Alcove

Alcoves

Alcoves

To study how the groups ${\hat{W}}_a$ and $W_a$ permute the hyperplanes in $\mathcal{H}$, we examine how they permute the collection $\mathcal{A}$ of connected components of $V^{\circ }:=V\setminus {\bigcup }_{H\in \mathcal{H}}H$. Each element of $\mathcal{A}$ is called an alcove. Fix a set $\Delta$ of simple roots in $\Phi$. We assume moreover that $\Phi$ is irreducible. It is convenient to single out one particular alcove and define it to be the fundamental alcove: $A_0:=\{\lambda \in V\mid 0<(\lambda ,\alpha )<1\text{ for all }\alpha \in {\Phi }^{+}\}.$ Equivalently for $\Phi$ irreducible, we have the following definition: $A_0=\{\lambda \in V\mid 0<(\lambda ,\alpha )\text{ for all }\alpha \in \Delta ,(\lambda ,\tilde{\alpha })<1\},$ with $\tilde{\alpha }$ the highest root.

It is clear (since elements of $\mathrm{Aff}(V)$ act as homeomorphisms) that ${\hat{W}}_a$ does permute $\mathcal{A}$.

What do alcoves look like? Notice first that $V^{\circ }$ is open in $V$. Given $\lambda \in V^{\circ }$, for each root $\alpha$ there is some $k\in \mathbf{Z}$ such that $\lambda$ lies between $H_{\alpha ,k}$ and $H_{\alpha ,k+1}$, so we can find an open neighborhood $U_{\alpha }$ of $\lambda$ meeting no $\alpha$-hyperplane. Intersecting these neighborhoods for all roots $\alpha$ yields an open neighborhood of $\lambda$ in $V^{\circ }$. Since $V^{\circ }$ is open, its connected components are also open.

Any element outside $A_0$ is separated from it by one of the hyperplanes $H_{\alpha }$ or $H_{\alpha ,1}$; so $A_0$ is a connected component of $V^{\circ }$. In general, an alcove is defined by a set of inequalities (some of which may be redundant) of the form: $k_{\alpha }<(\lambda ,\alpha )<k_{\alpha }+1$, $\alpha \in {\Phi }^{+}$.

It is obvious that $A_0$ is included in this set. On the other hand, if $\lambda$ satisfies the indicated inequalities, note that also $(\lambda ,\alpha )>0$ for all positive $\alpha$. Since $\tilde{\alpha }-\alpha$ is a sum of simple roots, $(\lambda ,\tilde{\alpha }-\alpha )\ge 0$, so $(\lambda ,\alpha )\le (\lambda ,\tilde{\alpha })<1$ and thus $\lambda \in A_0$.

This description shows that $A_0$ is simply an intersection of open half-spaces. Moreover, it is a euclidean simplex (whereas if $\Phi$ has a number of irreducible components, $A_0$ will be a product of simplexes). (Question: What are the vertices of $A_0$?)

Walls of Alcoves

Define the walls of $A_0$ to be the hyperplanes $H_{\alpha }$, $\alpha \in \Delta$ and $H_{\tilde{\alpha },1}$, and define $S_a$ to be the corresponding set of reflections: $S_a:=\{s_{\alpha },\alpha \in \Delta \}\cup \{s_{\tilde{\alpha },1}\}.$ The walls of $w{A}_0$ can then be defined to be the images of these hyperplanes under $w$ for any $w\in W_a$. As soon as we show that $W_a$ acts transitively on $\mathcal{A}$, we will have well-defined walls for each alcove.

Proposition

The group $W_a$ permutes the collection $\mathcal{A}$ of all alcoves transitively, and is generated by the set $S_a$ of reflections with respect to the walls of the alcove $A_0$.