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Version 8 |
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A root system is a key notion in the the classification and the
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A root system is a key notion in the the classification and the
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| representation theory of reflection groups and of semi-simple Lie |
representation theory of reflection groups and of semi-simple Lie |
| algebras. Let $E$ be a Euclidean vector space with inner product |
algebras. Let $E$ be a Euclidean vector space with inner product |
| $(\cdot,\cdot)$. A root system is a finite spanning set $R\subset E$ |
$(\cdot,\cdot)$. A root system is a finite spanning set $R\subset E$ |
| such that for every $u\in R$, the orthogonal reflection $$v\mapsto |
such that for every $u\in R$, the orthogonal reflection $$v\mapsto |
| v-2\frac{(u,v)}{(u,u)},\quad v\in E$$ |
v-2\frac{(u,v)}{(u,u)},\quad v\in E$$ |
| preserves $R$. |
preserves $R$. |
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| A root system is called \emph{crystallographic} if |
A root system is called \emph{crystallographic} if |
| $2\frac{(u,v)}{(u,u)}$ is an integer for all $u,v\in R$. |
$2\frac{(u,v)}{(u,u)}$ is an integer for all $u,v\in R$. |
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| A root system is called {\em reduced} if for all $u\in R$, we have |
A root system is called {\em reduced} if for all $u\in R$, we have |
| $ku\in R$ for $k=\pm 1$ only. |
$ku\in R$ for $k=\pm 1$ only. |
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| We call a root system {\em indecomposable} if there is no proper |
We call a root system {\em indecomposable} if there is no proper |
| decomposition $R=R'\cup R''$ such that every vector in $R'$ is orthogonal to |
decomposition $R=R'\cup R''$ such that every vector in $R'$ is orthogonal to |
| every vector in $R''$. |
every vector in $R''$. |
