---- > [!definition] Definition. ([[character ring of a root system]]) > Let $\Phi$ be a [[root system]], $X$ its [[root and weight lattice of a root system|weight lattice]]. Let $\mathbb{Z}[X]$ be the [[free abelian group]] on the formal symbols $\{ e^{\lambda}: \lambda \in X \}$. An element of $\mathbb{Z}[X]$ is given by $\sum_{\lambda \in X}c_{\lambda} e^{\lambda} $ with all but finitely many $c_{\lambda} \in \mathbb{Z}$ zero. The assignment $e^{\lambda} \cdot e^{\mu}:=e^{\lambda+\mu}$ extends to a product turning $\mathbb{Z}[X]$ into a [[commutative ring|commutative]] [[ring]] called the **character ring of $\Phi$**. ^definition > [!basicexample] > If $\Phi=\{ \pm \alpha \}$, then $X=\mathbb{Z}\left\{ \frac{\alpha}{2} \right\}$, and an element of $\mathbb{Z}[X]$ is a $\mathbb{Z}$-[[linear combination]] of the elements $e^{n\alpha/2}$. With $t=e^{\alpha / 2}$, $\mathbb{Z}[X]$ is [[Laurent polynomial|Laurent polynomials]] $\mathbb{Z}[X]=\mathbb{Z}[t, t^{-1}]$. In general, choosing an isomorphism $X \xrightarrow{\cong}\mathbb{Z}^{\ell}$ of [[abelian group|abelian groups]] determines an isomorphism $\mathbb{Z}[X] \xrightarrow{\cong}\mathbb{Z}[t_{1},t_{1}^{-1},\dots, t_{\ell}, t_{\ell}^{-1}]$. But we don't usually want to choose an isomorphism. ^basic-example ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```