Year: 2015
Type: article
Abstract: We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline–Parshal–Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra that is free of finite dimension over a noetherian integral domain and that splits over the fraction field of this ring are generically trivial, that is, they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way, we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.
DOI: 10.1093/imrn/rnv204
Author: Ulrich Thiel
Institution: University of Stuttgart
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W2199319736