On minimal lengths of expressions of Coxeter group elements as products of reflections (Q2716101)

Let \(W\) be a Coxeter group and let \(T\) be the set of all reflections (not only simple reflections) in \(W\). Let \(w\in W\). The length \(\ell'(w)\) of \(w\) is the minimal number of factors occurring amongst all expressions of \(w\) as a product of elements in \(T\). If \(W\) is finite, then \(\ell'(w)\) is the codimension of the \(1\)-eigenspace of \(w\).NEWLINENEWLINENEWLINEThe author shows that \(\ell'(w)\) is equal to the minimum number of simple reflections that must be deleted from a fixed reduced expression of \(w\) so that the resulting product of simple reflections is equal to the identity element of \(W\). He also characterizes the length \(\ell'(w)\) in terms of combinatorial data related to the Bruhat graph and to the \(R\)-polynomial \(R_{e,w}\).