Stratifying endomorphism algebras

Abstract

Suppose that R is a finite dimensional algebra and T is a right R-module. Let A = End R (T) be the endomorphism algebra of T. This paper presents a systematic study of the relationships between the representation theories of R and A, especially those involving actual or potential quasi-hereditary structures on the latter algebra. Our original motivation comes from the theory of Schur algebras, work of Soergel on the Bernstein-Gelfand-Gelfand category O, and recent results of Dlab-Heath-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 involve new results in the theory of tilting modules, an abstract "Specht/Weyl module" correspondence, a new theory of stratified algebras, and a deformation theory based on the study of orders in semisimple algebras. Our approach reconceptualizes most of the modular representation theory of symmetric groups involving Specht modules and places that theory in a broader context. Finally, we formulate some conjectures involving the theory of stratified algebras and finite Coxeter groups.