PreprintSeminormal forms and Gram determinants for cellular algebrasAndrew Mathas and Marcos Soriano (appendix)AbstractThis paper develops an abstract framework for constructing `seminormal forms' for cellular algebras. That is, given a cellular Ralgebra A which is equipped with a family of JMelements we give a general technique for constructing orthogonal bases for A, and for all of its irreducible representations, when the JMelements 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 Amodule is equal to a product of certain structure constants coming from the seminormal basis of A. In the nonseparated case we use our seminormal forms to give an explicit basis for a block decomposition of A. <p> The appendix, by Marcos Soriano, gives a general construction of a complete set of orthogonal idempotents for an algera starting from a set of elements which act on the algebra in an upper triangular fashion. The appendix shows that constructions with "JucysMurphy elements"depend, ultimately, on the CayleyHamilton theorem.Keywords: Cellular algebras; seminormal forms; blocks. AMS Subject Classification: Primary 20C99.
