# Cartan matrix

Let $R\subset E$ be a reduced root system^{}, with $E$ a Euclidean vector space, with inner product
$(\cdot ,\cdot )$, and let $\mathrm{\Pi}=\{{\alpha}_{1},\mathrm{\cdots},{\alpha}_{n}\}$ be a base of this root system. Then the
Cartan matrix^{} of the root system is the matrix

$${C}_{i,j}=\left(\frac{2({\alpha}_{i},{\alpha}_{j})}{({\alpha}_{i},{\alpha}_{i})}\right).$$ |

The Cartan matrix uniquely determines the root system, and is unique up to simultaneous
permutation^{} of the rows and columns. It is also the basis change matrix from the basis
of fundamental weights to the basis of simple roots in $E$.

Title | Cartan matrix |
---|---|

Canonical name | CartanMatrix |

Date of creation | 2013-03-22 13:17:56 |

Last modified on | 2013-03-22 13:17:56 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B20 |