Seminormal Forms and Gram Determinants for Cellular Algebras

This paper develops an abstract framework for constructing “seminormal forms” for cellular algebras. That is, given a cellular R–algebra A which is equipped with a family of JM–elements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JM–elements separate A. The seminormal forms for A are defined over the field of fractions of R. Significantly, we show that the Gram determinant of each irreducible A–module is equal to a product of certain structure constants coming from the seminormal basis of A. In the non–separated case we use our seminormal forms to give an explicit basis for a block decomposition of A.