Seminormal forms and Gram determinants for cellular algebras

Details Seminormal forms and Gram determinants for cellular algebras Seminormal forms and Gram determinants for cellular algebras

2006-04-05

arxiv.org/abs/math/0604108v4

Mathematics

Representation Theory

Final version. To appear J. Reine Angew. Math. Appendix by Marcos Soriano

Scientific paper

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. 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 "Jucys-Murphy elements"depend, ultimately, on the Cayley-Hamilton theorem.

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