Besides synthesizing common features of all these examples, we go beyond them in a number of new directions. Some examples involv e new results in the theory of tilting modules, an abstract ''Specht/W eyl module'' correspondence, a new theory of stratified algebras, and a deformation theory based on the study of orders in semisimple algebr as. Our approach reconceptualizes most of the modular representation t heory of symmetric groups involving Specht modules and places that the ory in a broader context. Finally, we formulate some conjectures invol ving the theory of stratified algebras and finite Coxeter groups.