A criterion is given to show that a k–algebra A is quasi–hereditary if it can be defined over an integral domain R, and if there is a certain commutative semisimple subalgebra satisfying a technical but easily verified condition (which roughly states that over the field of fractions K of R, the formal characters of the semisimple K–algebra generated by the R–algebra defining A satisfy an ordering condition). This applies in particular to Schur algebras (where various proofs of quasi–hereditary are known, by de Concini, Eisenbud and Procesi, by Donkin, by Parshall, and by J.A. Green), generalized Schur algebras (covering a result of Donkin), q–Schur algebras (Dipper and James, Parshall and Wang), and Temperley–Lieb algebras (Westbury). The second application of this point of view is an abstract straightening formula for the algebras satisfying the assumptions of the first theorem.
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