Dual geometric representation of a Coxeter group (${\sigma }^{\ast }$)

The dual geometric representation acts on $V^{\ast }$ byThe open fundamental chamber consists of the functionals positive on every simple basis vector.

For the closed fundamental chamber $D\subseteq V^{\ast }$, the Tits cone isIt equals all of $V^{\ast }$ exactly when the finite-rank Coxeter group $W$ is finite.

The Coxeter complex has faces represented by cosets $w{W}_J$ of standard parabolic subgroups, ordered by reverse inclusion. Geometrically, its maximal simplices are the chambers of the Coxeter arrangement intersected with a sphere.