Extending $π$-systems to bases of root systems

Details Extending $π$-systems to bases of root systems Extending $π$-systems to bases of root systems

2004-10-15

arxiv.org/abs/math/0410357v3

Journal of Algebra 287 (2005), 496-500

Mathematics

Representation Theory

6 pages, LaTeX. Corrected typo in statement of theorem and clarified proof

Scientific paper

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 of two or more roots to a basis $B$ of $R$? A $\pi$-system is a linearly independent set of roots, $C$, such that if $\alpha$ and $\beta$ are in $C$, then $\alpha - \beta$ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, $A_3 \subset B_n$ and $A_7 \subset E_8$, an indecomposable $\pi$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$ can always be extended to a basis of $R$.

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