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