  We introduce and study a family of operators which act in the span of a Weyl
group $W$ and provide a multi-parameter solution to the quantum Yang-Baxter
equations of the corresponding type. Our operators generalize the "quantum
Bruhat operators" that appear in the explicit description of the multiplicative
structure of the (small) quantum cohomology ring of $G/B$.
  The main combinatorial applications concern the "tilted Bruhat order," a
graded poset whose unique minimal element is an arbitrarily chosen element
$w\in W$. (The ordinary Bruhat order corresponds to the case $w=1$.) Using the
mixed Bruhat operators, we prove that these posets are lexicographically
shellable, and every interval in a tilted Bruhat order is Eulerian. This
generalizes well known results of Verma, Bjorner, Wachs, and Dyer.
