Coxeter arrangements are hereditarily free (Q1313226)

An arrangement is a finite set of hyperplanes of a real finite- dimensional vectorspace. Let \(L(A)\) denote the set of intersections of elements of \(A\). For \(X\in L(A)\) one has an arrangement \(A^ X:= \{X\cap H\mid H\in A\), \(X\not\subset H\}\) (restriction to \(X\)). Each hyperplane \(H\) of \(V\) defines (up to a constant) \(\alpha_ H\in V^*\) (dual vector space) such that kernel \((\alpha_ H)= H\). Let \(S\) denote the symmetric algebra of \(V^*\) and \(Q:= \prod \alpha_ H\) (over \(H\in A\)) the ``defining polynomial of \(A''\) and \(\text{Der} (S)\) the \(S\)-module of derivations \(S\to S\). Then the ``module of \(A\)-derivations'' \(D(A):= \{\Theta\in \text{Der}(S)\mid \Theta(Q)\in QS\}\) is an \(S\)-submodule of \(\text{Der} (S)\). Call \(A\) free iff \(D(A)\) is a free \(S\)-module. Call \(A\) heridetarily free if \(A^ X\) is free for each \(X\in L(A)\). A Coxeter arrangement is an arrangement \(A\) such that at each hyperplane of \(A\) a reflection is defined and the group generated by these reflections in \(\text{GL}(V)\) is finite (a Coxeter group). Coxeter arrangements are always free. A restriction \(A^ X\) of a Coxeter arrangement is not always a Coxeter arrangement. Result: Coxeter arrangements are hereditarily free. The proof is based on the classification of Coxeter groups.