Finite Coxeter group

A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.

The longest element $w_0$ is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w)$.

The $E_7$ Coxeter graph is the simply-laced seven-vertex tree whose three arms have lengths $1$, $2$, and $3$. Its Coxeter Gram matrix is positive definite and has determinant $2$.

The signed symmetric group consists of permutations of $\{\pm 1,\dots ,\pm n\}$ commuting with sign change. It is the Coxeter group of type $B_n$.

The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type $D_n$.