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Given a simple lie algebra $\mathfrak{g}$ with root system $R$, the Weyl group $W$ acts on $R$ by definition. The action of any simple reflection $r_i \in W$ on any simple root $\alpha_j \in R$ is very easy to write down.

This question tells us that we can write simple roots in terms of fundamental weights via the Cartan matrix, and so accordingly, using the inverse of the Cartan matrix, we should be able to do the opposite and write down each fundamental weight as a sum of simple roots.

Given such an expression, we could then figure out the action of any $r_i$ on a fundamental weight $\omega_p$. Is there a more natural way to think about this action? Is there a simpler formula (say, for $r_i(\omega_p)$) we can use to write down the action of $W$ on fundamental weights without going through this process?

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1 Answers1

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Yes: the action of the simple reflection $r_i$ is given by $$r_i(x)=x-(\alpha_i^\vee,x) \alpha_i,$$ where $\alpha_i$ is the $i$th simple root and $\alpha_i^\vee=\frac{2 \alpha_i}{(\alpha_i,\alpha_i)}$ is the $i$th simple coroot. The fundamental weights $\varpi_i$ are defined by the requirements $$(\alpha_i^\vee,\varpi_j)=\delta_{ij},$$ where $\delta_{ij}$ is the Kronecker delta, equal to $1$ if $i=j$ and $0$ otherwise, so that the $\varpi_i$'s are the dual basis to the simple coroots. Hence $$r_i(\varpi_j)=\begin{cases} \varpi_j \quad \hbox{if $i \neq j$, and} \\ \varpi_i-\alpha_i \quad \hbox{if $i=j$.} \end{cases} $$

As an aside: the I personally prefer the notation $\varpi$ to $\omega$. The reason is that $\varpi$ ("variation on pi") is intended to stand for poids, or weight.

Stephen
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