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| Title of object: |
positive root |
| Canonical Name: |
PositiveRoot |
| Type: |
Definition |
| Created on: |
2002-12-04 16:48:18.486952-05 |
| Modified on: |
2002-12-04 16:48:18.486952-05 |
Preamble:
% 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}
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% define commands here |
Content:
If $R\subset E$ is a root system, with $E$ a Euclidean vector space, then a subset
$R^\plus\subset R$ is called a set of positive roots if there is a vector $v\in E$ such that
$(\alpha,v)>0$ if $\alpha\in R^\plus$, and $(\alpha,v)<0$ if $\alpha\in R\backslash R^\plus$.
Roots which are not positive are called {\em negative}. Since $-\alpha$ is negative exactly
when $\alpha$ is positive, exactly half the roots must be positive. |
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