SimpleRoot 2002-12-04 16:53:36 2004-03-28 14:02:10 Definition base % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{simple} \PMlinkescapephrase{vector space} Let $R\subseteq E$ be a root system, with $E$ a Euclidean \PMlinkname{vector space}{VectorSpace}. If $R^+$ is a set of positive roots, then a root is called {\em simple} if it is positive, and not the sum of any two positive roots. The simple roots form a basis of the vector space $E$, and any positive root is a positive integer linear combination of simple roots. A set of roots which is simple with respect to some choice of a set of positive roots is called a {\em base}. The Weyl group of the root system acts simply transitively on the set of bases.