Extending π-systems to bases of root systems

Abstract

Let R be an indecomposable root system. It is well known that any root is part of a basis B of R. But when can you extend a set, C, of two or more roots to a basis B of R? A π-system is a linearly independent set of roots such that if α and β are in C, then α - β is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, A3Bn and A7E8, an indecomposable π-system whose Dynkin diagram is a subdiagram of the Dynkin diagrams of R can always be extended to a basis of R. © 2005 Published by Elsevier Inc.