The nil Hecke ring and Deodhar's conjecture on Bruhat intervals (Q1802545)

The author uses the properties of the ``nil Hecke ring'' proved by \textit{B. Kostant} and \textit{M. Kumar} [Adv. Math. 62, 187-237 (1986; Zbl 0641.17008)] to prove the following: Theorem. Let \((W,R)\) be a Coxeter system and \(\ell : W \to \mathbb{N}\) be the associated length function. Denote the Bruhat order on \(W\) by \(\leq\). Then for any \(v \leq y \leq w\) one has \(\#\{t \in T\mid v\leq ty \leq w\} \geq \ell(w) - \ell(v)\) where \(T\) denotes the set of reflections of \(W\), i.e. the conjugates in \(W\) of the simple reflections.