On Coxeter Arrangements and the Coxeter Number

Abstract

Let $(G, V)$ be an irreducible Coxeter group and let $\mathscr{A}$ be the corresponding Coxeter arrangement. Let $H \in \mathscr{A}$ be a hyperplane and let $\mathscr{A}^H$ be the restriction of $\mathscr{A}$ to $H$. Let $h$ be the Coxeter number. We prove that \[|\mathscr{A}^H|=|\mathscr{A}|-h+1\] and show that $\mathscr{A}^H$ is a free arrangement whose degrees are $m_1, \cdots, m_{l-1}$ the first $l-1$ exponents $G$.