Standardly stratified algebras and cellular algebras

Abstract

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A -modules. Suppose S 1, S 2, …, S n form a complete list of all non-isomorphic simple A -modules and we fix this ordering of simple modules. Let P i and Q i be the projective cover and the injective envelope of S i respectively. With this order of simple modules we define for each i the standard module Δ( i ) to be the maximal quotient of P i with composition factors S j with j [les ] i . Let Δ be the set of all these standard modules Δ( i ). We denote by [Fscr ](Δ) the subcategory of A -mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain 0 = M 0 ⊂ M 1 ⊂ M 2 ⊂ … ⊂ M t = M of submodules of M such that M i / M i −1 is isomorphic to a module in Δ for all i . The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇( i ) to be the maximal submodule of Q i with composition factors S j with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A -mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ( n ) is always projective and ∇( n ) is always injective.