2006/04/05 by Andrew Mathas, Mathas, Andrew, Marcos Soriano +1 · 2 citations
Mathematics · #20C99 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:20C99
paper · pdf · doi:10.48550/arxiv.math/0604108
Final version. To appear J. Reine Angew. Math. Appendix by Marcos Soriano
arxiv created 2007/03/09 · arxiv updated 2009/12/01
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.