ABSTRACT. 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.
We solve the modular isomorphism problem for small group rings, i.e., we determine, for a given finite p-group H, precisely which central Frattini extensions of H give rise to isomorphic small group rings over the field with p elements.
This paper classifies the blocks of the truncated q-Schur algebras of type A which have as weight poset an arbitrary cosaturated set of partitions.2000 Mathematics Subject Classification. 20C08, 20C30, 05E10.
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